I am having a bit of a complicated maths problem and am hoping someone might help or have a suggestion.
My problem concerns finding the location of a point P in 3D. From this point there are three lines to point A,B and C, respectively lines pa, pb and pc. Given are:
- The locations (x,y,z) of points A,B and C
- The distances between A-B, B-C and C-A: d1, d2 and d3
- The angles between pa-pb, pb-pc and pc-pa: alpha_1, alpha_2, alpha_3
I have made a schematic overview here: https://imgur.com/FfZMpMS
I am hoping any one has a suggestion how to find point P from this information, or perhaps even that sweet analytical solution.
I already have a start, but it seems more complicated then necessary. First find a solution for two points in 3D.
In 2D you can find a circle with center C, for point P if you have a distance d1 and angle_1. This image shows the result: https://imgur.com/iyEGR9n
If you expand this problem to 3D, you have to rotate it around the line of d1. The circle center C now forms a new circle with center C2 in the middle of the line of d2. The results is shown here: https://imgur.com/Ux5wo6a
When the 2D circle with center C is rotated around line d1, a torus is found. On this torus lie all points P for which angle_1 and distance d1 are constant : https://imgur.com/BAZIZMF
Now, if you have more points and angles, you have to find an intersection of toruses, which is really not nice.
That's how far I've gotten. Please let me know if you have a suggestion.
A bit of side note, in reality I have 6-10 distances and angles, but I would expect that for 3 points the solution is easier.
Regards, Matlab M.