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.