There are some special cases with well-known results. If the radii of both circles are reduced to 0, so that the angle is wedged between two points,
sweeps out part of a circle, since these are the points where the subtended angle of the line segment between the points is constant. The angle would then be an inscribed angle of the circle [1, p. 283]. A circle is, in fact, a special case of a limaçon.
If only one of the radii is reduced to 0, so that there is a point and a circle, and the angle is
, this gives the definition of the pedal curve
of the circle. In general, the pedal curve of a plane curve
and given fixed pedal point
is the locus of all orthogonal projections of the point
onto the tangent lines of the curve
. The pedal curve of a circle is known to be a limaçon [2, p. 163].
The centers of the two circles are placed
units to the left and right of the origin. (In the figure,
is always 2.) Denote the radii of the circles by
be the angle of the wedge (the two green lines) between the circles. The parameter
is introduced so that the tangent point on the left circle is at an angle of
. Using simple geometry, the standard angle for the tangent point on the right circle is
, the figure becomes symmetric, so the intersection point is on the
The coordinates of the point of tangency for the left circle are
and the slope of the tangent line is
Thus, the left green tangent line can be expressed as
Likewise, the tangent point on the right circle is
so the right green tangent line can be expressed by
These green lines intersect at the point
can never be 0 or
, so the denominator is never zero. Thus, this can be scaled by
to remove the denominators. Also,
can be shifted by the constant
can be found such that
Then the parametric equations become
It is already starting to look like the equation of a limaçon. To finish, rotate the coordinate system by letting
Finally, change the parameter by letting
, to produce the equations
which is the equation of a limaçon [3, p. 113].
 D. C. Alexander and G. M. Koeberlein, Elementary Geometry for College Students
, 5th ed., Belmont, CA: Brooks/Cole, 2011.
 J. Edwards, An Elementary Treatise on the Differential Calculus
, 2nd ed., London: Macmillan and Co., 1892.
 J. D. Lawrence, A Catalog of Special Plane Curves
, New York: Dover Publications, 1972.