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list-style-type="inherit" tabs="inherit"><f family="Arial" charset="0"><u><inlineAttr line-through="false"><c val="#808000"><link href="./examples.xmcd" title="Links to Section Contents" popup="false"><sp count="4" />EXAMPLES</link></c></inlineAttr></u></f></p></text></region><region region-id="2324" left="36" top="84.75" width="234" height="13.5" align-x="36" align-y="96" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Sous-titre" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Complex Cubic Splines of Prescribed Length</p></text><indexes><phrase>complex cubic splines of prescribed length</phrase><phrase>cubic splines</phrase></indexes></region><region region-id="2569" left="36" top="98.25" width="88.5" height="12" align-x="36" align-y="108" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">by Alex Kushkuley</p></text></region><region region-id="2325" left="36" top="140.25" width="333.75" height="156" align-x="36" align-y="150" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit"><b><u><link href="./splines_1.xmcd" title="Links to section" popup="false">Cubic splines</link></u></b> are a common method of approximating a curve with cubic polynomials. This article discusses a method for specifying the length of a section of the cubic spline fit between two endpoints. Splines are used to describe a curve without a generating function. Spline interpolation can also be used to approximate a curve which has an exceptionally complicated function associated with it. The utility of splines is based on the fact that a cubic polynomial is completely determined if its value and the value of its derivative are specified at two distinct points. This means that one can prescribe directions of the curve defined by a cubic polynomial at its endpoints. This property of cubic polynomials is used for creating continuous and well-behaved interpolation curves. </p></text></region><region region-id="2331" left="36" top="314.25" width="292.5" height="12" align-x="36" align-y="324" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit">To be more precise, examine the third-order polynomial</p></text></region><region region-id="2156" left="66" top="339" width="129.75" height="24" align-x="87.75" align-y="354" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:apply><ml:equal /><ml:apply><ml:id xml:space="preserve">p</ml:id><ml:id 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align-x="36" align-y="384" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit">and assume the initial and boundary conditions</p></text></region><region region-id="2158" left="66" top="399" width="33.75" height="12.75" align-x="88.5" align-y="408" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:apply><ml:equal /><ml:apply><ml:id xml:space="preserve">p</ml:id><ml:real>0</ml:real></ml:apply><ml:real>0</ml:real></ml:apply></math><rendering item-idref="5" /></region><region region-id="2159" left="138" top="399" width="33.75" height="12.75" align-x="160.5" align-y="408" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:apply><ml:equal /><ml:apply><ml:id xml:space="preserve">p</ml:id><ml:real>1</ml:real></ml:apply><ml:real>0</ml:real></ml:apply></math><rendering item-idref="6" /></region><region region-id="2160" left="204" top="391.5" width="43.5" height="29.25" align-x="236.25" align-y="408" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:apply><ml:equal /><ml:apply><ml:derivative /><ml:lambda><ml:boundVars><ml:id xml:space="preserve">t</ml:id></ml:boundVars><ml:apply><ml:id xml:space="preserve">p</ml:id><ml:real>0</ml:real></ml:apply></ml:lambda></ml:apply><ml:id xml:space="preserve">c</ml:id></ml:apply></math><rendering item-idref="7" /></region><region region-id="2161" left="282" top="391.5" width="44.25" height="29.25" align-x="314.25" align-y="408" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:apply><ml:equal /><ml:apply><ml:derivative /><ml:lambda><ml:boundVars><ml:id xml:space="preserve">t</ml:id></ml:boundVars><ml:apply><ml:id xml:space="preserve">p</ml:id><ml:real>1</ml:real></ml:apply></ml:lambda></ml:apply><ml:id xml:space="preserve">b</ml:id></ml:apply></math><rendering item-idref="8" /></region><region region-id="2333" left="36" top="428.25" width="315" height="24" align-x="36" align-y="438" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit">Plugging these values back into the cubic equation, we get a system of linear equations </p></text></region><region region-id="2163" 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xml:space="preserve">a</ml:id><ml:real>1</ml:real></ml:apply><ml:apply><ml:mult style="default" /><ml:real>2</ml:real><ml:apply><ml:indexer /><ml:id xml:space="preserve">a</ml:id><ml:real>2</ml:real></ml:apply></ml:apply></ml:apply><ml:apply><ml:mult style="default" /><ml:real>3</ml:real><ml:apply><ml:indexer /><ml:id xml:space="preserve">a</ml:id><ml:real>3</ml:real></ml:apply></ml:apply></ml:apply><ml:id xml:space="preserve">b</ml:id></ml:apply></math><rendering item-idref="12" /></region><region region-id="2334" left="36" top="512.25" width="327" height="24" align-x="36" align-y="522" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit">which yields the equation of an approximating cubic spline in the form </p></text></region><region region-id="2168" left="72" top="537" width="189" height="18.75" align-x="131.25" align-y="552" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:apply><ml:equal /><ml:apply><ml:id xml:space="preserve">spline</ml:id><ml:sequence><ml:id xml:space="preserve">x</ml:id><ml:id xml:space="preserve">c</ml:id><ml:id xml:space="preserve">b</ml:id></ml:sequence></ml:apply><ml:apply><ml:plus /><ml:apply><ml:minus /><ml:apply><ml:mult style="default" /><ml:id xml:space="preserve">c</ml:id><ml:id xml:space="preserve">x</ml:id></ml:apply><ml:apply><ml:mult style="default" /><ml:parens><ml:apply><ml:plus /><ml:apply><ml:mult style="default" /><ml:real>2</ml:real><ml:id xml:space="preserve">c</ml:id></ml:apply><ml:id xml:space="preserve">b</ml:id></ml:apply></ml:parens><ml:apply><ml:pow /><ml:id xml:space="preserve">x</ml:id><ml:real>2</ml:real></ml:apply></ml:apply></ml:apply><ml:apply><ml:mult style="default" /><ml:parens><ml:apply><ml:plus /><ml:id xml:space="preserve">c</ml:id><ml:id xml:space="preserve">b</ml:id></ml:apply></ml:parens><ml:apply><ml:pow /><ml:id xml:space="preserve">x</ml:id><ml:real>3</ml:real></ml:apply></ml:apply></ml:apply></ml:apply></math><rendering item-idref="13" /></region><region region-id="2335" left="36" top="572.25" width="324" height="24" align-x="36" align-y="582" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit">The curve defined by this equation passes through points (0,0) and (1,0), and its slopes at these points are equal to <b>c</b> and <b>b</b>.</p></text></region><region region-id="2336" left="0" top="606" width="6000" height="6" align-x="0" align-y="606" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><pageBreak /></region><region region-id="2357" left="36" top="614.25" width="317.25" height="48" align-x="36" align-y="624" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit">On the other hand, suppose that we want the arc of our curve between the endpoints to have a prescribed length - this requirement is also quite natural for various CAD/CAM problems. Now we have a nonlinear equation</p></text></region><region region-id="2377" left="18" top="698.25" width="15" height="12" align-x="18" align-y="708" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit">(1)</p></text></region><region region-id="2171" left="66" top="699" width="43.5" height="12.75" align-x="98.25" align-y="708" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:apply><ml:equal /><ml:apply><ml:id xml:space="preserve">L</ml:id><ml:sequence><ml:id xml:space="preserve">c</ml:id><ml:id xml:space="preserve">b</ml:id></ml:sequence></ml:apply><ml:id xml:space="preserve">s</ml:id></ml:apply></math><rendering item-idref="14" /></region><region region-id="2172" left="144" top="698.25" width="30" height="12" align-x="144" align-y="708" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit">where</p></text></region><region region-id="2173" left="210" top="675" width="168.75" height="57" align-x="242.25" align-y="708" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:apply><ml:equal /><ml:apply><ml:id xml:space="preserve">L</ml:id><ml:sequence><ml:id xml:space="preserve">c</ml:id><ml:id xml:space="preserve">b</ml:id></ml:sequence></ml:apply><ml:apply><ml:integral auto-algorithm="true" /><ml:lambda><ml:boundVars><ml:id xml:space="preserve">x</ml:id></ml:boundVars><ml:apply><ml:sqrt /><ml:apply><ml:plus /><ml:real>1</ml:real><ml:apply><ml:pow /><ml:parens><ml:apply><ml:derivative /><ml:lambda><ml:boundVars><ml:id xml:space="preserve">x</ml:id></ml:boundVars><ml:apply><ml:id xml:space="preserve">spline</ml:id><ml:sequence><ml:id xml:space="preserve">x</ml:id><ml:id xml:space="preserve">c</ml:id><ml:id xml:space="preserve">b</ml:id></ml:sequence></ml:apply></ml:lambda></ml:apply></ml:parens><ml:real>2</ml:real></ml:apply></ml:apply></ml:apply></ml:lambda><ml:bounds><ml:real>0</ml:real><ml:real>1</ml:real></ml:bounds></ml:apply></ml:apply></math><rendering item-idref="15" /></region><region region-id="2378" left="36" top="746.25" width="339.75" height="84" align-x="36" align-y="756" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit">and<b> s</b> is a specified length. Fixing <b>c</b>, we can solve this equation for <b>b</b> in order to obtain a cubic curve which has a given length and which satisfies all the conditions of the "plain" cubic spline, except that now we can't prescribe the slope of the curve at its destination point. The integral involved is an elliptic one, so it is impossible to solve equation (1) analytically (see [1], for example ). Mathcad can be used to get a numeric solution. </p></text></region><region region-id="2359" left="36" top="848.25" width="327.75" height="96" align-x="36" align-y="858" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit">The important thing to understand is that the length function on the left-hand side of equation (1) is a convex function of its second argument,<sp count="2" />that is, the second derivative of <b>L </b>in<b> b</b> is positive and hence, <b>dL/db</b> is a monotonically increasing function of <b>b</b>. This property means that the function has no more than two solutions. This is a rather general property of such functions (see [2]) but in our simple case we can use live symbolics to verify that it is correct. </p></text></region><region region-id="2360" left="36" top="956.25" width="324.75" height="48" align-x="36" align-y="966" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">The integrand as a function of its second argument looks like the following expression on the left, where <b>x</b> and <b>y </b>are arbitrary constants. Using the symbolic processor, differentiate twice on <b>b</b> to get<sp count="2" /></p></text></region><region