Summer Session 7
July 25

#### 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).

Construct the line segments joining points of the curve with the corresponding center of curvature for each of the following: the nephroid, the astroid, the deltoid and the cardioid.

Construct an animation displaying the various positions of the osculating circle of the following curves:
• The astroid: (3 cos t + cos 3t, 3 sin t - sin 3t)

• The deltoid: (2 cos t + cos 2t, 2 sin t - sin 2t)

• The cardioid: (2 cos t + cos 2t, 2 sin t - sin 2t)

• The cycloid: (t + sin t, cos t)

• The ellipse: (5 cos t, 3 sin t)

• The "egg": (5*cos(t)-cos(2t), 3*sin(t)-sin(2t)

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

The curve (5 cos(t)2 + sin(2 t), 3 cos(3 t) - sin(4 t))

The spiral: