Determine coordinates of 3D point base on other points

Question

Is there any easy way to determine a coordinates of point that is marked as BLUE SPHERE at image which one is attached into that thread. We have only coordinates of P1,P2 and P3, we dont have a information about plane that contains those points and also we dont have information about normal point to that plane. Is there any option to solve that problem ??

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Answer 1

Consider slant edges are of length $(p,q,r)$ meeting at cuboid corner with known base triangle positioned as a triangular pyramid.

Given sides $(a,b,c)$ of this particular triangular pyramid base are related by the Pythagoras theorem:

$$ p^2+q^2= a^2, q^2+r^2=b^2,r^2+p^2=c^2 \,;$$

Solving these simultaneous equations

$$ p^2= (a^2-b^2+ c^2)/2, q^2 = (a^2+b^2-c^2)/2, r^2=(-a^2+b^2+c^2)/2 ;$$

Also we can find height of the pyramid using Volume and base area $\Delta$ (Brahmagupta/Heron) relation:

$$ \frac13 \Delta h = \frac16 pqr ;\, \rightarrow h= \frac{pqr}{2 \Delta}$$

With slant edge dimensions and height we can find projected slant edge lengths by Pythagoras:

$$ \sqrt{p^2-h^2},\sqrt{q^2-h^2},\sqrt{r^2-h^2}; $$

so that all coordinates of blue sphere center are now found in this configuration.