region-id="2179" left="66" top="1020" width="263.25" height="75" align-x="203.25" align-y="1056" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:symEval style="default" hide-keywords="false" hide-lhs="false"><ml:apply><ml:derivative /><ml:lambda><ml:boundVars><ml:id xml:space="preserve">b</ml:id></ml:boundVars><ml:apply><ml:pow /><ml:parens><ml:apply><ml:plus /><ml:real>1</ml:real><ml:apply><ml:pow /><ml:parens><ml:apply><ml:plus /><ml:id xml:space="preserve">x</ml:id><ml:apply><ml:mult style="default" /><ml:id xml:space="preserve">b</ml:id><ml:id xml:space="preserve">y</ml:id></ml:apply></ml:apply></ml:parens><ml:real>2</ml:real></ml:apply></ml:apply></ml:parens><ml:apply><ml:div /><ml:real>1</ml:real><ml:real>2</ml:real></ml:apply></ml:apply></ml:lambda><ml:degree><ml:real>2</ml:real></ml:degree></ml:apply><ml:command><ml:id xml:space="preserve">simplify</ml:id></ml:command><ml:symResult><ml:apply><ml:div /><ml:apply><ml:pow /><ml:id xml:space="preserve">y</ml:id><ml:real font="0">2</ml:real></ml:apply><ml:apply><ml:pow /><ml:parens><ml:apply><ml:plus /><ml:apply><ml:plus /><ml:apply><ml:plus /><ml:real font="0">1</ml:real><ml:apply><ml:pow /><ml:id xml:space="preserve">x</ml:id><ml:real font="0">2</ml:real></ml:apply></ml:apply><ml:apply><ml:mult style="default" /><ml:real font="0">2</ml:real><ml:apply><ml:mult style="default" /><ml:id xml:space="preserve">x</ml:id><ml:apply><ml:mult style="default" /><ml:id xml:space="preserve">b</ml:id><ml:id xml:space="preserve">y</ml:id></ml:apply></ml:apply></ml:apply></ml:apply><ml:apply><ml:mult style="default" /><ml:apply><ml:pow /><ml:id xml:space="preserve">b</ml:id><ml:real font="0">2</ml:real></ml:apply><ml:apply><ml:pow /><ml:id xml:space="preserve">y</ml:id><ml:real font="0">2</ml:real></ml:apply></ml:apply></ml:apply></ml:parens><ml:apply><ml:div /><ml:real font="0">3</ml:real><ml:real font="0">2</ml:real></ml:apply></ml:apply></ml:apply></ml:symResult></ml:symEval></math><rendering item-idref="16" /></region><region region-id="2361" left="36" top="1118.25" width="333.75" height="48" align-x="36" align-y="1128" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit">This expression confirms that the second derivative of the length function is positive, and therefore that equation (1) has no more than two solutions. We can find these solutions by the following procedure. </p></text></region><region region-id="2379" left="0" top="1176" width="6000" height="6" align-x="0" align-y="1176" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><pageBreak /></region><region region-id="2362" left="36" top="1184.25" width="121.5" height="12" align-x="36" align-y="1194" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit">First, solve the equation</p></text></region><region region-id="2182" left="210" top="1177.5" width="46.5" height="29.25" align-x="245.25" align-y="1194" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:apply><ml:equal /><ml:apply><ml:derivative /><ml:lambda><ml:boundVars><ml:id xml:space="preserve">b</ml:id></ml:boundVars><ml:apply><ml:id xml:space="preserve">L</ml:id><ml:id xml:space="preserve">b</ml:id></ml:apply></ml:lambda></ml:apply><ml:real>0</ml:real></ml:apply></math><rendering item-idref="17" /></region><region region-id="2363" left="36" top="1220.25" width="328.5" height="36" align-x="36" align-y="1230" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit">which has only one solution since the left hand side is a monotone function. Using the previously derived expression for<b> spline(x,c,b)</b> we get</p></text><indexes><phrase>spline</phrase></indexes></region><region region-id="2184" left="66" top="1269" width="236.25" height="67.5" align-x="99" align-y="1320" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define warning="WarnRedefinedBIUnit"><ml:function><ml:id xml:space="preserve">L</ml:id><ml:boundVars><ml:id xml:space="preserve">c</ml:id><ml:id xml:space="preserve">b</ml:id></ml:boundVars></ml:function><ml:apply><ml:integral auto-algorithm="true" algorithm="adaptive" /><ml:lambda><ml:boundVars><ml:id xml:space="preserve">x</ml:id></ml:boundVars><ml:apply><ml:pow /><ml:parens><ml:apply><ml:plus /><ml:real>1</ml:real><ml:apply><ml:pow /><ml:parens><ml:apply><ml:plus /><ml:apply><ml:minus /><ml:id xml:space="preserve">c</ml:id><ml:apply><ml:mult style="default" /><ml:apply><ml:mult style="default" /><ml:real>2</ml:real><ml:parens><ml:apply><ml:plus /><ml:apply><ml:mult style="default" /><ml:real>2</ml:real><ml:id xml:space="preserve">c</ml:id></ml:apply><ml:id xml:space="preserve">b</ml:id></ml:apply></ml:parens></ml:apply><ml:id xml:space="preserve">x</ml:id></ml:apply></ml:apply><ml:apply><ml:mult style="default" /><ml:apply><ml:mult style="default" /><ml:real>3</ml:real><ml:parens><ml:apply><ml:plus /><ml:id xml:space="preserve">c</ml:id><ml:id xml:space="preserve">b</ml:id></ml:apply></ml:parens></ml:apply><ml:apply><ml:pow /><ml:id xml:space="preserve">x</ml:id><ml:real>2</ml:real></ml:apply></ml:apply></ml:apply></ml:parens><ml:real>2</ml:real></ml:apply></ml:apply></ml:parens><ml:apply><ml:div /><ml:real>1</ml:real><ml:real>2</ml:real></ml:apply></ml:apply></ml:lambda><ml:bounds><ml:real>0</ml:real><ml:real>1</ml:real></ml:bounds></ml:apply></ml:define></math><rendering item-idref="18" /></region><region region-id="2364" left="36" top="1352.25" width="17.25" height="12" align-x="36" align-y="1362" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit">Fix </p></text></region><region region-id="2186" left="90" top="1353" width="27.75" height="12.75" align-x="100.5" align-y="1362" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define warning="WarnRedefinedBIUnit"><ml:id xml:space="preserve">c</ml:id><ml:real>-4</ml:real></ml:define></math><rendering item-idref="19" /></region><region region-id="2187" left="156" top="1352.25" width="55.5" height="12" align-x="156" align-y="1362" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit">and define </p></text></region><region region-id="2188" left="240" top="1353" width="63" height="12.75" align-x="268.5" align-y="1362" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define><ml:function><ml:id xml:space="preserve">L1</ml:id><ml:boundVars><ml:id xml:space="preserve">b</ml:id></ml:boundVars></ml:function><ml:apply><ml:id xml:space="preserve">L</ml:id><ml:sequence><ml:id xml:space="preserve">c</ml:id><ml:id xml:space="preserve">b</ml:id></ml:sequence></ml:apply></ml:define></math><rendering item-idref="20" /></region><region region-id="2365" left="36" top="1382.25" width="155.25" height="12" align-x="36" align-y="1392" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit">The derivative of <b>L(c,b)</b> in <b>b</b> is</p></text></region><region region-id="2190" left="72" top="1421.25" width="279.75" height="87" align-x="126.75" align-y="1452" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define><ml:function><ml:id xml:space="preserve">DerivL</ml:id><ml:boundVars><ml:id xml:space="preserve">c</ml:id><ml:id xml:space="preserve">b</ml:id></ml:boundVars></ml:function><ml:apply><ml:integral auto-algorithm="true" algorithm="adaptive" /><ml:lambda><ml:boundVars><ml:id xml:space="preserve">x</ml:id></ml:boundVars><ml:apply><ml:div /><ml:apply><ml:mult style="default" /><ml:parens><ml:apply><ml:plus /><ml:apply><ml:minus /><ml:id xml:space="preserve">c</ml:id><ml:apply><ml:mult style="default" /><ml:apply><ml:mult style="default" /><ml:real>2</ml:real><ml:parens><ml:apply><ml:plus /><ml:apply><ml:mult style="default" /><ml:real>2</ml:real><ml:id xml:space="preserve">c</ml:id></ml:apply><ml:id xml:space="preserve">b</ml:id></ml:apply></ml:parens></ml:apply><ml:id xml:space="preserve">x</ml:id></ml:apply></ml:apply><ml:apply><ml:mult style="default" /><ml:apply><ml:mult style="default" /><ml:real>3</ml:real><ml:parens><ml:apply><ml:plus /><ml:id xml:space="preserve">c</ml:id><ml:id xml:space="preserve">b</ml:id></ml:apply></ml:parens></ml:apply><ml:apply><ml:pow /><ml:id xml:space="preserve">x</ml:id><ml:real>2</ml:real></ml:apply></ml:apply></ml:apply></ml:parens><ml:parens><ml:apply><ml:plus /><ml:apply><ml:mult style="default" /><ml:real>-2</ml:real><ml:id xml:space="preserve">x</ml:id></ml:apply><ml:apply><ml:mult style="default" /><ml:real>3</ml:real><ml:apply><ml:pow /><ml:id xml:space="preserve">x</ml:id><ml:real>2</ml:real></ml:apply></ml:apply></ml:apply></ml:parens></ml:apply><ml:apply><ml:pow /><ml:parens><ml:apply><ml:plus /><ml:real>1</ml:real><ml:apply><ml:pow /><ml:parens><ml:apply><ml:plus /><ml:apply><ml:minus /><ml:id xml:space="preserve">c</ml:id><ml:apply><ml:mult style="default" /><ml:apply><ml:mult style="default" /><ml:real>2</ml:real><ml:parens><ml:apply><ml:plus /><ml:apply><ml:mult style="default" /><ml:real>2</ml:real><ml:id xml:space="preserve">c</ml:id></ml:apply><ml:id xml:space="preserve">b</ml:id></ml:apply></ml:parens></ml:apply><ml:id xml:space="preserve">x</ml:id></ml:apply></ml:apply><ml:apply><ml:mult style="default" /><ml:apply><ml:mult style="default" /><ml:real>3</ml:real><ml:parens><ml:apply><ml:plus /><ml:id xml:space="preserve">c</ml:id><ml:id xml:space="preserve">b</ml:id></ml:apply></ml:parens></ml:apply><ml:apply><ml:pow /><ml:id xml:space="preserve">x</ml:id><ml:real>2</ml:real></ml:apply></ml:apply></ml:apply></ml:parens><ml:real>2</ml:real></ml:apply></ml:apply></ml:parens><ml:apply><ml:div /><ml:real>1</ml:real><ml:real>2</ml:real></ml:apply></ml:apply></ml:apply></ml:lambda><ml:bounds><ml:real>0</ml:real><ml:real>1</ml:real></ml:bounds></ml:apply></ml:define></math><rendering item-idref="21" /></region><region region-id="2366" left="36" top="1526.25" width="34.5" height="12" align-x="36" align-y="1536" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit">Define </p></text></region><region region-id="2192" left="138" top="1527" width="106.5" height="12.75" align-x="188.25" align-y="1536" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define><ml:function><ml:id xml:space="preserve">DerivL1</ml:id><ml:boundVars><ml:id xml:space="preserve">b</ml:id></ml:boundVars></ml:function><ml:apply><ml:id xml:space="preserve">DerivL</ml:id><ml:sequence><ml:id xml:space="preserve">c</ml:id><ml:id xml:space="preserve">b</ml:id></ml:sequence></ml:apply></ml:define></math><rendering item-idref="22" /></region><region region-id="2367" left="36" top="1556.25" width="126.75" height="12" align-x="36" align-y="1566" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit">and take the initial value </p></text></region><region region-id="2194" left="210" top="1557" width="23.25" height="12.75" align-x="221.25" align-y="1566" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define><ml:id xml:space="preserve">b</ml:id><ml:real>0</ml:real></ml:define></math><rendering item-idref="23" /></region><region region-id="2368" left="36" top="1586.25" width="300.75" height="24" align-x="36" align-y="1596" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit">and compute the resulting value of the first derivative at the endpoint: </p></text></region><region region-id="2196" left="66" top="1623" width="114" height="12.75" align-x="94.5" align-y="1632" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define><ml:id