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How to Compute Distance from a Point to Three Points 3-D
Onur Cagirici, Izmir University of Economics
10 years ago
Assume that we have three points (
) in a 3-D space.
Those three points determine a plane.
How do I calculate the distance from a point
to those point,
given only the distances between four points
Sean Clarke, Wolfram Research
10 years ago
There's probably a ton of ways to do this. Are you looking for a clever answer to this by any chance?
A straightforward approach would be to use
the law of cosines
, which will give you all the angles in the triangles from their lengths. If you assume a is at the origin and that b is on the x axis an appropiate distance away from a, then you can use basic trig to find vector coordinates of all the points in that embedding. Once you do this, you can find
the equation of the plane
solve for the distance from point d to that plane
Alternatively, you could first find the volume of the tetrahedron ABCD using the Cayley-Menger determinant (I didn't know about this until I just read it). You can also find the are of the triangle ABC using Heron's formula. The area of a tetrahedron relates these to each other using the distance between d and the triangle ABC:
(Volume of the tetrahedron ABCD) = 1/3 * (distance from d to triangle ABC) * (Area of Triangle ABC)
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