\hfill \thepage} %} \input{tcilatex} \begin{document} \section{Special plane Curves Report:} \section{Equiangular Spiral} \vspace{1pt} \begin{center} Aaron Ambrosini \vspace{1pt} Math 50C: \ Calculus III \end{center} \vspace{1pt} \subparagraph{\qquad Abstract: \ Description of a plane curve; \ it's system of equations; \ the curve's graphs and images; description of curve's usefulness.} \vspace{1pt} \subsection{Description} \vspace{1pt} \qquad An equiangular spiral, also known as a logorithmic spiral or logistique, describes a famous series of sprials in logarithmic (exponential) notation. \ Each equiangular spiral can be written in a Cartesian, paremetric, and polar equation. \ There are two distiguishing aspects of equiangular spirals: \begin{itemize} \item As the angle $\theta $ on the plane increases, so does the radius for all values of r($\theta $). \item The equiangular spiral cuts all radii at a constant angle. \ What that means is this: \ if the angle between the vector connecting a point \textit{P} on r($\theta $) to the spiral's center and the line tangent to the line at that point will be the same angle for any other value \textit{P}. \end{itemize} \vspace{1pt}\FRAME{dtbpF}{4.3742in}{3.243in}{0pt}{}{}{% equiangularspiraltitleg.gif}{\special{language "Scientific Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display "USEDEF";valid_file "F";width 4.3742in;height 3.243in;depth 0pt;original-width 312.4375pt;original-height 231.125pt;cropleft "0";croptop "1";cropright "1";cropbottom "0";filename 'equiangularSpiralTitleG.gif';file-properties "XNPEU";}} \subsection{Equations} \vspace{1pt} \textbf{Paremetric}: \ t is variable; \ $\alpha $\ is the constant angle, where\ $0\prec \alpha \prec \pi /2$. \vspace{1pt} \[ x=e^{t\ast cot(\alpha )}cost \] \[ y=e^{t\ast cot(\alpha )}sint \] \vspace{1pt} In \textbf{Cartesian} form the equation is: \[ x^{2}+y^{2}=e^{tan^{-1}(y/x)cot(\alpha )} \] And the \textbf{polar} equation is: \[ r(\theta )=e^{\theta cot(\alpha )} \] \vspace{1pt} \subsection{\protect\vspace{1pt}The graph of the equiangular spiral} \qquad An equiangular spiral, in parametric form, will have an initial position of (1,0) at t = 0. \ If a line was drawn from the center of the spiral to any point on the graph, a line which is called the radial vector, the angle formed by the radial vector and the tangent line is constant: \ the value entered for $\alpha $. \vspace{1pt} \begin{center} \textbf{Some examples of equiangular spirals with different angle values:} \end{center} \FRAME{dtbpFU}{1.5039in}{1.5039in}{0pt}{\Qcb{$\alpha $ = 40$^{\circ }$}}{}{% equiangularspiralf1.gif}{\special{language "Scientific Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display "USEDEF";valid_file "F";width 1.5039in;height 1.5039in;depth 0pt;original-width 105.375pt;original-height 105.375pt;cropleft "0";croptop "1";cropright "1";cropbottom "0";filename 'equiangularSpiralF1.gif';file-properties "XNPEU";}} \vspace{1pt} \FRAME{dtbpFU}{1.5039in}{1.5039in}{0pt}{\Qcb{$\alpha $ = 60$^{\circ }$}}{}{% equiangularspiralf3.gif}{\special{language "Scientific Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display "USEDEF";valid_file "F";width 1.5039in;height 1.5039in;depth 0pt;original-width 105.375pt;original-height 105.375pt;cropleft "0";croptop "1";cropright "1";cropbottom "0";filename 'equiangularSpiralF3.gif';file-properties "XNPEU";}} \vspace{1pt}\FRAME{dtbpFU}{1.4935in}{1.4935in}{0pt}{\Qcb{$\alpha $ = 70$% ^{\circ }$}}{}{equiangularspiralf4.gif}{\special{language "Scientific Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display "USEDEF";valid_file "F";width 1.4935in;height 1.4935in;depth 0pt;original-width 105.375pt;original-height 105.375pt;cropleft "0";croptop "1";cropright "1";cropbottom "0";filename 'equiangularSpiralF4.gif';file-properties "XNPEU";}} \vspace{1pt} \FRAME{dtbpFU}{1.4935in}{1.4935in}{0pt}{\Qcb{$\alpha $ = 85$^{\circ }$}}{}{% equiangularspiralf6.gif}{\special{language "Scientific Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display "USEDEF";valid_file "F";width 1.4935in;height 1.4935in;depth 0pt;original-width 105.375pt;original-height 105.375pt;cropleft "0";croptop "1";cropright "1";cropbottom "0";filename 'equiangularSpiralF6.gif';file-properties "XNPEU";}} \vspace{1pt} \qquad The Matlab-generated sample graph on the next page has a given value of 80$^{\circ }$ for $\alpha $\ . \subsection{History and Usefulness of Curve} \qquad The equiangular spiral was discovered in 1638 by Descartes. \ Later that century, James (Jacob) Bernoulli also studied the curve for its self-reproducing properties. \qquad The curve has a shape which is analogous to shapes found in nature; some examples are the arrangement of seeds in a sunflower, the three-dimensional spiral shape of seashells, and the formation of pine cones . \ It appears that the gestative and/or physical development of these organisms corresponds to the outward curve on the graph of an equiangular spiral graph. \vspace{1pt} \subsection{\protect\vspace{1pt}\textbf{Bibliography}} \vspace{1pt} \qquad ''Equiangular Spiral''. \ Source: Internet via NETSCAPE. \ Address: \qquad \qquad \qquad \qquad \qquad \qquad \qquad http://www.best.com\symbol{% 126}xah/SpecialPlaneCurv...inangularSpiral\_dir/equiangularSpiral.html \vspace{1pt} \qquad ''Curves and Their Properties''. \ Yates, Robert C. \ Source: \ David Arnold \vspace{1pt} \qquad ''A Book of Curves''. \ Lockwood, E.H. \ Source: \ David Arnold \end{document} %%%%%%%%%%%%%%%%%%%%%% End /document/report1.tex %%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%% Start /document/equiangularSpiralTitleG.gif %%%%%%%%%%%%% GedQxdSX_FpLAXO@@|}w{oygow^ouVosNoqF_o}~_mu~_km~_ie~Og\~OeT~O cL~OaD~^{}\s}Zk}Xc}oVZ}oTR}oRJ}oPB}_Ny|_Lq|_Ji|_Ha|OFX|ODP|OD@|_H@|_N@| oR@|oV@|\@|Oa@|Og@|_k@|_o@|ou@|y@|@\@\~@x|@t{@tz@Py@Px@lv@hu@ ht@Ds@Dr@`p@@p@@pP@pi@py@pJApcApsApLBp\BpmBpFCpVCpoCpCp} C`wC`sCPoCPiC@eCp^CpZC`VC`PCPLC@FC@BC@@cA@gB@KD@KE@OF@oG 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