xml:space="preserve">Slope</ml:id><ml:apply><ml:id xml:space="preserve">root</ml:id><ml:sequence><ml:apply><ml:id xml:space="preserve">DerivL1</ml:id><ml:id xml:space="preserve">b</ml:id></ml:apply><ml:id xml:space="preserve">b</ml:id></ml:sequence></ml:apply></ml:define></math><rendering item-idref="24" /></region><region region-id="2197" left="210" top="1623" width="79.5" height="12.75" align-x="237.75" align-y="1632" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:eval><ml:id xml:space="preserve">Slope</ml:id><result xmlns="http://schemas.mathsoft.com/math20"><ml:real>-0.83163940984283446</ml:real></result></ml:eval></math><rendering item-idref="25" /></region><region region-id="2369" left="36" top="1646.25" width="324.75" height="12" align-x="36" align-y="1656" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit">And find the length corresponding to this value of the derivative:</p></text></region><region region-id="2199" left="66" top="1677" width="91.5" height="12.75" align-x="111" align-y="1686" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:eval><ml:apply><ml:id xml:space="preserve">L1</ml:id><ml:id xml:space="preserve">Slope</ml:id></ml:apply><result xmlns="http://schemas.mathsoft.com/math20"><ml:real>1.5992833223091068</ml:real></result></ml:eval></math><rendering item-idref="26" /></region><region region-id="2200" left="210" top="1671" width="155.25" height="18.75" align-x="276.75" align-y="1686" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:eval><ml:apply><ml:id xml:space="preserve">DerivL1</ml:id><ml:id xml:space="preserve">Slope</ml:id></ml:apply><result xmlns="http://schemas.mathsoft.com/math20"><ml:real>-2.5170837636423471E-15</ml:real></result></ml:eval></math><rendering item-idref="27" /></region><region region-id="2383" left="36" top="1706.25" width="339" height="72" align-x="36" align-y="1716" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">Note that<b> L1(Slope)</b> is the minimum possible length of the cubic curve which satisfies three given conditions (values at the endpoints and the slope at the initial point). If we want our curve to have a given length, this length should be no less than <b>L1(Slope)</b>. In this case the number <b>Slope</b> separates the two roots of our equation (1). </p></text></region><region region-id="2385" left="0" top="1788" width="6000" height="6" align-x="0" align-y="1788" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><pageBreak /></region><region region-id="2384" left="36" top="1796.25" width="227.25" height="12" align-x="36" align-y="1806" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit">Let's say we look for a spline with the length </p></text></region><region region-id="2202" left="66" top="1821" width="34.5" height="12.75" align-x="81.75" align-y="1830" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define><ml:id xml:space="preserve">L0</ml:id><ml:real>1.8</ml:real></ml:define></math><rendering item-idref="28" /></region><region region-id="2386" left="36" top="1844.25" width="236.25" height="12" align-x="36" align-y="1854" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit">then we have one root on the left of <b>Slope</b></p></text></region><region region-id="2204" left="66" top="1875" width="75.75" height="12.75" align-x="96" align-y="1884" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define><ml:id xml:space="preserve">guess</ml:id><ml:apply><ml:minus /><ml:id xml:space="preserve">Slope</ml:id><ml:real>1</ml:real></ml:apply></ml:define></math><rendering item-idref="29" /></region><region region-id="2400" left="222" top="1875" width="138" height="12.75" align-x="237.75" align-y="1884" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define><ml:id xml:space="preserve">b1</ml:id><ml:apply><ml:id xml:space="preserve">root</ml:id><ml:sequence><ml:apply><ml:minus /><ml:apply><ml:id xml:space="preserve">L1</ml:id><ml:id xml:space="preserve">guess</ml:id></ml:apply><ml:id xml:space="preserve">L0</ml:id></ml:apply><ml:id xml:space="preserve">guess</ml:id></ml:sequence></ml:apply></ml:define></math><rendering item-idref="30" /></region><region region-id="2387" left="36" top="1910.25" width="86.25" height="12" align-x="36" align-y="1920" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit">Check the result:</p></text></region><region region-id="2403" left="144" top="1911" width="66.75" height="12.75" align-x="159" align-y="1920" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:eval><ml:id xml:space="preserve">b1</ml:id><result xmlns="http://schemas.mathsoft.com/math20"><ml:real>-3.2648781082675593</ml:real></result></ml:eval></math><rendering item-idref="31" /></region><region region-id="2401" left="222" top="1911" width="56.25" height="12.75" align-x="254.25" align-y="1920" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:eval><ml:apply><ml:id xml:space="preserve">L1</ml:id><ml:id xml:space="preserve">b1</ml:id></ml:apply><result xmlns="http://schemas.mathsoft.com/math20"><ml:real>1.7999999999988809</ml:real></result></ml:eval></math><rendering item-idref="32" /></region><region region-id="2373" left="36" top="1946.25" width="202.5" height="12" align-x="36" align-y="1956" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit">The second root is on the right of <b>Slope</b></p></text></region><region region-id="2210" left="66" top="1983" width="76.5" height="12.75" align-x="96" align-y="1992" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define warning="WarnRedefinedUDScalar"><ml:id xml:space="preserve">guess</ml:id><ml:apply><ml:plus /><ml:id xml:space="preserve">Slope</ml:id><ml:real>1</ml:real></ml:apply></ml:define></math><rendering item-idref="33" /></region><region region-id="2402" left="222" top="1983" width="138" height="12.75" align-x="237.75" align-y="1992" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define><ml:id xml:space="preserve">b2</ml:id><ml:apply><ml:id xml:space="preserve">root</ml:id><ml:sequence><ml:apply><ml:minus /><ml:apply><ml:id xml:space="preserve">L1</ml:id><ml:id xml:space="preserve">guess</ml:id></ml:apply><ml:id xml:space="preserve">L0</ml:id></ml:apply><ml:id xml:space="preserve">guess</ml:id></ml:sequence></ml:apply></ml:define></math><rendering item-idref="34" /></region><region region-id="2374" left="36" top="2024.25" width="86.25" height="12" align-x="36" align-y="2034" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit">Check the result:</p></text></region><region region-id="2404" left="144" top="2025" width="52.5" height="12.75" align-x="159" align-y="2034" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:eval><ml:id xml:space="preserve">b2</ml:id><result xmlns="http://schemas.mathsoft.com/math20"><ml:real>1.3398001354308793</ml:real></result></ml:eval></math><rendering item-idref="35" /></region><region region-id="2407" left="222" top="2025" width="56.25" height="12.75" align-x="254.25" align-y="2034" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:eval><ml:apply><ml:id xml:space="preserve">L1</ml:id><ml:id xml:space="preserve">b2</ml:id></ml:apply><result xmlns="http://schemas.mathsoft.com/math20"><ml:real>1.7999999999999767</ml:real></result></ml:eval></math><rendering item-idref="36" /></region><region region-id="2375" left="36" top="2054.25" width="337.5" height="12" align-x="36" align-y="2064" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit">This gives us two curves, corresponding to the two roots of<b> L1(b)</b>:</p></text></region><region region-id="2216" left="66" top="2079" width="192.75" height="18.75" align-x="111.75" align-y="2094" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define><ml:function><ml:id xml:space="preserve">spline1</ml:id><ml:boundVars><ml:id xml:space="preserve">x</ml:id></ml:boundVars></ml:function><ml:apply><ml:plus /><ml:parens><ml:apply><ml:minus /><ml:apply><ml:mult style="default" /><ml:id xml:space="preserve">c</ml:id><ml:id xml:space="preserve">x</ml:id></ml:apply><ml:apply><ml:mult style="default" /><ml:parens><ml:apply><ml:plus /><ml:apply><ml:mult style="default" /><ml:real>2</ml:real><ml:id xml:space="preserve">c</ml:id></ml:apply><ml:id xml:space="preserve">b1</ml:id></ml:apply></ml:parens><ml:apply><ml:pow /><ml:id xml:space="preserve">x</ml:id><ml:real>2</ml:real></ml:apply></ml:apply></ml:apply></ml:parens><ml:apply><ml:mult style="default" /><ml:parens><ml:apply><ml:plus /><ml:id xml:space="preserve">c</ml:id><ml:id xml:space="preserve">b1</ml:id></ml:apply></ml:parens><ml:apply><ml:pow /><ml:id xml:space="preserve">x</ml:id><ml:real>3</ml:real></ml:apply></ml:apply></ml:apply></ml:define></math><rendering item-idref="37" /></region><region region-id="2217" left="66" top="2109" width="192.75" height="18.75" align-x="111.75" align-y="2124" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define><ml:function><ml:id xml:space="preserve">spline2</ml:id><ml:boundVars><ml:id xml:space="preserve">x</ml:id></ml:boundVars></ml:function><ml:apply><ml:plus /><ml:parens><ml:apply><ml:minus /><ml:apply><ml:mult style="default" /><ml:id xml:space="preserve">c</ml:id><ml:id xml:space="preserve">x</ml:id></ml:apply><ml:apply><ml:mult style="default" /><ml:parens><ml:apply><ml:plus /><ml:apply><ml:mult style="default" /><ml:real>2</ml:real><ml:id xml:space="preserve">c</ml:id></ml:apply><ml:id xml:space="preserve">b2</ml:id></ml:apply></ml:parens><ml:apply><ml:pow /><ml:id xml:space="preserve">x</ml:id><ml:real>2</ml:real></ml:apply></ml:apply></ml:apply></ml:parens><ml:apply><ml:mult style="default" /><ml:parens><ml:apply><ml:plus /><ml:id xml:space="preserve">c</ml:id><ml:id xml:space="preserve">b2</ml:id></ml:apply></ml:parens><ml:apply><ml:pow /><ml:id xml:space="preserve">x</ml:id><ml:real>3</ml:real></ml:apply></ml:apply></ml:apply></ml:define></math><rendering item-idref="38" /></region><region region-id="2376" left="36" top="2144.25" width="101.25" height="12" align-x="36" align-y="2154" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit">which we plot below</p></text></region><region region-id="2219" left="210" top="2145" width="49.5" height="12.75" align-x="219.75" align-y="2154" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define><ml:id xml:space="preserve">x</ml:id><ml:range><ml:sequence><ml:real>0</ml:real><ml:real>.01</ml:real></ml:sequence><ml:real>1</ml:real></ml:range></ml:define></math><rendering item-idref="39" /></region><region region-id="2408" left="24" top="2178" width="315" height="173.25" align-x="24" align-y="2178" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><plot disable-calc="false" item-idref="40" /><rendering item-idref="41" /></region><region region-id="2435" left="36" top="2366.25" width="321" height="60" align-x="36" align-y="2376" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit">So far our methodology is of limited use, since we cannot change the endpoints of the curve, which, for the sake of simplicity, were fixed at (0,0) and (1,0). Below is a more general formulation. After the formulations given above, only a few changes are necessary.