Question

Maybe this fits in math.stackexchange.com, but since I am programing it in OpenGL, I shall ask it here.

I had the idea of an spaceship game where the world is confined in the surface of an 4-D hypersphere (also called a 3-sphere). Thus, in seeing it from inside, it would look like a 3-D world, but by navigating in every direction, I would never leave the limited volume of the 3-sphere.

To represent the 3-shpere as a “flat” 3-D space, I use an stereographic projection, which is very simple to implement as a GLSL shader, just need to divide the input vector by one minus its w coordinate.

To represent the vertices of the objects I am using normalized 4d vectors, such that x²+y²+z²+w²=1, thus keeping them inside the 3-sphere.

The first problem to solve was rotation. But I soon figured out that ordinary 3d rotation matrices would suffice to rotate the world around the viewer in the 3d projection, since it does not mess up with the w coordinate (pretty much like rotating a sphere around the z-axis would also rotate its stereographic projection).

Then I figured out that rotating along w-axis would be equivalent of translation inside the 3d projection (just not commutative, as ordinary 3d translations on “flat” spaces), then I could translate along the axis by using a simple around axis rotation matrix (x’, y’) = (x * cos a – y * sin a, x * sin a + y * cos a), but variating w along with another axis.

This is so far where I got, and I could not figure out how to navigate forward, based on the position the viewer is facing from the projection. I can apply the inverse transform to derive the normalized 4-D vector (called F) the viewer is facing in the hypersphere coordinates, but I don’t know how to navigate in that direction by using a 4×4 matrix (what is optimal in OpenGL). I could think on a hackish solution: for every vertex V, do V’ = normalize(d*F + V), where d is the distance moved forward (in some strange unit I can not exactly precise). This way only works for small values of d, there is no direct correlation between d and the angle variation.

Thus the question is: how to move forward (using a 4×4 matrix transform) being in the surface of a 4-D hypersphere?

Answer 1

I did ask the same question on math.stackexchange.com, because the question is too math related and I could possibly get a better answer there. Look here if you are interested in the original answer I’ve actually used.

Being in 4D, there is not a single vector perpendicular to any rotation, but actually a whole plane of perpendicular vectors, and that was confusing me.

Also, in OpenGL, I soon figured out that I could simplify the computation of the given solution, because I could always assume I am at the 4D projection origin (0,0,0,-1), always facing (0,0,-1,0), then translate a given angle along z and w, and multiply it to the already accumulated GL_MODELVIEW_MATRIX from the previous frame. So I did MODELVIEW <= M x MODELVIEW instead of MODELVIEW <= MODELVIEW x M I would get if I simply called glMultMatrix(M).