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);

> plot([w$n=1..100],color=black,axes=none);

The **evolute** of a curve is the locus of its center of
curvature. All tangents to the evolute are normal to the given curve. The
evolute is the envelope of normals to the given curve.

Draw the line segments joining with the corresponding center of curvature for each of the following curves:

(a) the cycloid:

(b) the epicycloid:

for the cases

(c) the hypocycloid:

for the cases

(d) the parabola:

(e)

(f)

(g)

(h)

An **involute** of a curve is the path of a point of a string
tautly unwound from the curve. The curve itself is the evolute of its
involute.

Show that the following parametric equation describes the involute of the
circle with center
radius
:

`> restart;`

`> x:=cos(t);`

`> y:=sin(t);`

`> xx:=x+t*sin(t);`

`> yy:=y-t*cos(t);`

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

`> m:=[[x,y],[xx,yy]];`

`> w:=evalf(m);`

`> plot([w$n=1..100],color=black,axes=none);`

>

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);`