</p></text></region><region region-id="2411" left="0" top="2436" width="6000" height="6" align-x="0" align-y="2436" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><pageBreak /></region><region region-id="2436" left="36" top="2443.5" width="172.5" height="12.75" align-x="36" align-y="2454" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="subsubtitle" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Arbitrary splines of prescribed length</p></text><indexes><phrase>arbitrary splines of prescribed length</phrase></indexes></region><region region-id="2422" left="36" top="2468.25" width="329.25" height="48" align-x="36" align-y="2478" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit">Points on the plane will now be represented as complex numbers, and cubic curves as cubic polynomials with complex coefficients. So, in specifying the initial conditions, use complex numbers as follows:</p></text></region><region region-id="2423" left="36" top="2528.25" width="71.25" height="12" align-x="36" align-y="2538" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit">starting point:</p></text></region><region region-id="2432" left="144" top="2529" width="50.25" height="12.75" align-x="168" align-y="2538" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define><ml:id xml:space="preserve">start</ml:id><ml:apply><ml:plus /><ml:real>1</ml:real><ml:imag symbol="i">1</ml:imag></ml:apply></ml:define></math><rendering item-idref="42" /></region><region region-id="2424" left="36" top="2558.25" width="51" height="12" align-x="36" align-y="2568" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit">end point:</p></text></region><region region-id="2433" left="144" top="2559" width="51.75" height="12.75" align-x="165" align-y="2568" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define><ml:id xml:space="preserve">end</ml:id><ml:apply><ml:minus /><ml:real>-1</ml:real><ml:imag symbol="i">1</ml:imag></ml:apply></ml:define></math><rendering item-idref="43" /></region><region region-id="2425" left="36" top="2594.25" width="166.5" height="36" align-x="36" align-y="2604" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit">derivative of the curve at the starting point (the velocity of the curve):</p></text></region><region region-id="2434" left="228" top="2595" width="49.5" height="12.75" align-x="238.5" align-y="2604" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define warning="WarnRedefinedUDScalar"><ml:id xml:space="preserve">c</ml:id><ml:apply><ml:plus /><ml:real>-1</ml:real><ml:apply><ml:mult style="default" /><ml:real>5</ml:real><ml:imag symbol="i">1</ml:imag></ml:apply></ml:apply></ml:define></math><rendering item-idref="44" /></region><region region-id="2426" left="36" top="2636.25" width="323.25" height="72" align-x="36" align-y="2646" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit">To solve for the derivative at the end point, we can fix the ratio between its imaginary and real parts and solve the length equation for the real part. Doing so also ensures that the curve has a prescribed tangent line at the end point. Note that we were unable to achieve this with real cubic polynomials. If we set, for example, the slope at the finishing point to be</p></text></region><region region-id="2230" left="66" top="2721" width="40.5" height="12.75" align-x="94.5" align-y="2730" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define warning="WarnRedefinedUDScalar"><ml:id xml:space="preserve">Slope</ml:id><ml:real>1</ml:real></ml:define></math><rendering item-idref="45" /></region><region region-id="2427" left="36" top="2744.25" width="324" height="87" align-x="36" align-y="2754" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit">then the velocity at the finishing point is <b>Slope · </b><b>x ·</b><b><sub><sp /></sub></b><b>i + x</b><f family="Verdana" charset="0"><inlineAttr font-weight="normal" line-through="false"><c val="#000">,</c></inlineAttr></f> where<b> x</b> is the real part of a complex variable. Define the coefficients of the complex spline by setting up a system of equations from the initial conditions, just as we did for the non-complex case. In this instance, the second and third order coefficients are functions of the real variable, <b>x</b>, since the velocity at the endpoint is written in terms of <b>x</b>, as above. This yields:</p></text></region><region region-id="2232" left="66" top="2841" width="189" height="12.75" align-x="90.75" align-y="2850" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define warning="WarnRedefinedBIUnit"><ml:function><ml:id xml:space="preserve">A</ml:id><ml:boundVars><ml:id xml:space="preserve">x</ml:id></ml:boundVars></ml:function><ml:parens><ml:apply><ml:minus /><ml:apply><ml:minus /><ml:apply><ml:minus /><ml:apply><ml:plus /><ml:apply><ml:mult style="default" /><ml:real>-3</ml:real><ml:id xml:space="preserve">start</ml:id></ml:apply><ml:apply><ml:mult style="default" /><ml:real>3</ml:real><ml:id xml:space="preserve">end</ml:id></ml:apply></ml:apply><ml:apply><ml:mult style="default" /><ml:real>2</ml:real><ml:id xml:space="preserve">c</ml:id></ml:apply></ml:apply><ml:id xml:space="preserve">x</ml:id></ml:apply><ml:apply><ml:mult style="default" /><ml:apply><ml:mult style="default" /><ml:id xml:space="preserve">Slope</ml:id><ml:id xml:space="preserve">x</ml:id></ml:apply><ml:imag symbol="i">1</ml:imag></ml:apply></ml:apply></ml:parens></ml:define></math><rendering item-idref="46" /></region><region region-id="2233" left="66" top="2871" width="176.25" height="12.75" align-x="89.25" align-y="2880" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define><ml:function><ml:id xml:space="preserve">B</ml:id><ml:boundVars><ml:id xml:space="preserve">x</ml:id></ml:boundVars></ml:function><ml:parens><ml:apply><ml:plus /><ml:apply><ml:minus /><ml:apply><ml:plus /><ml:apply><ml:plus /><ml:id xml:space="preserve">x</ml:id><ml:apply><ml:mult style="default" /><ml:apply><ml:mult style="default" /><ml:id xml:space="preserve">Slope</ml:id><ml:id xml:space="preserve">x</ml:id></ml:apply><ml:imag symbol="i">1</ml:imag></ml:apply></ml:apply><ml:apply><ml:mult style="default" /><ml:real>2</ml:real><ml:id xml:space="preserve">start</ml:id></ml:apply></ml:apply><ml:apply><ml:mult style="default" /><ml:real>2</ml:real><ml:id xml:space="preserve">end</ml:id></ml:apply></ml:apply><ml:id xml:space="preserve">c</ml:id></ml:apply></ml:parens></ml:define></math><rendering item-idref="47" /></region><region region-id="2428" left="36" top="2900.25" width="250.5" height="12" align-x="36" align-y="2910" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit">so that the curve we are looking for has the form </p></text></region><region region-id="2235" left="66" top="2919" width="177.75" height="18.75" align-x="114.75" align-y="2934" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define><ml:function><ml:id xml:space="preserve">spline</ml:id><ml:boundVars><ml:id xml:space="preserve">x</ml:id><ml:id xml:space="preserve">t</ml:id></ml:boundVars></ml:function><ml:apply><ml:plus /><ml:apply><ml:plus /><ml:apply><ml:plus /><ml:id xml:space="preserve">start</ml:id><ml:apply><ml:mult style="default" /><ml:id xml:space="preserve">c</ml:id><ml:id xml:space="preserve">t</ml:id></ml:apply></ml:apply><ml:apply><ml:mult style="default" /><ml:apply><ml:id xml:space="preserve">A</ml:id><ml:id xml:space="preserve">x</ml:id></ml:apply><ml:apply><ml:pow /><ml:id xml:space="preserve">t</ml:id><ml:real>2</ml:real></ml:apply></ml:apply></ml:apply><ml:apply><ml:mult style="default" /><ml:apply><ml:id xml:space="preserve">B</ml:id><ml:id xml:space="preserve">x</ml:id></ml:apply><ml:apply><ml:pow /><ml:id xml:space="preserve">t</ml:id><ml:real>3</ml:real></ml:apply></ml:apply></ml:apply></ml:define></math><rendering item-idref="48" /></region><region region-id="2429" left="36" top="2948.25" width="283.5" height="12" align-x="36" align-y="2958" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit">where the real parameter <b>t </b>belongs to the interval <b>[0,1]</b>.</p></text></region><region region-id="2430" left="36" top="2978.25" width="155.25" height="24" align-x="36" align-y="2988" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit">The velocity vector is equal to the derivative of <b>spline(x, t)</b>: </p></text></region><region region-id="2438" left="222" top="2985" width="134.25" height="18.75" align-x="252.75" align-y="3000" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define warning="WarnRedefinedBIUnit"><ml:function><ml:id xml:space="preserve">C</ml:id><ml:boundVars><ml:id xml:space="preserve">x</ml:id><ml:id xml:space="preserve">t</ml:id></ml:boundVars></ml:function><ml:apply><ml:plus /><ml:apply><ml:plus /><ml:id xml:space="preserve">c</ml:id><ml:apply><ml:mult style="default" /><ml:apply><ml:mult style="default" /><ml:real>2</ml:real><ml:apply><ml:id xml:space="preserve">A</ml:id><ml:id xml:space="preserve">x</ml:id></ml:apply></ml:apply><ml:id xml:space="preserve">t</ml:id></ml:apply></ml:apply><ml:apply><ml:mult style="default" /><ml:apply><ml:mult style="default" /><ml:apply><ml:id xml:space="preserve">B</ml:id><ml:id xml:space="preserve">x</ml:id></ml:apply><ml:apply><ml:pow /><ml:id xml:space="preserve">t</ml:id><ml:real>2</ml:real></ml:apply></ml:apply><ml:real>3</ml:real></ml:apply></ml:apply></ml:define></math><rendering item-idref="49" /></region><region region-id="2439" left="0" top="3018" width="6000" height="6" align-x="0" align-y="3018" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><pageBreak /></region><region region-id="2431" left="36" top="3026.25" width="213.75" height="12" align-x="36" align-y="3036" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit">Hence, the length of the curve is equal to </p></text></region><region region-id="2240" left="66" top="3055.5" width="88.5" height="39" align-x="88.5" align-y="3078" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define warning="WarnRedefinedUDFunction"><ml:function><ml:id xml:space="preserve">L</ml:id><ml:boundVars><ml:id xml:space="preserve">x</ml:id></ml:boundVars></ml:function><ml:apply><ml:integral auto-algorithm="true" algorithm="adaptive" /><ml:lambda><ml:boundVars><ml:id xml:space="preserve">t</ml:id></ml:boundVars><ml:apply><ml:absval /><ml:apply><ml:id xml:space="preserve">C</ml:id><ml:sequence><ml:id xml:space="preserve">x</ml:id><ml:id xml:space="preserve">t</ml:id></ml:sequence></ml:apply></ml:apply></ml:lambda><ml:bounds><ml:real>0</ml:real><ml:real>1</ml:real></ml:bounds></ml:apply></ml:define></math><rendering item-idref="50" /></region><region region-id="2440" left="36" top="3104.25" width="328.5" height="108" align-x="36" align-y="3114" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit">There are two ways to proceed at this juncture. One way to find the minimum possible length is to use a solve block and the<b> Minerr</b> function. The accuracy of the value for the root of the function <b>L(x)</b> will be poor with the low tolerance used, but it will calculate quickly, and allow you to play with the values <b>start</b>,<b> end</b>,<b> c</b> and <b>slope</b> above. The second, more familiar way is to find the derivative of <b>L(x)</b>, set it to zero, and solve for <b>x</b>. This method will be covered later, since it raises some interesting issues.