Lemma 1

Posted by Fried et al. on December 12, 2001 at 14:15:56:

In Reply to: Re: math question posted by Fried on December 11, 2001 at 14:28:33:

: : 3-D geometry
: : if you know the 4 vertices of a tetrahedron, how do you find the center of the insphere?

: The following assumes that the definition of insphere is the largest possible sphere contained within the tetrahedron that touches all four faces.

: If its a regular tetrahedron, then its easy. You take the lines determined by the norm of the 4 faces of the tetrahedron, set them equal to each other, and where they meet, that's the center of the insphere.

: If its not regular, its a bit tougher. Note: My only experience with 3D geometry is Calc 3, so its not too extensive. Second Note: This requires writing a computer program, but its a simple computer program. Use the maximazation of volume formula on a sphere contained in the tetrahedron.
: Volume of a sphere is 4/3(PI)r^3. r is the distance between a typical point inside the tetrahedron and the closest face of the tetrahedron. The r that you get when volume is at its max is going to be the center of the insphere. the reason you need to write a computer program is because the maximazation formula doesn't exactly allow you to do a comparison between the typical point and the face.

: There's probably another way of doing it, but it is unknown to me.

K, after doing some thinking and discussing with Isaac, we came up with what we think is the other simpler way. Take the line formed by the norm of a vertex in the direction of the opposite face. (Isaac, "Everything you didn't want to know about Tetrahedrons and Inspheres Volume 54",pg 54,line 54, 1954) Where this line intersects the face should be where the insphere touches the face. Do this for all four verteces and you'll have the 4 points where the insphere touches the tetrahedron. Derive the equation for the sphere from those 4 points, and then take the center of the sphere.

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