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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?


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 

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

- Doctor George, The Math Forum 
Associated Topics:
College Higher-Dimensional Geometry

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