</p></text></region><region region-id="2242" left="66" top="3231" width="60" height="12.75" align-x="75.75" align-y="3240" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define warning="WarnRedefinedUDScalar"><ml:id xml:space="preserve">x</ml:id><ml:range><ml:sequence><ml:real>-5</ml:real><ml:real>-4.5</ml:real></ml:sequence><ml:real>5</ml:real></ml:range></ml:define></math><rendering item-idref="51" /></region><region region-id="2444" left="180" top="3252" width="164.25" height="95.25" align-x="180" align-y="3252" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><plot disable-calc="false" item-idref="52" /><rendering item-idref="53" /></region><region region-id="2441" left="36" top="3260.25" width="143.25" height="84" align-x="36" align-y="3270" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit">To find a guess for the minimum possible length for the spline, plot<b> L</b> over a range of <b>x</b> values (you may need to change the range of <b>x</b> above to see the minimum.)</p></text></region><region region-id="2442" left="36" top="3368.25" width="86.25" height="72" align-x="36" align-y="3378" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit">Pick a value of<b> x</b> near the minimum and enter it as a guess for the solve block.</p></text></region><region region-id="2445" left="144" top="3369" width="21.75" height="12.75" align-x="153.75" align-y="3378" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define warning="WarnRedefinedUDScalar"><ml:id xml:space="preserve">x</ml:id><ml:real>0</ml:real></ml:define></math><rendering item-idref="54" /></region><region region-id="2446" left="144" top="3399" width="24" height="12.75" align-x="155.25" align-y="3408" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:id xml:space="preserve">Given</ml:id></math><rendering item-idref="55" /></region><region region-id="2451" left="222" top="3399" width="33" height="12.75" align-x="243.75" align-y="3408" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:apply><ml:equal /><ml:apply><ml:id xml:space="preserve">L</ml:id><ml:id xml:space="preserve">x</ml:id></ml:apply><ml:real>0</ml:real></ml:apply></math><rendering item-idref="56" /></region><region region-id="2448" left="144" top="3429" width="57" height="12.75" align-x="155.25" align-y="3438" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define><ml:id xml:space="preserve">d</ml:id><ml:apply><ml:Minerr auto-method="false" method="levenberg" derivative-estimation="central" variable-estimation="tangent" linear-check="false" multistart="false" evolutionary="false" /><ml:id xml:space="preserve">x</ml:id></ml:apply></ml:define></math><rendering item-idref="57" /></region><region region-id="2452" left="222" top="3429" width="57" height="12.75" align-x="232.5" align-y="3438" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:eval><ml:id xml:space="preserve">d</ml:id><result xmlns="http://schemas.mathsoft.com/math20"><ml:real>0.13744317397006042</ml:real></result></ml:eval></math><rendering item-idref="58" /></region><region region-id="2453" left="294" top="3429" width="69.75" height="12.75" align-x="317.25" align-y="3438" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:eval><ml:apply><ml:id xml:space="preserve">L</ml:id><ml:id xml:space="preserve">d</ml:id></ml:apply><result xmlns="http://schemas.mathsoft.com/math20"><ml:real>3.6017858442317507</ml:real></result></ml:eval></math><rendering item-idref="59" /></region><region region-id="2454" left="36" top="3452.25" width="324.75" height="24" align-x="36" align-y="3462" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit">The value of <b>L(d)</b> found this way is still good because the minimum is shallow. </p></text></region><region region-id="2455" left="36" top="3494.25" width="342" height="24" align-x="36" align-y="3504" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit"><b>Method 2</b>:</p><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit">Differentiating on <b>x </b>we get the derivative of the length function </p></text></region><region region-id="2458" left="66" top="3533.25" width="230.25" height="54" align-x="110.25" align-y="3564" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define warning="WarnRedefinedUDFunction"><ml:function><ml:id xml:space="preserve">DerivL</ml:id><ml:boundVars><ml:id xml:space="preserve">x</ml:id></ml:boundVars></ml:function><ml:apply><ml:integral auto-algorithm="true" /><ml:lambda><ml:boundVars><ml:id xml:space="preserve">t</ml:id></ml:boundVars><ml:apply><ml:div /><ml:apply><ml:mult style="default" /><ml:apply><ml:id xml:space="preserve">Re</ml:id><ml:apply><ml:mult style="default" /><ml:parens><ml:apply><ml:plus /><ml:real>1</ml:real><ml:apply><ml:mult style="default" /><ml:id xml:space="preserve">Slope</ml:id><ml:imag symbol="i">1</ml:imag></ml:apply></ml:apply></ml:parens><ml:apply><ml:conjugate /><ml:apply><ml:id xml:space="preserve">C</ml:id><ml:sequence><ml:id xml:space="preserve">x</ml:id><ml:id xml:space="preserve">t</ml:id></ml:sequence></ml:apply></ml:apply></ml:apply></ml:apply><ml:parens><ml:apply><ml:plus /><ml:apply><ml:mult style="default" /><ml:real>-2</ml:real><ml:id xml:space="preserve">t</ml:id></ml:apply><ml:apply><ml:mult style="default" /><ml:real>3</ml:real><ml:apply><ml:pow /><ml:id xml:space="preserve">t</ml:id><ml:real>2</ml:real></ml:apply></ml:apply></ml:apply></ml:parens></ml:apply><ml:apply><ml:absval /><ml:apply><ml:id xml:space="preserve">C</ml:id><ml:sequence><ml:id xml:space="preserve">x</ml:id><ml:id xml:space="preserve">t</ml:id></ml:sequence></ml:apply></ml:apply></ml:apply></ml:lambda><ml:bounds><ml:real>0</ml:real><ml:real>1</ml:real></ml:bounds></ml:apply></ml:define></math><rendering item-idref="60" /></region><region region-id="2459" left="0" top="3606" width="6000" height="6" align-x="0" align-y="3606" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><pageBreak /></region><region region-id="2456" left="36" top="3632.25" width="330.75" height="84" align-x="36" align-y="3642" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit"><b>DerivL(x) </b>is differentiable even when<b> |C(x,t)| =0</b>, and in many such cases Mathcad will compute this integral correctly, despite the apparent singularity. To avoid this issue and still get a good approximation for <b>DerivL(x), </b>remove small intervals of integration in the neighborhood of the roots of <b>|C(x,t)|. </b>The following expressions calculate the roots of <b>|C(x,t)</b><b>|</b>, and create three integrals that leave out the intervals <f family="Symbol" charset="2"><b>±e </b></f>around each root.</p></text></region><region region-id="2460" left="36" top="3728.25" width="253.5" height="12" align-x="36" align-y="3738" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit">To do this, compute the roots<b> t1 </b>and<b> t2 </b>of <b>C(x,t)</b>:</p></text></region><region region-id="2257" left="66" top="3757.5" width="105" height="20.25" align-x="90" align-y="3774" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define><ml:function><ml:id xml:space="preserve">D</ml:id><ml:boundVars><ml:id xml:space="preserve">x</ml:id></ml:boundVars></ml:function><ml:apply><ml:sqrt /><ml:apply><ml:minus /><ml:apply><ml:pow /><ml:apply><ml:id xml:space="preserve">A</ml:id><ml:id xml:space="preserve">x</ml:id></ml:apply><ml:real>2</ml:real></ml:apply><ml:apply><ml:mult style="default" /><ml:apply><ml:mult style="default" /><ml:real>3</ml:real><ml:id xml:space="preserve">c</ml:id></ml:apply><ml:apply><ml:id xml:space="preserve">B</ml:id><ml:id xml:space="preserve">x</ml:id></ml:apply></ml:apply></ml:apply></ml:apply></ml:define></math><rendering item-idref="61" /></region><region region-id="2461" left="66" top="3800.25" width="93.75" height="27.75" align-x="95.25" align-y="3816" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define><ml:function><ml:id xml:space="preserve">t1a</ml:id><ml:boundVars><ml:id xml:space="preserve">x</ml:id></ml:boundVars></ml:function><ml:apply><ml:div /><ml:apply><ml:plus /><ml:apply><ml:neg /><ml:apply><ml:id xml:space="preserve">A</ml:id><ml:id xml:space="preserve">x</ml:id></ml:apply></ml:apply><ml:apply><ml:id xml:space="preserve">D</ml:id><ml:id xml:space="preserve">x</ml:id></ml:apply></ml:apply><ml:apply><ml:mult style="default" /><ml:real>3</ml:real><ml:apply><ml:id xml:space="preserve">B</ml:id><ml:id xml:space="preserve">x</ml:id></ml:apply></ml:apply></ml:apply></ml:define></math><rendering item-idref="62" /></region><region region-id="2464" left="222" top="3800.25" width="93" height="27.75" align-x="251.25" align-y="3816" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define><ml:function><ml:id xml:space="preserve">t2a</ml:id><ml:boundVars><ml:id xml:space="preserve">x</ml:id></ml:boundVars></ml:function><ml:apply><ml:div /><ml:apply><ml:minus /><ml:apply><ml:neg /><ml:apply><ml:id xml:space="preserve">A</ml:id><ml:id xml:space="preserve">x</ml:id></ml:apply></ml:apply><ml:apply><ml:id xml:space="preserve">D</ml:id><ml:id xml:space="preserve">x</ml:id></ml:apply></ml:apply><ml:apply><ml:mult style="default" /><ml:real>3</ml:real><ml:apply><ml:id xml:space="preserve">B</ml:id><ml:id xml:space="preserve">x</ml:id></ml:apply></ml:apply></ml:apply></ml:define></math><rendering item-idref="63" /></region><region region-id="2474" left="36" top="3854.25" width="333" height="24" align-x="36" align-y="3864" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit">We look for those roots which are real and belong to the interval [0,1]:</p></text></region><region region-id="2261" left="66" top="3891" width="250.5" height="12.75" align-x="90" align-y="3900" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define><ml:function><ml:id xml:space="preserve">Q</ml:id><ml:boundVars><ml:id xml:space="preserve">x</ml:id></ml:boundVars></ml:function><ml:apply><ml:id xml:space="preserve">if</ml:id><ml:sequence><ml:apply><ml:mult style="default" /><ml:apply><ml:mult style="default" /><ml:parens><ml:apply><ml:lessThan /><ml:apply><ml:absval /><ml:apply><ml:id xml:space="preserve">Im</ml:id><ml:parens><ml:id xml:space="preserve">x</ml:id></ml:parens></ml:apply></ml:apply><ml:id xml:space="preserve">TOL</ml:id></ml:apply></ml:parens><ml:parens><ml:apply><ml:greaterOrEqual /><ml:apply><ml:id xml:space="preserve">Re</ml:id><ml:id xml:space="preserve">x</ml:id></ml:apply><ml:real>0</ml:real></ml:apply></ml:parens></ml:apply><ml:parens><ml:apply><ml:lessOrEqual /><ml:apply><ml:id xml:space="preserve">Re</ml:id><ml:id xml:space="preserve">x</ml:id></ml:apply><ml:real>1</ml:real></ml:apply></ml:parens></ml:apply><ml:apply><ml:id xml:space="preserve">Re</ml:id><ml:id xml:space="preserve">x</ml:id></ml:apply><ml:real>1</ml:real></ml:sequence></ml:apply></ml:define></math><rendering item-idref="64" /></region><region region-id="2262" left="66" top="3921" width="69.75" height="12.75" align-x="90.75" align-y="3930" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define><ml:function><ml:id