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Parametric Curves and Curvature

A curve $in$ the plane can $be$ defined $by$ a pair $of$ parametric equations

$x=f(t)$

$y=g(t)$

$In$ Mathematica, parametric curves vec$(r)(t)=($:$f(t),g(t)$:$)$ can $be$ plotted with the ParametricPlot command.

For example, the curve

$x=cos(2t)cos(t)$

$y=cos(2t)sin(t)$

where the parameter $t$ goes from $0to2,$ can $be$ plotted $by$

ParametricPlot$[\{Cos[2t]Cos[t],$ Cos$[2t]Sin[t]\},\{t,0,2Pi\}]$

$to$ get

For two points $on$ a parametric curve vec$(r)(t),$ say when $t=$a and $t=b,$ the arc length between the points $is$

${\displaystyle \underset{a}{\overset{b}{}}}\sqrt[\phantom{2}]{(f^{\text{'}}(t){)}^2+(g^{\text{'}}(t){)}^2}dt={\displaystyle \underset{a}{\overset{b}{}}}||r^{\text{'}}(t)||dt$

where $||r^{\text{'}}(t)||is$ the magnitude $of$ the tangent vector $r^{\text{'}}(t)=($:$f^{\text{'}}(t),g^{\text{'}}(t)$:$).$

$By$ letting the upper $limitof$ this integral $be$ variable, $we$ define the arc length function for the curve,

starting $ata$ : $s(t)={\displaystyle \underset{a}{\overset{t}{}}}||r^{\text{'}}(u)||du$

$if$ the arc length function has $an$ inverse, then $we$ can define the parameter $tin$ terms $ofs$ and

parameterize the curve $by$ its arc length $as$ vec$(r)(s).$ The Fundamental Theorem $of$ Calculus says that

$s^{\text{'}}(t)=||r^{\text{'}}(t)||$ and the Chain Rule says that $r^{\text{'}}(s)s^{\text{'}}(t)=r^{\text{'}}(t),so||r^{\text{'}}(s)||=1.$ That $is,if$ a curve $is$

parameterized $by$ its arc length then the parameterization moves along the curve with unit speed $(the$

curve $is$ a unit speed curve$).$

For a general parametric curve vec$(r)(t)$ the unit tangent vector $attis$

vec$(T)(t)=\frac{\text{v}\text{e}\text{c}(r{)}^{\text{'}}(t)}{||vec(r{)}^{\text{'}}(t)||}$

For example, the unit tangent vector $att=0\mathrm{.}1$ for the curve above $is$ shown $in$ the following figure:

For a curve parameterized $by$ arc length, $||r^{\text{'}}(s)||=1,so$ vec$(T)(s)=$vec$(r{)}^{\text{'}}(s).$

Note that $aswe$ move along the curve the unit tangent vector will change direction, and that the greater

the curvature, the faster the direction will change, $asin$ this animation.

Given a curve parameterized $by$ arc length, $we$ define the curvature function $(s)as$ the function that

keeps track $of$ the magnitude $of$ change $in$ direction $of$ the unit tangent vector:

$(s)=||vec(T{)}^{\text{'}}(s)||$ Since vec$(T)(s)=$vec$(r{)}^{\text{'}}(s)we$ also have $(s)=||vec(r{)}^{\text{'}\text{'}}(s)||.$

Question

Use ParametricPlot $to$ plot the parametric curve

$x=cos(5t)-cos(t)$

$y=sin(3t)-sin(5t)$

for $0t2.$ Which curve $do$ you get?