is called the radius of curvature. The point (x_{c},y_{c})
with coordinates given by
x_{c} = x-
(x^{¢})^{2}+(y^{¢})^{2
}x^{¢}y^{¢¢}-x^{¢¢}y^{¢}
y^{¢},
y_{c}
= y+
(x^{¢})^{2}+(y^{¢})^{2
}x^{¢}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).
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-cos^{2}t), sin t
cos t / (2-cos^{2}t))