xml:space="preserve">t1</ml:id><ml:boundVars><ml:id xml:space="preserve">x</ml:id></ml:boundVars></ml:function><ml:apply><ml:id xml:space="preserve">Q</ml:id><ml:apply><ml:id xml:space="preserve">t1a</ml:id><ml:id xml:space="preserve">x</ml:id></ml:apply></ml:apply></ml:define></math><rendering item-idref="65" /></region><region region-id="2477" left="222" top="3921" width="69.75" height="12.75" align-x="246.75" align-y="3930" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define><ml:function><ml:id xml:space="preserve">t2</ml:id><ml:boundVars><ml:id xml:space="preserve">x</ml:id></ml:boundVars></ml:function><ml:apply><ml:id xml:space="preserve">Q</ml:id><ml:apply><ml:id xml:space="preserve">t2a</ml:id><ml:id xml:space="preserve">x</ml:id></ml:apply></ml:apply></ml:define></math><rendering item-idref="66" /></region><region region-id="2475" left="36" top="3956.25" width="332.25" height="24" align-x="36" align-y="3966" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit">Order the roots in such a way that <b>T1 &lt; T2</b>, using the intermediate values <b>t1</b> and <b>t2</b>:</p></text></region><region region-id="2265" left="66" top="3999" width="144" height="12.75" align-x="93.75" align-y="4008" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define><ml:function><ml:id xml:space="preserve">T1</ml:id><ml:boundVars><ml:id xml:space="preserve">x</ml:id></ml:boundVars></ml:function><ml:apply><ml:id xml:space="preserve">if</ml:id><ml:sequence><ml:apply><ml:lessThan /><ml:apply><ml:id xml:space="preserve">t1</ml:id><ml:id xml:space="preserve">x</ml:id></ml:apply><ml:apply><ml:id xml:space="preserve">t2</ml:id><ml:id xml:space="preserve">x</ml:id></ml:apply></ml:apply><ml:apply><ml:id xml:space="preserve">t1</ml:id><ml:id xml:space="preserve">x</ml:id></ml:apply><ml:apply><ml:id xml:space="preserve">t2</ml:id><ml:id xml:space="preserve">x</ml:id></ml:apply></ml:sequence></ml:apply></ml:define></math><rendering item-idref="67" /></region><region region-id="2266" left="66" top="4023" width="144" height="12.75" align-x="93.75" align-y="4032" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define><ml:function><ml:id xml:space="preserve">T2</ml:id><ml:boundVars><ml:id xml:space="preserve">x</ml:id></ml:boundVars></ml:function><ml:apply><ml:id xml:space="preserve">if</ml:id><ml:sequence><ml:apply><ml:greaterThan /><ml:apply><ml:id xml:space="preserve">t2</ml:id><ml:id xml:space="preserve">x</ml:id></ml:apply><ml:apply><ml:id xml:space="preserve">t1</ml:id><ml:id xml:space="preserve">x</ml:id></ml:apply></ml:apply><ml:apply><ml:id xml:space="preserve">t2</ml:id><ml:id xml:space="preserve">x</ml:id></ml:apply><ml:apply><ml:id xml:space="preserve">t1</ml:id><ml:id xml:space="preserve">x</ml:id></ml:apply></ml:sequence></ml:apply></ml:define></math><rendering item-idref="68" /></region><region region-id="2476" left="36" top="4058.25" width="117.75" height="36" align-x="36" align-y="4068" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit">Define the size of the intervals to be removed from the integral:</p></text></region><region region-id="2482" left="222" top="4064.25" width="27.75" height="13.5" align-x="233.25" align-y="4074" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define><ml:id xml:space="preserve">λ</ml:id><ml:real>10</ml:real></ml:define></math><rendering item-idref="69" /></region><region region-id="2483" left="300" top="4064.25" width="44.25" height="13.5" align-x="310.5" align-y="4074" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define warning="WarnRedefinedBIFunction"><ml:id xml:space="preserve">ε</ml:id><ml:apply><ml:mult style="default" /><ml:id xml:space="preserve">λ</ml:id><ml:id xml:space="preserve">TOL</ml:id></ml:apply></ml:define></math><rendering item-idref="70" /></region><region region-id="2485" left="0" top="4104" width="6000" height="6" align-x="0" align-y="4104" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><pageBreak /></region><region region-id="2493" left="36" top="4118.25" width="285" height="24" align-x="36" align-y="4128" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit">and rewrite the expression for the derivative in the following way:</p></text></region><region region-id="2492" left="66" top="4163.25" width="261" height="54" align-x="114.75" align-y="4194" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define warning="WarnRedefinedUDFunction"><ml:function><ml:id xml:space="preserve">DerivL1</ml:id><ml:boundVars><ml:id xml:space="preserve">x</ml:id></ml:boundVars></ml:function><ml:apply><ml:integral auto-algorithm="true" algorithm="adaptive" /><ml:lambda><ml:boundVars><ml:id xml:space="preserve">t</ml:id></ml:boundVars><ml:apply><ml:div /><ml:apply><ml:mult style="default" /><ml:apply><ml:id xml:space="preserve">Re</ml:id><ml:apply><ml:mult style="default" /><ml:parens><ml:apply><ml:plus /><ml:real>1</ml:real><ml:apply><ml:mult style="default" /><ml:id xml:space="preserve">Slope</ml:id><ml:imag symbol="i">1</ml:imag></ml:apply></ml:apply></ml:parens><ml:apply><ml:conjugate /><ml:apply><ml:id xml:space="preserve">C</ml:id><ml:sequence><ml:id xml:space="preserve">x</ml:id><ml:id xml:space="preserve">t</ml:id></ml:sequence></ml:apply></ml:apply></ml:apply></ml:apply><ml:parens><ml:apply><ml:plus /><ml:apply><ml:mult style="default" /><ml:real>-2</ml:real><ml:id xml:space="preserve">t</ml:id></ml:apply><ml:apply><ml:mult style="default" /><ml:real>3</ml:real><ml:apply><ml:pow /><ml:id xml:space="preserve">t</ml:id><ml:real>2</ml:real></ml:apply></ml:apply></ml:apply></ml:parens></ml:apply><ml:apply><ml:absval /><ml:apply><ml:id xml:space="preserve">C</ml:id><ml:sequence><ml:id xml:space="preserve">x</ml:id><ml:id xml:space="preserve">t</ml:id></ml:sequence></ml:apply></ml:apply></ml:apply></ml:lambda><ml:bounds><ml:real>0</ml:real><ml:apply><ml:minus /><ml:apply><ml:id xml:space="preserve">T1</ml:id><ml:id xml:space="preserve">x</ml:id></ml:apply><ml:id xml:space="preserve">ε</ml:id></ml:apply></ml:bounds></ml:apply></ml:define></math><rendering item-idref="71" /></region><region region-id="2491" left="66" top="4247.25" width="261" height="54" align-x="114.75" align-y="4278" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define><ml:function><ml:id xml:space="preserve">DerivL2</ml:id><ml:boundVars><ml:id xml:space="preserve">x</ml:id></ml:boundVars></ml:function><ml:apply><ml:integral auto-algorithm="true" algorithm="adaptive" /><ml:lambda><ml:boundVars><ml:id xml:space="preserve">t</ml:id></ml:boundVars><ml:apply><ml:div /><ml:apply><ml:mult style="default" /><ml:apply><ml:id xml:space="preserve">Re</ml:id><ml:apply><ml:mult style="default" /><ml:parens><ml:apply><ml:plus /><ml:real>1</ml:real><ml:apply><ml:mult style="default" /><ml:id xml:space="preserve">Slope</ml:id><ml:imag symbol="i">1</ml:imag></ml:apply></ml:apply></ml:parens><ml:apply><ml:conjugate /><ml:apply><ml:id xml:space="preserve">C</ml:id><ml:sequence><ml:id xml:space="preserve">x</ml:id><ml:id xml:space="preserve">t</ml:id></ml:sequence></ml:apply></ml:apply></ml:apply></ml:apply><ml:parens><ml:apply><ml:plus /><ml:apply><ml:mult style="default" /><ml:real>-2</ml:real><ml:id xml:space="preserve">t</ml:id></ml:apply><ml:apply><ml:mult style="default" /><ml:real>3</ml:real><ml:apply><ml:pow /><ml:id xml:space="preserve">t</ml:id><ml:real>2</ml:real></ml:apply></ml:apply></ml:apply></ml:parens></ml:apply><ml:apply><ml:absval /><ml:apply><ml:id xml:space="preserve">C</ml:id><ml:sequence><ml:id xml:space="preserve">x</ml:id><ml:id xml:space="preserve">t</ml:id></ml:sequence></ml:apply></ml:apply></ml:apply></ml:lambda><ml:bounds><ml:apply><ml:plus /><ml:apply><ml:id xml:space="preserve">T1</ml:id><ml:id xml:space="preserve">x</ml:id></ml:apply><ml:id xml:space="preserve">ε</ml:id></ml:apply><ml:apply><ml:minus /><ml:apply><ml:id xml:space="preserve">T2</ml:id><ml:id xml:space="preserve">x</ml:id></ml:apply><ml:id xml:space="preserve">ε</ml:id></ml:apply></ml:bounds></ml:apply></ml:define></math><rendering item-idref="72" /></region><region region-id="2490" left="66" top="4331.25" width="259.5" height="54" align-x="114.75" align-y="4362" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define><ml:function><ml:id xml:space="preserve">DerivL3</ml:id><ml:boundVars><ml:id xml:space="preserve">x</ml:id></ml:boundVars></ml:function><ml:apply><ml:integral auto-algorithm="true" algorithm="adaptive" /><ml:lambda><ml:boundVars><ml:id xml:space="preserve">t</ml:id></ml:boundVars><ml:apply><ml:div /><ml:apply><ml:mult style="default" /><ml:apply><ml:id xml:space="preserve">Re</ml:id><ml:apply><ml:mult style="default" /><ml:parens><ml:apply><ml:plus /><ml:real>1</ml:real><ml:apply><ml:mult style="default" /><ml:id xml:space="preserve">Slope</ml:id><ml:imag symbol="i">1</ml:imag></ml:apply></ml:apply></ml:parens><ml:apply><ml:conjugate /><ml:apply><ml:id xml:space="preserve">C</ml:id><ml:sequence><ml:id xml:space="preserve">x</ml:id><ml:id xml:space="preserve">t</ml:id></ml:sequence></ml:apply></ml:apply></ml:apply></ml:apply><ml:parens><ml:apply><ml:plus /><ml:apply><ml:mult style="default" /><ml:real>-2</ml:real><ml:id xml:space="preserve">t</ml:id></ml:apply><ml:apply><ml:mult style="default" /><ml:real>3</ml:real><ml:apply><ml:pow /><ml:id xml:space="preserve">t</ml:id><ml:real>2</ml:real></ml:apply></ml:apply></ml:apply></ml:parens></ml:apply><ml:apply><ml:absval /><ml:apply><ml:id xml:space="preserve">C</ml:id><ml:sequence><ml:id xml:space="preserve">x</ml:id><ml:id xml:space="preserve">t</ml:id></ml:sequence></ml:apply></ml:apply></ml:apply></ml:lambda><ml:bounds><ml:apply><ml:plus /><ml:apply><ml:id xml:space="preserve">T2</ml:id><ml:id xml:space="preserve">x</ml:id></ml:apply><ml:id xml:space="preserve">ε</ml:id></ml:apply><ml:real>1</ml:real></ml:bounds></ml:apply></ml:define></math><rendering item-idref="73" /></region><region region-id="2499" left="36" top="4412.25" width="242.25" height="12" align-x="36" align-y="4422" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit">The new expression for the derivative becomes:</p></text></region><region region-id="2275" left="66" top="4442.25" width="366" height="13.5" align-x="110.25" align-y="4452" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define warning="WarnRedefinedUDFunction"><ml:function><ml:id xml:space="preserve">DerivL</ml:id><ml:boundVars><ml:id xml:space="preserve">x</ml:id></ml:boundVars></ml:function><ml:apply><ml:plus /><ml:apply><ml:plus /><ml:apply><ml:id xml:space="preserve">DerivL1</ml:id><ml:id xml:space="preserve">x</ml:id></ml:apply><ml:apply><ml:mult style="default" /><ml:parens><ml:apply><ml:lessThan /><ml:apply><ml:plus /><ml:apply><ml:id xml:space="preserve">T1</ml:id><ml:id xml:space="preserve">x</ml:id></ml:apply><ml:id xml:space="preserve">ε</ml:id></ml:apply><ml:apply><ml:minus /><ml:apply><ml:id xml:space="preserve">T2</ml:id><ml:id xml:space="preserve">x</ml:id></ml:apply><ml:id xml:space="preserve">ε</ml:id></ml:apply></ml:apply></ml:parens><ml:apply><ml:id xml:space="preserve">DerivL2</ml:id><ml:id