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Semicubical parabola
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<h2 id="properties-of-semicubical-parabolas">Properties of semicubical parabolas</h2> <h3>Similarity</h3> <p>Any semicubical parabola ( t 2 , a t 3 ) {\displaystyle (t^{2},at^{3})} is <a href="/facts/Similarity_(geometry)/Sbd6hEAN">similar</a> to the <i>semicubical unit parabola</i> ( u 2 , u 3 ) {\displaystyle (u^{2},u^{3})} . </p><p><i>Proof:</i> The similarity ( x , y ) → ( a 2 x , a 2 y ) {\displaystyle (x,y)\rightarrow (a^{2}x,a^{2}y)} (uniform scaling) maps the semicubical parabola ( t 2 , a t 3 ) {\displaystyle (t^{2},at^{3})} onto the curve ( ( a t ) 2 , ( a t ) 3 ) = ( u 2 , u 3 ) {\displaystyle ((at)^{2},(at)^{3})=(u^{2},u^{3})} with u = a t {\displaystyle u=at} . </p> <h3>Singularity</h3> <p>The parametric representation ( t 2 , a t 3 ) {\displaystyle (t^{2},at^{3})} is <i><a href="/facts/Regular_curve/xBQ0pTBW">regular</a> except</i> at point ( 0 , 0 ) {\displaystyle (0,0)} . At point ( 0 , 0 ) {\displaystyle (0,0)} the curve has a <i><a href="/facts/Singularity_(mathematics)/Y5sMQY4I">singularity</a></i> (cusp). The <i>proof</i> follows from the tangent vector ( 2 t , 3 t 2 ) {\displaystyle (2t,3t^{2})} . Only for t = 0 {\displaystyle t=0} this vector has zero length. </p> <h3>Tangents</h3> <p>Differentiating the <i>semicubical unit parabola</i> y = ± x 3 / 2 {\displaystyle y=\pm x^{3/2}} one gets at point ( x 0 , y 0 ) {\displaystyle (x_{0},y_{0})} of the <i>upper</i> branch the equation of the tangent: </p> y = x 0 2 ( 3 x − x 0 ) . {\displaystyle y={\frac {\sqrt {x_{0}}}{2}}\left(3x-x_{0}\right).} <p>This tangent intersects the <i>lower</i> branch at exactly one further point with coordinates <a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a> </p> ( x 0 4 , − y 0 8 ) . {\displaystyle \left({\frac {x_{0}}{4}},-{\frac {y_{0}}{8}}\right).} <p>(Proving this statement one should use the fact, that the tangent meets the curve at ( x 0 , y 0 ) {\displaystyle (x_{0},y_{0})} twice.) </p> <h3>Arclength</h3> <p>Determining the <a href="/facts/Arclength/QgTrmLxY">arclength</a> of a curve ( x ( t ) , y ( t ) ) {\displaystyle (x(t),y(t))} one has to solve the integral ∫ x ′ ( t ) 2 + y ′ ( t ) 2 d t . {\textstyle \int {\sqrt {x'(t)^{2}+y'(t)^{2}}}\;dt.} For the semicubical parabola ( t 2 , a t 3 ) , 0 ≤ t ≤ b , {\displaystyle (t^{2},at^{3}),\;0\leq t\leq b,} one gets </p> ∫ 0 b x ′ ( t ) 2 + y ′ ( t ) 2 d t = ∫ 0 b t 4 + 9 a 2 t 2 d t = ⋯ = [ 1 27 a 2 ( 4 + 9 a 2 t 2 ) 3 2 ] 0 b . {\displaystyle \int _{0}^{b}{\sqrt {x'(t)^{2}+y'(t)^{2}}}\;dt=\int _{0}^{b}t{\sqrt {4+9a^{2}t^{2}}}\;dt=\cdots =\left[{\frac {1}{27a^{2}}}\left(4+9a^{2}t^{2}\right)^{\frac {3}{2}}\right]_{0}^{b}\;.} <p>(The integral can be solved by the <a href="/facts/Integration_by_substitution/NvXfTTaf">substitution</a> u = 4 + 9 a 2 t 2 {\displaystyle u=4+9a^{2}t^{2}} .) </p><p><i>Example:</i> For <i>a</i> = 1 (semicubical unit parabola) and <i>b</i> = 2, which means the length of the arc between the origin and point (4,8), one gets the arc length 9.073. </p> <h3>Evolute of the unit parabola</h3> <p>The <a href="/facts/Evolute/TmNQdwFY">evolute of the <i>parabola</i></a> ( t 2 , t ) {\displaystyle (t^{2},t)} is a semicubical parabola shifted by 1/2 along the <i>x</i>-axis: ( 1 2 + t 2 , 4 3 3 t 3 ) . {\textstyle \left({\frac {1}{2}}+t^{2},{\frac {4}{{\sqrt {3}}^{3}}}t^{3}\right).