Fast calculation of circumcentre coordinates and triangle altitude

Question

I have three points in 3D space. I'd like to use these to construct a circle (it is always possible to do so, see Circumscribed_circle. For two of the points, I'd also like to find the triangle altitude formed by these points and the coordinate origin.

The calculations need to be fast - the equations I've found so far involve lots of square roots, so I was wondering if anyone knew of suitable algorithms or ways to restate the problem.

Thanks in advance!

score 2 · Answer 1 · edited May 23 '17 at 11:44

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You can add a 4 dimension and lift the points to a paraboloid. Then get the convex hull and project back to the surface. This works with a delaunay triangulation. Here is a similar question but for 2 dimension: How does this code for delaunay triangulation work?.

edited May 23 '17 at 11:44

Community

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answered Sep 13 '13 at 10:42

Micromega

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Do you have any more details or link to an example? – invisiblerhino Sep 13 '13 at 10:49
I only know how it works with 2 dimension: http://stackoverflow.com/questions/5825089/how-does-this-code-for-delaunay-triangulation-work?rq=1. – Micromega Sep 13 '13 at 10:56

score 0 · Answer 2 · answered Dec 17 '15 at 04:00

For the circumcircle, it can be done using this formula

m = a + (|c-a|^2 [(b-a)x(c-a)]x(b-a) + |b-a|^2 (c-a)x[(b-a)x(c-a)]) / (2 | (b-a)x(c-a) |^2)

Where a, b, and c are points in R^3, and m is the center of the circumcircle you are looking for. No sqrt needed.

Source: https://www.ics.uci.edu/~eppstein/junkyard/circumcenter.html

Fast calculation of circumcentre coordinates and triangle altitude

2 Answers2