Let a Cone of opening parameter and vertex at intersect a Sphere of
Radius centered at
, with the Cone oriented such that its axis does not pass through the
center of the Sphere. Then the equations of the curve of intersection are
Combining (1) and (2) gives
|
(3) |
|
|
|
(4) |
Therefore, and are connected by a complicated Quartic Equation, and , , and by a Quadratic
Equation.
If the Cone-Sphere intersection is on-axis so that a Cone of opening parameter and vertex at
is oriented with its Axis along a radial of the Sphere of radius centered at , then
the equations of the curve of intersection are
Combining (5) and (6) gives
|
(7) |
|
(8) |
|
(9) |
Using the Quadratic Equation gives
So the curve of intersection is planar. Plugging (10) into (5) shows that the curve is actually a Circle,
with Radius given by
|
(11) |
© 1996-9 Eric W. Weisstein
1999-05-26