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/
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