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Sneftel

Lowest point on a circle

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Alright, I'm sure this is easy, but I'm having a bad brain day. Given a unit circle at the origin on the XY plane, which is then transformed by an arbitrary orthonormal transformation M, what's the simplest way to find the lowest (minimum Z) point on the circle?

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                          ^ Z
M.z \ |
\ |
\ | __/
\ | __/
\|__/
<-------------+-----------> XY
__/
__/
C /
Oops. Here's what I really meant: -sqrt( 1 - M.zz2 ). My thinking is that the distance from C to XY plane is the same as the distance from M.z to Z (since it's a unit circle).

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It looks like the question has already been answered, but I'll go ahead and give the support mapping for a circle in 3D here (this will probably just confirm what John posted above):
S(v) = C + r(v-(v.n)n)\|v-(v.n)n|

Where:

C is the circle center
n is the normal to the circle plane
r is the circle radius
That's off the top of my head, but I think it's right. You could also of course develop the special-case version for one of the cardinal planes (xy in your example), and then use the support mapping affine transform theorem (as described in Gino's book) to find the support point given a particular transform. However, the above is probably simpler.

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