Mathematical Experiment 6


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
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¢)
x¢y¢¢-x¢¢y¢
y¢,      yc = y+ (x¢)2+(y¢)
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).

Construct an animation displaying the various positions of the osculating circle of the following curves:

[Maple Plot]



The cycloid: (t + sin t, cos t)

The ellipse: (5 cos t, 3 sin t)

The lemniscate of Bernoulli: (cos t  / (2-cos2t), sin t cos t / (2-cos2t))

The spiral: