I'm trying to run a simulation in Python of a linkage system. The problem I'm working with returns results that don't make sense (like one of the linkages changes size). I've reduced the issue I'm confused about to the following.
Suppose I have the following diagram:
Suppose I am given $\mathbf{c}$, $\lVert \mathbf{a}\rVert$, $\lVert \mathbf{b}\rVert$.
Are $\mathbf{a}$ and $\mathbf{b}$ unique? If so, how do I find $\mathbf{a}$ and $\mathbf{b}$? If they are not unique, what other information would I need to make them unique (e.g. the angle between $\mathbf{a}$ and $\mathbf{b}$)?
I feel like the SSS theorem I learned about waaaaay back in junior high might be relevant. By SSS, I think if two triangles have sides that are proportional to each other, then the angles are the same. Could this be use to show $\mathbf{a}$ and $\mathbf{b}$ are unique? If so, how would I then derive $\mathbf{a}$ and $\mathbf{b}$?
SSS determines the shape of the triangle, but the direction can vary.
You may think of $\vec{c}$ as an unoriented segment $AB$ of length $||c||$. Given $||a||$ and $||b||$, the possible locus of $a$ and $b$ are two circles centered at $A$ and $B$ respectively, with radii of $||a||$ and $||b||$. Geometrically, two circles in this situation intersect at exactly two points symmetric about the segment $AB$ according to the triangle inequality.
This does not break the SSS theorem, for the shape of these two triangle are actually the same, but the orientations of them are different.