# Curvature

For a plane curve its curvature is given by

The quantity

is called the radius of curvature. The point with coordinates given by

is called the center of curvature. The circle with center radius is called the osculating circle or the circle of curvature. The center of curvature lies on the normal of the curve at the point the line segment joining with is perpendicular to the tangent at

The following procedure draws the line segment joining with the corresponding center of curvature, where , are equally spaced points in the interval in the domain of the ellipse , 3 ,

> restart;

> x:=5*cos(t);

> y:=3*sin(t);

> x1:=diff(x,t);

> y1:=diff(y,t);

> x2:=diff(x1,t);

> y2:=diff(y1,t);

> rr:=(x1^2+y1^2)/(x1*y2-y1*x2);

> xc:=x-rr*y1;

> yc:=y+rr*x1;

> m:=[[x,y],[xc,yc]];

> t:=n*2*Pi/100;

> w:=evalf(m);

>

# Circle of Curvature

To construct an animation of the circle of curvature of the ellipse

the following steps are taken:

> restart;

> with(plots):

> x:=5*cos(t);

> y:=3*sin(t);

> x1:=diff(x,t);

> y1:=diff(y,t);

> x2:=diff(x1,t);

> y2:=diff(y1,t);

> rr:=(x1^2+y1^2)^(3/2)/abs(x1*y2-y1*x2);

> r:=(x1^2+y1^2)/abs(x1*y2-y1*x2);

> xx:=x-r*y1;

> yy:=y+r*x1;

> animate({[5*cos(s),3*sin(s),s=0..2*Pi],[xx+rr*cos(s),yy+rr*sin(s),s=

0..2*Pi]},t=0..2*Pi,color=black,axes=none);

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