Intersections with a Sphere

Marc A. Murison

Astronomical Applications Department
U.S. Naval Observatory
Washington, DC

murisonATusno.navy.mil
http://www.alpheratz.net/murison/

July 1, 1998

A Plane and a Sphere

Define a plane and a sphere:

Plot them, along with the projections of their intersection.

The Projection of the Intersection is an Ellipse

One Ellipse

The explicit solutions for ( p , q ), parametrized by r , of the intersection curve are

p and q are interchangable. Plot both sets:

This is, of course, just the ellipse that is the plane-sphere intersection projected onto the ( p , q ) plane.

Another Ellipse

Let's look at the intersection parametrized by ( p , q ).

which is an equation for an ellipse. Plotting this equation implicitly produces the same ellipse as above.

Ellipses the Pun ...

We can remove the cross term with a rotation:

We can easily put the equation into standard form:

An Interesting Surface

Recall the sphere and plane equations:

Now, consider

This is a quadratic surface whose appearance is

We've doubled our plane. Substitute the sphere equation into this to get

Another way to view this is

So we have a hyperboloid. A plot of this equation is

Let's put this in standard form. First, rotate in the ( q , r ) plane.

Next, rotate in the ( p , v ) plane.

Much better. Plot this: