\noindent \thepage\hfill \QTR{bf}{A Scientific Report}} %} %TCIDATA{OddPages= %H=36,\PARA{038
\QTR{bf}{A Scientific Report}\hfill \thepage} %} %TCIDATA{FirstPage= %F=36,\PARA{035
\thepage} %} \input{tcilatex} \begin{document} \section{The Conchoid of Nicomedes} \begin{center} \textbf{Athanasios Thomaides} College of the Redwoods February 9, 1998 \FRAME{dtbpF}{2.2719in}{2.7371in}{0pt}{}{}{conchoid1.gif}{\special{language "Scientific Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display "USEDEF";valid_file "F";width 2.2719in;height 2.7371in;depth 0pt;original-width 228.875pt;original-height 276.3125pt;cropleft "0";croptop "1";cropright "1";cropbottom "0";filename 'AThomaides/Conchoid1.gif';file-properties "XNPEU";}% } \end{center} \textbf{Abstract: }The purpose of this activity is to gain some insights on the plane curve known as the Conchoid. \subsection{Definition} \noindent \textbf{Conchoid: }Let C be a curve and O a fixed point. \ Let P and P' be points on a line from O to C meeting it at Q where P'Q=QP=k, where k is a given constant. \ If C is a circle and O is on C then the conchoid is a limacon, while in the special case that k is the diameter of C, then the conchoid is a cardiod. \begin{center} \FRAME{dtbpF}{2.0124in}{1.6155in}{0pt}{}{}{firemountain.gif}{\special% {language "Scientific Word";type "GRAPHIC";display "USEDEF";valid_file "F";width 2.0124in;height 1.6155in;depth 0pt;original-width 218.3125pt;original-height 201.75pt;cropleft "0";croptop "1";cropright "1";cropbottom "0";filename 'AThomaides/FireMountain.gif';file-properties "XNPEU";}% }\FRAME{dtbpF}{2.0427in}{2.2321in}{0pt}{}{}{limaconofpascalcon.gif}{\special% {language "Scientific Word";type "GRAPHIC";display "USEDEF";valid_file "F";width 2.0427in;height 2.2321in;depth 0pt;original-width 191.1875pt;original-height 214.5625pt;cropleft "0";croptop "1";cropright "1";cropbottom "0";filename 'AThomaides/limaconOfPascalCon.gif';file-properties "XNPEU";}% } \end{center} \subsection{History} The name Conchoid means 'shell form' and was invented by the Greek Nicomedes The Conchoid was used in the construction of ancient buildings. The vertical section of columns was made in the shape of the loop of the conchoid. \ Some of the applications of this curve was to solve the problems of cube duplication and angle trisection. In the 17th century, Newton said that the Conchoid ought to be a 'standard' curve. \ \subsection{Description} $\circ $ Given a curve C and a fixed point O, draw a line passing O and any point P on the curve C. $\circ $ On this line, mark points Q1 andQ2 such that distance \lbrack P,Q1\rbrack ==distance\lbrack P,Q2\rbrack ==k. $\circ $ Repeat this for other point P on the curve. $\circ $ The locus of Q1 and Q2 is the conchoid of the curve with respect to O and offset k.\FRAME{dtbpF}{3.3261in}{1.5567in}{0pt}{}{}{conchoidsingen.gif% }{\special{language "Scientific Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display "USEDEF";valid_file "F";width 3.3261in;height 1.5567in;depth 0pt;original-width 237.125pt;original-height 109.9375pt;cropleft "0";croptop "1";cropright "1";cropbottom "0";filename 'AThomaides/conchoidSinGen.gif';file-properties "XNPEU";}% }The Conchoid of Nicodemes is the conchoid of a straight line with respect to a point not on the line. \ There are three different cases where k (the fixed distance) can be less than, equal to or greater than the distance from the point to the line.\FRAME{dtbpF}{5.6204in}{1.7772in}{0pt}{}{}{conc.gif}{% \special{language "Scientific Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display "USEDEF";valid_file "F";width 5.6204in;height 1.7772in;depth 0pt;original-width 402pt;original-height 125.75pt;cropleft "0";croptop "1";cropright "1";cropbottom "0";filename 'AThomaides/conc.gif';file-properties "XNPEU";}} \vspace{1pt} \FRAME{dtbpF}{289.625pt}{198.0625pt}{0pt}{}{}{Figure }{\special{language "Scientific Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display "USEDEF";valid_file "T";width 289.625pt;height 198.0625pt;depth 0pt;original-width 286.0625pt;original-height 195pt;cropleft "0.002624";croptop "1";cropright "1.002624";cropbottom "0";tempfilename 'EO4GWF00.wmf';tempfile-properties "XPR";}} Parametrization of this curve was not a very hard thing to do. \ As you can see from the figure above the distance between O and the origin is $L$ and the fixed distance on either side of the line is $k$. \ When we extend a line from point O and through the origin $y$ $=0$ and the $x$ $=$ $k$. \ As we move this line segment up and down the vertical line, point M traces out a conchoid. \ This is the way I went figured out the parametric equations for the Conchoid of Nicomedes. \begin{center} $\cos (w)=\frac{x}{k}\Longrightarrow $ \ \vspace{1pt}$x=\pm k\cos (w)$ \vspace{1pt} \vspace{1pt}$y=L\tan (w)+k\sin (w)$ \vspace{1pt} \vspace{1pt} If $k>L$, there is a node at the origin; if \ $k=L$, there is a cusp; if \ $k