xml:space="preserve">x</ml:id></ml:apply></ml:apply></ml:apply><ml:apply><ml:mult style="default" /><ml:parens><ml:apply><ml:lessThan /><ml:apply><ml:plus /><ml:apply><ml:id xml:space="preserve">T2</ml:id><ml:id xml:space="preserve">x</ml:id></ml:apply><ml:id xml:space="preserve">ε</ml:id></ml:apply><ml:real>1</ml:real></ml:apply></ml:parens><ml:apply><ml:id xml:space="preserve">DerivL3</ml:id><ml:id xml:space="preserve">x</ml:id></ml:apply></ml:apply></ml:apply></ml:define></math><rendering item-idref="74" /></region><region region-id="2500" left="36" top="4478.25" width="220.5" height="12" align-x="36" align-y="4488" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit">Now we can find the root of the derivative. </p></text></region><region region-id="2277" left="66" top="4509" width="52.5" height="12.75" align-x="99.75" align-y="4518" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define><ml:id xml:space="preserve">guessx</ml:id><ml:real>0.7</ml:real></ml:define></math><rendering item-idref="75" /></region><region region-id="2504" left="144" top="4509" width="137.25" height="12.75" align-x="155.25" align-y="4518" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define warning="WarnRedefinedUDScalar"><ml:id xml:space="preserve">d</ml:id><ml:apply><ml:id xml:space="preserve">root</ml:id><ml:sequence><ml:apply><ml:id xml:space="preserve">DerivL</ml:id><ml:id xml:space="preserve">guessx</ml:id></ml:apply><ml:id xml:space="preserve">guessx</ml:id></ml:sequence></ml:apply></ml:define></math><rendering item-idref="76" /></region><region region-id="2279" left="66" top="4539" width="57" height="12.75" align-x="76.5" align-y="4548" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:eval><ml:id xml:space="preserve">d</ml:id><result xmlns="http://schemas.mathsoft.com/math20"><ml:real>0.21072136502466818</ml:real></result></ml:eval></math><rendering item-idref="77" /></region><region region-id="2505" left="144" top="4533" width="123.75" height="18.75" align-x="189" align-y="4548" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:eval><ml:apply><ml:id xml:space="preserve">DerivL</ml:id><ml:id xml:space="preserve">d</ml:id></ml:apply><result xmlns="http://schemas.mathsoft.com/math20"><ml:real>6.9997202478688791E-12</ml:real></result></ml:eval></math><rendering item-idref="78" /></region><region region-id="2501" left="36" top="4568.25" width="311.25" height="24" align-x="36" align-y="4578" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit">With either method, the minimal possible length for our curve is approximately </p></text></region><region region-id="2282" left="66" top="4605" width="69.75" height="12.75" align-x="89.25" align-y="4614" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:eval><ml:apply><ml:id xml:space="preserve">L</ml:id><ml:id xml:space="preserve">d</ml:id></ml:apply><result xmlns="http://schemas.mathsoft.com/math20"><ml:real>3.602274779449044</ml:real></result></ml:eval></math><rendering item-idref="79" /></region><region region-id="2502" left="36" top="4634.25" width="223.5" height="12" align-x="36" align-y="4644" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit">Choose a length <b>L0</b> to be greater than <b>L(d):</b></p></text></region><region region-id="2510" left="270" top="4635" width="34.5" height="12.75" align-x="285.75" align-y="4644" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define warning="WarnRedefinedUDScalar"><ml:id xml:space="preserve">L0</ml:id><ml:real>4.2</ml:real></ml:define></math><rendering item-idref="80" /></region><region region-id="2566" left="0" top="4656" width="6000" height="6" align-x="0" align-y="4656" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><pageBreak /></region><region region-id="2503" left="36" top="4664.25" width="189.75" height="12" align-x="36" align-y="4674" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit">Find two roots of equation <b>L(x) = L0.</b></p></text></region><region region-id="2286" left="66" top="4695" width="63.75" height="12.75" align-x="101.25" align-y="4704" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define><ml:id xml:space="preserve">guessp</ml:id><ml:apply><ml:minus /><ml:id xml:space="preserve">d</ml:id><ml:real>5</ml:real></ml:apply></ml:define></math><rendering item-idref="81" /></region><region region-id="2514" left="144" top="4695" width="64.5" height="12.75" align-x="179.25" align-y="4704" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define><ml:id xml:space="preserve">guessq</ml:id><ml:apply><ml:plus /><ml:id xml:space="preserve">d</ml:id><ml:real>5</ml:real></ml:apply></ml:define></math><rendering item-idref="82" /></region><region region-id="2511" left="270" top="4694.25" width="34.5" height="12" align-x="270" align-y="4704" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit">Check:</p></text></region><region region-id="2289" left="66" top="4725" width="46.5" height="12.75" align-x="75.75" align-y="4734" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define warning="WarnRedefinedUDScalar"><ml:id xml:space="preserve">x</ml:id><ml:id xml:space="preserve">guessp</ml:id></ml:define></math><rendering item-idref="83" /></region><region region-id="2515" left="144" top="4725" width="90.75" height="12.75" align-x="157.5" align-y="4734" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define><ml:id xml:space="preserve">r1</ml:id><ml:apply><ml:id xml:space="preserve">root</ml:id><ml:sequence><ml:apply><ml:minus /><ml:apply><ml:id xml:space="preserve">L</ml:id><ml:id xml:space="preserve">x</ml:id></ml:apply><ml:id xml:space="preserve">L0</ml:id></ml:apply><ml:id xml:space="preserve">x</ml:id></ml:sequence></ml:apply></ml:define></math><rendering item-idref="84" /></region><region region-id="2512" left="270" top="4725" width="49.5" height="12.75" align-x="295.5" align-y="4734" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:eval><ml:apply><ml:id xml:space="preserve">L</ml:id><ml:id xml:space="preserve">r1</ml:id></ml:apply><result xmlns="http://schemas.mathsoft.com/math20"><ml:real>4.2000000000001512</ml:real></result></ml:eval></math><rendering item-idref="85" /></region><region region-id="2292" left="66" top="4761" width="46.5" height="12.75" align-x="75.75" align-y="4770" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define warning="WarnRedefinedUDScalar"><ml:id xml:space="preserve">x</ml:id><ml:id xml:space="preserve">guessq</ml:id></ml:define></math><rendering item-idref="86" /></region><region region-id="2516" left="144" top="4761" width="90.75" height="12.75" align-x="157.5" align-y="4770" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define><ml:id xml:space="preserve">r2</ml:id><ml:apply><ml:id xml:space="preserve">root</ml:id><ml:sequence><ml:apply><ml:minus /><ml:apply><ml:id xml:space="preserve">L</ml:id><ml:id xml:space="preserve">x</ml:id></ml:apply><ml:id xml:space="preserve">L0</ml:id></ml:apply><ml:id xml:space="preserve">x</ml:id></ml:sequence></ml:apply></ml:define></math><rendering item-idref="87" /></region><region region-id="2513" left="270" top="4761" width="49.5" height="12.75" align-x="295.5" align-y="4770" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:eval><ml:apply><ml:id xml:space="preserve">L</ml:id><ml:id xml:space="preserve">r2</ml:id></ml:apply><result xmlns="http://schemas.mathsoft.com/math20"><ml:real>4.2000000000000073</ml:real></result></ml:eval></math><rendering item-idref="88" /></region><region region-id="2521" left="36" top="4796.25" width="102.75" height="12" align-x="36" align-y="4806" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit">So the first curve is </p></text></region><region region-id="2296" left="66" top="4821" width="181.5" height="18.75" align-x="111" align-y="4836" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define warning="WarnRedefinedUDFunction"><ml:function><ml:id xml:space="preserve">spline1</ml:id><ml:boundVars><ml:id xml:space="preserve">t</ml:id></ml:boundVars></ml:function><ml:apply><ml:plus /><ml:apply><ml:plus /><ml:apply><ml:plus /><ml:id xml:space="preserve">start</ml:id><ml:apply><ml:mult style="default" /><ml:id xml:space="preserve">c</ml:id><ml:id xml:space="preserve">t</ml:id></ml:apply></ml:apply><ml:apply><ml:mult style="default" /><ml:apply><ml:id xml:space="preserve">A</ml:id><ml:id xml:space="preserve">r1</ml:id></ml:apply><ml:apply><ml:pow /><ml:id xml:space="preserve">t</ml:id><ml:real>2</ml:real></ml:apply></ml:apply></ml:apply><ml:apply><ml:mult style="default" /><ml:apply><ml:id xml:space="preserve">B</ml:id><ml:id xml:space="preserve">r1</ml:id></ml:apply><ml:apply><ml:pow /><ml:id xml:space="preserve">t</ml:id><ml:real>3</ml:real></ml:apply></ml:apply></ml:apply></ml:define></math><rendering item-idref="89" /></region><region region-id="2522" left="36" top="4856.25" width="110.25" height="12" align-x="36" align-y="4866" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit">and the second one is</p></text></region><region region-id="2298" left="66" top="4881" width="181.5" height="18.75" align-x="111" align-y="4896" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define warning="WarnRedefinedUDFunction"><ml:function><ml:id xml:space="preserve">spline2</ml:id><ml:boundVars><ml:id xml:space="preserve">t</ml:id></ml:boundVars></ml:function><ml:apply><ml:plus /><ml:apply><ml:plus /><ml:apply><ml:plus /><ml:id xml:space="preserve">start</ml:id><ml:apply><ml:mult style="default" /><ml:id xml:space="preserve">c</ml:id><ml:id xml:space="preserve">t</ml:id></ml:apply></ml:apply><ml:apply><ml:mult style="default" /><ml:apply><ml:id xml:space="preserve">A</ml:id><ml:id xml:space="preserve">r2</ml:id></ml:apply><ml:apply><ml:pow /><ml:id xml:space="preserve">t</ml:id><ml:real>2</ml:real></ml:apply></ml:apply></ml:apply><ml:apply><ml:mult style="default" /><ml:apply><ml:id xml:space="preserve">B</ml:id><ml:id xml:space="preserve">r2</ml:id></ml:apply><ml:apply><ml:pow /><ml:id xml:space="preserve">t</ml:id><ml:real>3</ml:real></ml:apply></ml:apply></ml:apply></ml:define></math><rendering item-idref="90" /></region><region region-id="2523" left="36" top="4916.25" width="328.5" height="24" align-x="36" align-y="4926" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit">To graph these splines we need to plot imaginary part of our polynomials against their real parts.