My son has been looking at space curves parametrized as $$ x\mapsto (\cos(ax), \sin(bx), \sin(cx)) $$ for various integer triples. When $(a,b,c)$ involve few, but distinct, primes, the image seems to lie on a surface that is something like an inflated tetrahedron. There is a lot to chew on there but my question is more well-defined.
A slight variant brings what looks like a parametrization of this surface: $$ (x,t) \mapsto (\cos(t+5x), \sin(t+7x), \sin(2x)) $$ Is it easy to determine the equation defining this surface?
I think this affine surface is given by $ u^2 + v^2 +w^2 - 2uvw =1 $. Of course, if we restrict to real points with the given parametrization, we impose $ -1 \le u,v,w \le 1 $ where it is indeed a bulging tetrahedron with vertices $$ (1,1,1), (-1,-1,1), (1,-1,-1),(-1,1,-1) $$
To get the equation, note that $$ \sin(t+7x) = \sin(t+5x+2x) = \sin(t+5x) \cos(2x) + \cos(t+5x) \sin(2x) $$
So $$ (v-uw)^2 = (1-u^2)(1-w^2) \implies u^2 + v^2 +w^2 - 2uvw =1 $$ giving the desired equation. I just haven't checked that any point on this surface with the given bounds has the above parametrization but this shouldn't be hard now.