} </p> <h3>Polar coordinates</h3> <p>In order to get the representation of the semicubical parabola ( t 2 , a t 3 ) {\displaystyle (t^{2},at^{3})} in polar coordinates, one determines the intersection point of the line y = m x {\displaystyle y=mx} with the curve. For m ≠ 0 {\displaystyle m\neq 0} there is one point different from the origin: ( m 2 a 2 , m 3 a 2 ) . {\textstyle \left({\frac {m^{2}}{a^{2}}},{\frac {m^{3}}{a^{2}}}\right).} This point has distance m 2 a 2 1 + m 2 {\textstyle {\frac {m^{2}}{a^{2}}}{\sqrt {1+m^{2}}}} from the origin. With m = tan φ {\displaystyle m=\tan \varphi } and sec 2 φ = 1 + tan 2 φ {\displaystyle \sec ^{2}\varphi =1+\tan ^{2}\varphi } ( see <a href="/facts/List_of_trigonometric_identities/Xw9Jx8Vz">List of identities</a>) one gets <a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a> </p> r = ( tan φ a ) 2 sec φ , − π 2 < φ < π 2 . {\displaystyle r=\left({\frac {\tan \varphi }{a}}\right)^{2}\sec \varphi \;,\quad -{\frac {\pi }{2}}<\varphi <{\frac {\pi }{2}}.} <h3>Relation between a semicubical parabola and a cubic function</h3> <p>Mapping the semicubical parabola ( t 2 , t 3 ) {\displaystyle (t^{2},t^{3})} by the <a href="/facts/Projective_map/wCkNzUjd">projective map</a> ( x , y ) → ( x y , 1 y ) {\textstyle (x,y)\rightarrow \left({\frac {x}{y}},{\frac {1}{y}}\right)} (<a href="/facts/Involution_(mathematics)/kKxYrUVa">involutory</a> <a href="/facts/Perspective_(geometry)/IKUbVuYw">perspectivity</a> with axis y = 1 {\displaystyle y=1} and center ( 0 , − 1 ) {\displaystyle (0,-1)} ) yields ( 1 t , 1 t 3 ) , {\textstyle \left({\frac {1}{t}},{\frac {1}{t^{3}}}\right),} hence the <i>cubic function</i> y = x 3 . {\displaystyle y=x^{3}.} The cusp (origin) of the semicubical parabola is exchanged with the point at infinity of the y-axis. </p><p>This property can be derived, too, if one represents the semicubical parabola by <i><a href="/facts/Homogeneous_coordinates/TG9ebKN4">homogeneous coordinates</a></i>: In equation (A) the replacement x = x 1 x 3 , y = x 2 x 3 {\displaystyle x={\tfrac {x_{1}}{x_{3}}},\;y={\tfrac {x_{2}}{x_{3}}}} (the line at infinity has equation x 3 = 0 {\displaystyle x_{3}=0} .) and the multiplication by x 3 3 {\displaystyle x_{3}^{3}} is performed. One gets the equation of the curve </p> <ul><li>in <i>homogeneous coordinates</i>: x 3 x 2 2 − x 1 3 = 0. {\displaystyle x_{3}x_{2}^{2}-x_{1}^{3}=0.} </li></ul> <p>Choosing line x 2 = 0 {\displaystyle x_{\color {red}2}=0} as line at infinity and introducing x = x 1 x 2 , y = x 3 x 2 {\displaystyle x={\tfrac {x_{1}}{x_{2}}},\;y={\tfrac {x_{3}}{x_{2}}}} yields the (affine) curve y = x 3 . {\displaystyle y=x^{3}.