Parametric Form of Circle Equation

Date: 08/04/2003 at 04:47:30
From: Francis Chong
Subject: Parametric form of circle equation in 3D Space

Dear Dr. Math, 

If I know the center and radius of a circle, and three points on the 
circle, can I find the parametric form of the circle equation in 3D 
space?

Date: 08/04/2003 at 12:51:03
From: Doctor George
Subject: Re: Parametric form of circle equation in 3D Space

Hi Francis,

To get a parametric equation follow this procedure.
1. Let N be a unit normal vector for the plane.
2. Let C be the circle center, and let R be the radius.
3. Let U be a unit vector from C toward a point on the circle.
4. Let V = N x U.
5. Let A be the paramter.
6. A point P is on the circle if...

          P = C + R cos(A) U + R sin(A) V

You can verify this result be checking the distance from P to the 
plane, and the distance from P to C.

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

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