I am dealing with marker data in 3D space. I have a right triangle made by the three points [A, B, C] where AC = Opposite, BC = Hypotenuse, AB = Adjacent. The objective is to find the coordinates of point C for the given :
- The x,y,z coordinates of the A and B are known
- The sides lengths AB, BC, and AC are also known
- Angle $\theta$ made between AB and BC are known
- The equation of the plane where the triangle lies in $ax+by+cz = d$ format
I would like to know the most optimal method to find the coordinates of point C in 3D space.
Expanding from my comment:
I think this problem hasn't a unique solution. You can construct infinitely many different triangles from the data given. Coordinates for all possible $C$ will form a circle on a plane perpendicular to the one that contains $A$ and $B$. Imagine the coordinates for $A$ and $B$ are $(0,1,0)$ and $(0,0,0)$ respectively, the lengths $AB$, $BC$ and $AC$ are $1$, $\sqrt{2}$ and $1$ and $\theta=45$. Then $(1,1,0)$, $(1,-1,0)$, $(1,0,1)$ and $(1,0,-1)$ (among infinitely many others) would all be valid positions for $C$. Even in $\Bbb R^2$ you'd have two solutions for $C$.