} </p> <h3>Isochrone curve</h3> <p>An additional defining property of the semicubical parabola is that it is an isochrone curve, meaning that a particle following its path while being pulled down by gravity travels equal vertical intervals in equal time periods. In this way it is related to the <a href="/facts/Tautochrone_curve/zrfIzYj1">tautochrone curve</a>, for which particles at different starting points always take equal time to reach the bottom, and the <a href="/facts/Brachistochrone_curve/zb7IVBQJ">brachistochrone curve</a>, the curve that minimizes the time it takes for a falling particle to travel from its start to its end. </p> <h2 id="history">History</h2> <p>The semicubical parabola was discovered in 1657 by <a href="/facts/William_Neile/29KzEf1S">William Neile</a> who computed its <a href="/facts/Arc_length/QgTrmLxY">arc length</a>. Although the lengths of some other non-algebraic curves including the <a href="/facts/Logarithmic_spiral/4rwkxL7B">logarithmic spiral</a> and <a href="/facts/Cycloid/8Qokqud5">cycloid</a> had already been computed (that is, those curves had been <i>rectified</i>), the semicubical parabola was the first <a href="/facts/Algebraic_curve/LnWsPmDP">algebraic curve</a> (excluding the <a href="/facts/Line_(geometry)/gJqsr4Gj">line</a> and <a href="/facts/Circle/9fy5ggKn">circle</a>) to be rectified.<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a>[<i>disputed (for: It appears that <a href="/facts/Parabola/Cweo3wgw">parabola</a> and other <a href="/facts/Conic_section/teqgLptX">conic sections</a> have been rectified a long time before) – discuss</i>] </p> <ul><li>August Pein: <i>Die semicubische oder Neil'sche Parabel, ihre Sekanten und Tangenten </i>, 1875, <a href="https://books.google.com/books?id=wFxaAAAAYAAJ&dq=neilsche+parabel&pg=PA9">Dissertation</a></li> <li><a href="https://books.google.com/books?id=JrslMKTgSZwC&pg=PA148">Clifford A. Pickover: <i>The Length of Neile's Semicubical Parabola</i></a></li></ul> <h2 id="external-links">External links</h2> <ul><li>O'Connor, John J.; <a href="/facts/Edmund_F._Robertson/jQMcnY79">Robertson, Edmund F.</a>, <a href="https://mathshistory.st-andrews.ac.uk/Curves/Neiles.html">"Neile's Semi-cubical Parabola"</a>, <i><a href="/facts/MacTutor_History_of_Mathematics_Archive/xtwLmN5q">MacTutor History of Mathematics Archive</a></i>, <a href="/facts/University_of_St_Andrews/IwXjmljw">University of St Andrews</a></li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Pickover, Clifford A. (2009), "The Length of Neile's Semicubical Parabola", The Math Book: From Pythagoras to the 57th Dimension, 250 Milestones in the History of Mathematics, Sterling Publishing Company, Inc., p. 148, ISBN 9781402757969. <a href="9781402757969" target="_blank">9781402757969</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>August Pein: Die semicubische oder Neil'sche Parabel, ihre Sekanten und Tangenten , p.2 <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>August Pein: Die semicubische oder Neil'sche Parabel, ihre Sekanten und Tangenten , p.26 <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>August Pein: Die semicubische oder Neil'sche Parabel, ihre Sekanten und Tangenten , p. 10 <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>Pickover, Clifford A. (2009), "The Length of Neile's Semicubical Parabola", The Math Book: From Pythagoras to the 57th Dimension, 250 Milestones in the History of Mathematics, Sterling Publishing Company, Inc., p. 148, ISBN 9781402757969. <a href="9781402757969" target="_blank">9781402757969</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> </ol>