Given the centers and radii of two overlapping spheres, what is the equation
for the circle where the two spheres intersect?  I need this for a project
I'm working on.

Mark

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Mark Wagner wrote:
> 
> Given the centers and radii of two overlapping spheres, what is the equation
> for the circle where the two spheres intersect?  I need this for a project
> I'm working on.
> 
> Mark

Perhaps the Sphere FAQ can answer your question for you -
http://www.math.niu.edu/~rusin/papers/known-math/index/spheres.html

Or the math and geometry links at povray.org -
http://www.povray.org/links/3D_Resources/Math_and_Geometry_Resources_and_Information/

-- 
Ken Tyler -  1200+ Povray, Graphics, 3D Rendering, and Raytracing Links:
http://home.pacbell.net/tylereng/index.html http://www.povray.org/links/

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Mark Wagner wrote:

> Given the centers and radii of two overlapping spheres, what is the equation
> for the circle where the two spheres intersect?  I need this for a project
> I'm working on.
>
> Mark

What form do you need this equation in?

If r1 and r2 are the radii of the two spheres and d is the distance between
their centers, then the radius of the circle of their intersection should be:

sqrt(-(d+r1+r2)*(d-r1+r2)*(d+r1-r2)*(d-r1-r2))/(2*d)

This circle will be centered on the line connecting the centers of the spheres
and will be

(d+r1-r2)/2

units away from the center of the first sphere and

(d+r2-r1)/2

units away from the center of the second sphere.  The circle will also be
oriented so that it is orthogonal to the line connecting the centers of the
spheres.

I hope that this is helpful.

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Kevin Wampler wrote in message <3876DA06.B2A9AC10@tapestry.tucson.az.us>...
>Mark Wagner wrote:
>
>> Given the centers and radii of two overlapping spheres, what is the
equation
>> for the circle where the two spheres intersect?  I need this for a
project
>> I'm working on.
>>
>> Mark
>
>What form do you need this equation in?
>
>If r1 and r2 are the radii of the two spheres and d is the distance between
>their centers, then the radius of the circle of their intersection should
be:
>
>sqrt(-(d+r1+r2)*(d-r1+r2)*(d+r1-r2)*(d-r1-r2))/(2*d)
>
>This circle will be centered on the line connecting the centers of the
spheres
>and will be
>
>(d+r1-r2)/2
>
>units away from the center of the first sphere and
>
>(d+r2-r1)/2
>
>units away from the center of the second sphere.  The circle will also be
>oriented so that it is orthogonal to the line connecting the centers of the
>spheres.
>
>I hope that this is helpful.


Thanks!  I should be able to work with this.

Mark

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Sorry, I just realized that I made a stupid mistake when I derived the distance
of the center of the circle from the centers of the spheres.  I'll try to
derive the correct equation in school today and will post it if nobody else
beats me to it.

Mark Wagner wrote:

> Kevin Wampler wrote in message <3876DA06.B2A9AC10@tapestry.tucson.az.us>...
> >Mark Wagner wrote:
> >
> >> Given the centers and radii of two overlapping spheres, what is the
> equation
> >> for the circle where the two spheres intersect?  I need this for a
> project
> >> I'm working on.
> >>
> >> Mark
> >
> >What form do you need this equation in?
> >
> >If r1 and r2 are the radii of the two spheres and d is the distance between
> >their centers, then the radius of the circle of their intersection should
> be:
> >
> >sqrt(-(d+r1+r2)*(d-r1+r2)*(d+r1-r2)*(d-r1-r2))/(2*d)
> >
> >This circle will be centered on the line connecting the centers of the
> spheres
> >and will be
> >
> >(d+r1-r2)/2
> >
> >units away from the center of the first sphere and
> >
> >(d+r2-r1)/2
> >
> >units away from the center of the second sphere.  The circle will also be
> >oriented so that it is orthogonal to the line connecting the centers of the
> >spheres.
> >
> >I hope that this is helpful.
>
> Thanks!  I should be able to work with this.
>
> Mark

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Ok, this should work better.  The distance of the center of the circle from the
center of the first sphere is:

(d^2+r1^2-r2^2)/(2*d)

And the distance from the center of the second sphere is:

(d^2-r1^2+r2^2)/(2*d)

This time I double checked these so they should be correct, I guess that teaches
me to do math when I should be sleeping.  Anyway, sorry about that, I home this
helps.

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