In two dimensions, another way of thinking about the central angle theorem is: "Q: Given points A and B, what is the locus of all points C such that the size of angle ACB is a given theta? A: The locus is two arcs of the two circles with centers O and O', where angles AOB and AO'B are size 2*theta, and O and O' are on the perpendicular bisector of AB." This question can certainly be generalized to three dimensions. Probably the most natural way is to ask: "Given points A,B,C, what is the locus of all points D such that the solid angle at D subtended by ABC has measure theta?" The answer is evidently not of the form "All points on some piece of some sphere", but it is certainly *some* surface... --Michael On Thu, Dec 31, 2015 at 1:26 PM, Dan Asimov <asimov@msri.org> wrote:
Just now trying a right circular cone, 45-degrees apart from its axis, whose vertex lies on a unit sphere.
If the axis of the cone is normal to the sphere, it subtends a hemisphere.
But at the other extreme, if the axis of the cone is tangent to the sphere, the cone subtends less than a full hemisphere.
So there can't be a simple relationship.
—Dan
P.S. Everytime I'm reminded of the inscribed angle theorem, I am struck by how counter-intuitive it is. Not so much the factor of 2 per se, but the fact that no matter how a given inscribed angle is situated, it always subtends the same amount of central angle.
On Dec 31, 2015, at 5:45 AM, James Propp <jamespropp@gmail.com> wrote:
Yes, your central angle theorem is the same as my central angle theorem; I was reformulating it in a non-standard (and, I now see, confusing) way that I thought would be likely to generalize.
Maybe I should just ask the question of whether there's ANY formulation of the central angle theorem that extends to solid angles...
On Thursday, December 31, 2015, Dan Asimov <asimov@msri.org> wrote:
On Dec 30, 2015, at 8:56 PM, James Propp <jamespropp@gmail.com <javascript:;>> wrote:
The central angle theorem (in one formulation) tells us that if C is a circle containing the point p and C' is a circle centered at p, then projection from C to C' through p halves all angles.
For some reason I'm having trouble imagining this projection properly.
Is this "central angle theorem" the same as the theorem that says:
An angle inscribed in a circle subtends a central angle that's twice its size
?
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