Re:Geometry from Another Universe
by Asooka on Thursday January 20, 2022 @03:18PM
(#30014174)
The w coordinate is there to aid the perspective transform and translations. Because in homogeneous coordinates everything becomes a matrix multiplication. At the end, before rasterisation, the coordinates are normalised by dividing by w. Yes, you can use it for whatever else you want, GPUs are just vector processors, as long as you keep in mind how the rasterisation step will happen.
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Thanks. I'm aware of all those uses. But that strongly suggests that he's constructed a simulation of projective / elliptic 3-space, not spherical 3-space. The image on the screen is a perspective projection thereof.
Additionally:
To clarify my understanding of projective vs elliptic geometry, because I keep conflating them: Projective space is a topological space, while elliptic space is projective space endowed with a certain metric, turning it into a metric space. Additionally, every elliptic isometry is a projective symmetry, while the other way round isn't true. On a purely topological level, there's no difference between elliptic geometry and projective geometry.