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))
![[Maple Plot]](astroid1.gif)
