Is there a way to develop a linear transformation which will always send solutions of one hyperboloid to another? (for example the hyperboloids:
$$a^2+b^2-c^2=4$$ and $$d^2+e^2-f^2=9$$
)I know that given a solution to one of them (say $(a,b,c)=(2,1,1)$ we can find another through applying a linear transformation used to develop the Tree of primitive Pythagorean triples. However, I'm wondering if there is a way to transform a solution of one hyperboloid to another. Of course, given a linear transformation building the tree mentioned above, we can apply the inverse, send a fundamental solution of the first hyperboloid to the second and then re-apply the transformation. However, I think this will only work for one specific solution and not arbitrary ones.