1.  Curvature

For a plane curve [x(t),y(t)] its curvature is given by
k =  |x¢y¢¢-x¢¢y¢| 
((x¢)2+(y¢)2)3/2		 . 
 
The quantity
r =  1 
k		 =  ((x¢)2+(y¢)2)3/2 
|x¢y¢¢-x¢¢y¢|
 
is called the radius of curvature. The point (xc,yc) with coordinates given by
xc = x- (x¢)2+(y¢)2 
x¢y¢¢-x¢¢y¢		 y¢, yc = y+ (x¢)2+(y¢)2 
x¢y¢¢-x¢¢y¢		 x¢
 
is called the center of curvature. The circle with center (xc,yc) radius r is called the osculating circle or the circle of curvature. The center of curvature (xc,yc) lies on the normal of the curve at the point (x,y): the line segment joining (x,y) with (xc,yc) is perpendicular to the tangent at (x,y).

The following procedure draws the line segment joining (x(tk),y(tk)) with the corresponding center of curvature, where t0, t1,···,t100 are equally spaced points in the interval [0,2p] in the domain of the ellipse [x(t),y(t)] = [5 cos t, 3  sin t], t Î [0,2p].

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,scaling=constrained);

2.  Evolute

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 (x,y) with the corresponding center of curvature (xc,yc) for each of the following curves:

(a) the cycloid:

[x(t),y(t)] = [t+sin t,1+cos t],tÎ [-3p,3p] 
 
(b) the epicycloid:
[x(t),y(t)] = [n cos t-cos nt,n sin t-sin nt],tÎ [0,2p] 
 
for the cases n = 2,3,4,5.



(c) the hypocycloid:

[x(t),y(t)] = [n cos t+cos nt,n sin t-sin nt],tÎ [0,2p] 
 
for the cases n = 2,3,4,5.



(d) the parabola:

y = x2,x Î [-1,1] 
 
(e) y = x3

(f) y3 = x5

(g) y2 = x3

(h) y2 = x5.

3.  Involute

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 (0,0) radius a:

[a(cos t+t sin t), a(sin t-t cos t)], t ³ 0. 
 
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);

4.  Circle of Curvature

To construct an animation of the circle of curvature of the ellipse
[x(t),y(t)] = [5 cos t,3 sin t],t Î [0,2p] 
 
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)/(x1*y2-y1*x2);

xx:=x-r*y1;

yy:=y+r*x1;

animate({[(1-s)*x+s*xx,(1-s)*y+s*yy,s=0..1],[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,scaling=constrained);