</p></text></region><region region-id="2524" left="36" top="4952.25" width="74.25" height="12" align-x="36" align-y="4962" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit">range variable:</p></text></region><region region-id="2525" left="144" top="4953" width="30" height="12.75" align-x="153" align-y="4962" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define><ml:id xml:space="preserve">I</ml:id><ml:real>100</ml:real></ml:define></math><rendering item-idref="91" /></region><region region-id="2526" left="222" top="4953" width="30.75" height="12.75" align-x="230.25" align-y="4962" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define><ml:id xml:space="preserve">i</ml:id><ml:range><ml:real>0</ml:real><ml:id xml:space="preserve">I</ml:id></ml:range></ml:define></math><rendering item-idref="92" /></region><region region-id="2527" left="306" top="4946.25" width="26.25" height="27.75" align-x="318.75" align-y="4962" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define warning="WarnRedefinedBIUnit"><ml:apply><ml:indexer /><ml:id xml:space="preserve">s</ml:id><ml:id xml:space="preserve">i</ml:id></ml:apply><ml:apply><ml:div /><ml:id xml:space="preserve">i</ml:id><ml:id xml:space="preserve">I</ml:id></ml:apply></ml:define></math><rendering item-idref="93" /></region><region region-id="2536" left="66" top="4983" width="79.5" height="18.75" align-x="79.5" align-y="4998" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define><ml:id xml:space="preserve">X</ml:id><ml:apply><ml:vectorize /><ml:apply><ml:id xml:space="preserve">Re</ml:id><ml:apply><ml:id xml:space="preserve">spline1</ml:id><ml:id xml:space="preserve">s</ml:id></ml:apply></ml:apply></ml:apply></ml:define></math><rendering item-idref="94" /></region><region region-id="2540" left="222" top="4983" width="77.25" height="18.75" align-x="234" align-y="4998" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define><ml:id xml:space="preserve">Y</ml:id><ml:apply><ml:vectorize /><ml:apply><ml:id xml:space="preserve">Im</ml:id><ml:apply><ml:id xml:space="preserve">spline1</ml:id><ml:id xml:space="preserve">s</ml:id></ml:apply></ml:apply></ml:apply></ml:define></math><rendering item-idref="95" /></region><region region-id="2538" left="66" top="5007" width="78.75" height="18.75" align-x="78.75" align-y="5022" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define><ml:id xml:space="preserve">U</ml:id><ml:apply><ml:vectorize /><ml:apply><ml:id xml:space="preserve">Re</ml:id><ml:apply><ml:id xml:space="preserve">spline2</ml:id><ml:id xml:space="preserve">s</ml:id></ml:apply></ml:apply></ml:apply></ml:define></math><rendering item-idref="96" /></region><region region-id="2541" left="222" top="5007" width="81" height="18.75" align-x="237.75" align-y="5022" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define warning="WarnRedefinedBIUnit"><ml:id xml:space="preserve">W</ml:id><ml:apply><ml:vectorize /><ml:apply><ml:id xml:space="preserve">Im</ml:id><ml:apply><ml:id xml:space="preserve">spline2</ml:id><ml:id xml:space="preserve">s</ml:id></ml:apply></ml:apply></ml:apply></ml:define></math><rendering item-idref="97" /></region><region region-id="2542" left="36" top="5046" width="288.75" height="174.75" align-x="36" align-y="5046" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><plot disable-calc="false" item-idref="98" /><rendering item-idref="99" /></region><region region-id="2545" left="36" top="5228.25" width="325.5" height="84" align-x="36" align-y="5238" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit">This is an "equilateral fish" — the red solid curve has the same length as the blue dotted one. Both curves start at the same point on the "fish tail." Their starting velocities are equal. They finish simultaneously at the same point. Their finishing velocities are different. However the curves have the same tangent line at the finishing point which has a prescribed angle with the <b>x</b>-axis (45 degrees in this case).</p></text></region><region region-id="2547" left="0" top="5322" width="6000" height="6" align-x="0" align-y="5322" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><pageBreak /></region><region region-id="2546" left="36" top="5330.25" width="328.5" height="36" align-x="36" align-y="5340" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit">To verify the results let's calculate the velocities of our curves at the endpoints (these match the complex value of <b>c </b>set at the start of this example):</p></text></region><region region-id="2311" left="66" top="5385" width="21" height="12.75" align-x="75" align-y="5394" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define><ml:id xml:space="preserve">t</ml:id><ml:real>0</ml:real></ml:define></math><rendering item-idref="100" /></region><region region-id="2548" left="144" top="5377.5" width="95.25" height="29.25" align-x="198" align-y="5394" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:eval><ml:apply><ml:derivative /><ml:lambda><ml:boundVars><ml:id xml:space="preserve">t</ml:id></ml:boundVars><ml:apply><ml:id xml:space="preserve">spline1</ml:id><ml:id xml:space="preserve">t</ml:id></ml:apply></ml:lambda></ml:apply><result xmlns="http://schemas.mathsoft.com/math20"><ml:complex><ml:real>-1.000000000000006</ml:real><ml:imag>5.000000000000008</ml:imag></ml:complex></result></ml:eval></math><rendering item-idref="101" /></region><region region-id="2556" left="270" top="5377.5" width="95.25" height="29.25" align-x="324" align-y="5394" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:eval><ml:apply><ml:derivative /><ml:lambda><ml:boundVars><ml:id xml:space="preserve">t</ml:id></ml:boundVars><ml:apply><ml:id xml:space="preserve">spline2</ml:id><ml:id xml:space="preserve">t</ml:id></ml:apply></ml:lambda></ml:apply><result xmlns="http://schemas.mathsoft.com/math20"><ml:complex><ml:real>-1.0000000000000071</ml:real><ml:imag>5.000000000000008</ml:imag></ml:complex></result></ml:eval></math><rendering item-idref="102" /></region><region region-id="2314" left="66" top="5433" width="21" height="12.75" align-x="75" align-y="5442" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define warning="WarnRedefinedUDScalar"><ml:id xml:space="preserve">t</ml:id><ml:real>1</ml:real></ml:define></math><rendering item-idref="103" /></region><region region-id="2549" left="144" top="5425.5" width="103.5" height="29.25" align-x="166.5" align-y="5442" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define><ml:id xml:space="preserve">vel1</ml:id><ml:apply><ml:id xml:space="preserve">arg</ml:id><ml:apply><ml:derivative /><ml:lambda><ml:boundVars><ml:id xml:space="preserve">t</ml:id></ml:boundVars><ml:apply><ml:id xml:space="preserve">spline1</ml:id><ml:id xml:space="preserve">t</ml:id></ml:apply></ml:lambda></ml:apply></ml:apply></ml:define></math><rendering item-idref="104" /></region><region region-id="2557" left="270" top="5425.5" width="103.5" height="29.25" align-x="292.5" align-y="5442" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define><ml:id xml:space="preserve">vel2</ml:id><ml:apply><ml:id xml:space="preserve">arg</ml:id><ml:apply><ml:derivative /><ml:lambda><ml:boundVars><ml:id xml:space="preserve">t</ml:id></ml:boundVars><ml:apply><ml:id xml:space="preserve">spline2</ml:id><ml:id xml:space="preserve">t</ml:id></ml:apply></ml:lambda></ml:apply></ml:apply></ml:define></math><rendering item-idref="105" /></region><region region-id="2559" left="36" top="5468.25" width="333" height="36" align-x="36" align-y="5478" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit">To see that the slope at the finishing point really is equal to 45 degrees, compare the tangents of the two velocities (the tangent will equal the value of <b>Slope</b>, set earlier):</p></text></region><region region-id="2318" left="66" top="5517" width="59.25" height="12.75" align-x="108" align-y="5526" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:eval><ml:apply><ml:id xml:space="preserve">tan</ml:id><ml:id xml:space="preserve">vel1</ml:id></ml:apply><result xmlns="http://schemas.mathsoft.com/math20"><ml:real>0.99999999999999845</ml:real></result></ml:eval></math><rendering item-idref="106" /></region><region region-id="2319" left="210" top="5517" width="59.25" height="12.75" align-x="252" align-y="5526" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:eval><ml:apply><ml:id xml:space="preserve">tan</ml:id><ml:id xml:space="preserve">vel2</ml:id></ml:apply><result xmlns="http://schemas.mathsoft.com/math20"><ml:real>1.00000000000002</ml:real></result></ml:eval></math><rendering item-idref="107" /></region><region region-id="2560" left="36" top="5540.25" width="336.75" height="276" align-x="36" align-y="5550" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">The reader is advised to play with all the parameters in this document. For example, setting the parameter <b>slope</b> to be a big number will force both curves to be perpendicular to the<b> x</b>-axis at the finishing point (the correct way of doing this is, of course, to reparametrize the problem in terms of<b> y </b>instead of <b>x</b>). Changing the velocity at the starting point can radically change the shape of the solutions even if the slope at the starting point remains the same. Try, for example, multiplying the starting velocity by 5 and changing length<b> L0 </b>to 12. Note also that if the required length approaches <b>L(d) </b>the curves will move closer and closer to each other until they will merge when the length will be approximately equal to <b>L(d)</b>. When the length is less then <b>L(d)</b> the root functions fail to converge (bear in mind, however, that some precision loss is inevitable).</p><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit" /><p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">Another experiment is to impose other linear initial conditions or to relax some of these conditions. This article can be used for the purposes of practical curve design. Some experimentation will be required in choosing initial guesses for the Mathcad <b>root</b> functions, but the information on the length function presented here is sufficient for doing this quickly. The calculations performed here are applicable to any parametric families of curves which depend linearly on a parameter which controls the length. (cf. [2]).</p></text></region><region region-id="2561" left="36" top="5828.25" width="341.25" height="24" align-x="36" align-y="5838" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit">The author is grateful to Frank Purcell, Paul Lorczak and Leslie Bondaryk for very useful comments.</p></text></region><region region-id="2573" left="36" top="5869.5" width="51.75" height="12.75" align-x="36" align-y="5880" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="subsubtitle" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">References</p></text></region><region region-id="2570" left="36" top="5888.25" width="334.5" height="60" align-x="36" align-y="5898" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">[1] Handbook of Mathematical Functions, edited by M. Abramovitz and A Stegun, National Bureau of Standards, 1964.</p><p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">[2] A. Kushkuley and S. Rozenberg. Length function on a parametric family of curves, <i>Latvian Math. Ezhegodnik</i>, vol. 27, Riga, pp. 154-159, 1983 (Russian). </p></text></region><region region-id="2564" left="6" top="5952" width="427.5" height="5.25" align-x="6" align-y="5952" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><png item-idref="108" display-width="426" display-height="3.75" /><rendering item-idref="109" /></region></regions><binaryContent><item item-id="1" content-encoding="gzip">H4sIAAAAAAAA/4yNvQ6CQBCEdwEVEMXGGhMbj5InsDBWJj6CnvifEA1SWPrm5+yCvUf2mJv7
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