Intersection Between Circle and Plane in 3D

Date: 05/03/2006 at 12:25:12
From: Mark
Subject: Intersection between circle and plane

I am trying to find the intersection point(s) between a circle and a 
plane in 3D.  The circle is defined parametrically in the form:

  x = acos(t) + bsin(t) + c
  y = dcos(t) + esin(t) + f
  z = gcos(t) + hsin(t) + i

and the plane is defined in the form: ax + by + cz = d.  Any ideas? 
Do I first need to convert the plane to parametric form?

Thanks.

Date: 05/10/2006 at 08:39:33
From: Doctor George
Subject: Re: Intersection between circle and plane

Hi Mark,

Thanks for writing to Doctor Math.

There are some additional constraints needed on your parameterization
to insure that it is actually a circle.  Here is how I would solve 
this with vectors.

Let C0 be the circle center.  Let U be a vector from C0 to some point
on the circle, and let V be another vector from C0 to a point on the
circle such that U.V = 0.  We can parameterize the circle as

  C = C0 + cos(t) U + sin(t) V

The vectors C0, U and V can be related to your parameterization.

  C0 = (c,f,i)
  U  = (a,d,g)
  V  = (b,e,h)

From this we can see the additional constraints that I mentioned
earlier.  We must have |U| = |V| = radius, and U.V = 0.

Now let P be a point on the plane with normal vector N.  The circle
intersects the plane where

  (C-P).N = 0

After substituting for C you can solve the equation for t, which will
lead to the intersection points.  To solve for t, this article should
be helpful.

  Obliterating Iterating
    http://mathforum.org/library/drmath/view/65138.html 

Does that make sense?  Write again if you need more help.

- Doctor George, The Math Forum
  http://mathforum.org/dr.math/