\hfill \thepage} %} \input{tcilatex} \begin{document} \begin{center} {\Large \vspace{1pt}The semicubic parabola} \vspace{1pt}by Arne Petersen 10/2/97 Math 50c\FRAME{dtbpFU}{2.9118in}{2.9118in}{0pt}{\Qcb{Figure 1}}{}{scp.gif}{% \special{language "Scientific Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display "USEDEF";valid_file "F";width 2.9118in;height 2.9118in;depth 0pt;original-width 259.75pt;original-height 259.75pt;cropleft "0";croptop "1";cropright "1";cropbottom "0";filename 'scp.gif';file-properties "XNPEU";}% } \end{center} \vspace{1pt} \vspace{1pt} \vspace{1pt} \vspace{1pt}The semicubic \ parabola has no real usefulness. It was the result of a problem introduced by Gilfried Wilhelm Leibniq. Leibniz for some reason wanted the path of a particle Descending due to gravity,falling equal vertical distance in equal time intervals with initial velocity not equal to zero.This problem caught the interest of a (Dutch) scientist Christian Huygens who gave the solution as a semicubic parabola. \vspace{1pt} \vspace{1pt} \textbf{The semicubical parabola}(the predecessor of the semicubic parabola) \vspace{1pt}This curve is the first curve to be rectified absolutely.John Wallis was given the credit for naming it and according to him William Neil(1637-1670) was the first to accomplish rectifying it .Although P.Fermat of France and Van Heuraet of Holland were also noted to be working on this at the same time and did accomplish the rectification independently. \vspace{1pt} \[ ay^{2}=x^{2} \] \vspace{1pt}The general equation has a cusp at the origin and two coincidental tangents with the equation $y^{2}$ $=0$. An intersecting property of the semicubical parabola is that it can not be streched.That is,the curve $y=bt^{2},x=at^{3}$ is equal to the curve $% y=(b^{3}/a^{2})t^{2},x=(b^{3}/a^{2})t^{2}$ \textbf{Example. }Graph $y=bt^{2},x=at^{3}$ when $a=2$ and $b=3.$ \textbf{Solution. }Substituting $a=2$ and $b=3$ into $y=bt^{2}$ and $% x=at^{3} $ yields \begin{eqnarray*} y &=&3t^{2} \\ x &=&2t^{3} \end{eqnarray*} \FRAME{dtbpF}{263.8125pt}{263.8125pt}{0pt}{}{}{Figure }{\special{language "Scientific Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display "USEDEF";valid_file "T";width 263.8125pt;height 263.8125pt;depth 0pt;original-width 65.5pt;original-height 65.5pt;cropleft "0";croptop "1";cropright "1";cropbottom "0";tempfilename 'spcf2.wmf';tempfile-properties "XNPR";}} Solution:Substituting $a=3$ and $b=2$ yields \begin{eqnarray*} y &=&(27/4)t^{2} \\ x &=&(27/4)t^{3} \end{eqnarray*} \FRAME{dtbpF}{261pt}{241.375pt}{0pt}{}{}{Figure }{\special{language "Scientific Word";type "GRAPHIC";display "USEDEF";valid_file "T";width 261pt;height 241.375pt;depth 0pt;original-width 59.5pt;original-height 59.5pt;cropleft "0";croptop "1";cropright "1";cropbottom "0";tempfilename 'spcf3.wmf';tempfile-properties "XNPR";}} \begin{center} \vspace{1pt} \end{center} If these axis were similar they would appear \ to be the same.The semicubic parabola has the same traits as the semicubical parabola. \begin{center} \vspace{1pt} \textbf{The semicubic parabola} \vspace{1pt}System of parametric equations \vspace{1pt}$x=t^{3}$ \vspace{1pt}$y=t^{2}$ \vspace{1pt}-$\infty