4 dimensional raytracer
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From: Thorsten Kiefer on 16 Jul 2010 06:10 Hi, I'm starting to program a 4-dimensional raytracer. But before I finish it I want to know if this is interesting for anyone, if not, I'll stop it. By the way is this the right newgroup ? I post it here, because the sense of the program is not amazing graphics, but to give a better visual insight into understanding the fourth dimension. It is easy with spheres. I put spheres at (0,0,0,0),(1,0,0,0),(0,1,0,0),(1,1,0,0) and so on. I.e. in the edges of the 4d hypercube. I even implement lighting, as the normal of a hypesphere's suface is just the normalized difference vector of the intersection point and it's middle point. The cross product does not exist in 4d space, so we have a problem computing normals of planes. My idea is to generate a square plane in the xy-plane and set it's normal parallel to the z-axis. Then rotate, scale and translate it in 4d-space until it fits the desired 4d plane. this way we have a 2-dimensional plane in 4d-space with even a normal vector, and now we can implement lighting ! What is that good for ? Imagine 2 simple 3d-scenes made up of triangles. We put these scenes on 2 different positions along the 4th axis. I guess with the right camera position and rotation we only see one scene. And if we move the camera along the 4th axis we can switch between the both scenes. Now another trick is to not use 2-dimensional triangles to make up the scenes but to use tetraeders to connect the both scenes. Adjusting the camera propperly would make the tetraeders look like triangles. Now the clou : rotating the camera around the both scenes in 4d-space would result in some kind of mixture of the both scenes. And that is what i want to see. And I hope people will have a highly educational purpose - i.e. understanding 4 dimensions. Best Regards Thorsten
From: Tim Little on 16 Jul 2010 06:41
On 2010-07-16, Thorsten Kiefer <t.kiefer(a)tokis-edv-service.de> wrote: > The cross product does not exist in 4d space, so we have a problem > computing normals of planes. More precisely, every plane has infinitely many normals. That's not actually a problem, as surfaces of 4D objects are tangent to hyperplanes, not planes. Instead of using triangles to construct approximations to surfaces, you would use tetrahedra. - Tim |