7

Summer Session 7
July 25

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. 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:

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 =

((x^¢)²+(y^¢)²)^3/2|x^¢y^¢¢-x^¢¢y^¢|

is called the radius of curvature. The point (x_c,y_c) with coordinates given by

x_c = x-

(x^¢)²+(y^¢)²x^¢y^¢¢-x^¢¢y^¢

y^¢, y_c = y+

(x^¢)²+(y^¢)²x^¢y^¢¢-x^¢¢y^¢

x^¢

is called the center of curvature. The circle with center (x_c,y_c) radius r is called the osculating circle or the circle of curvature. The center of curvature (x_c,y_c) lies on the normal of the curve at the point (x,y): the line segment joining (x,y) with (x_c,y_c) is perpendicular to the tangent at (x,y).