I’m amused by your subdivision of space note, given Figure 1 :). There are clearly some problems and transforms that are better on N-spheres (N-balls?) and others on rectangles / N-cubes. Do you have a deeper intuition for which? (The x^2 example was simple and cute).
Years ago [1], we applied interval arithmetic to tracing groups of rays (and compared to geometric bounding via planes/frustums). I’d be curious to think through an equivalent with your midpoint ball arithmetic, but I feel like it would “need” to be parameterized as a ball of origins (easy) and then something else for the cone of directions —- maybe theta/phi clusters or “cluster of points on the unit sphere” (but converting back and forth is more expensive than the gains).
Comments
Great write up!
I’m amused by your subdivision of space note, given Figure 1 :). There are clearly some problems and transforms that are better on N-spheres (N-balls?) and others on rectangles / N-cubes. Do you have a deeper intuition for which? (The x^2 example was simple and cute).
Years ago [1], we applied interval arithmetic to tracing groups of rays (and compared to geometric bounding via planes/frustums). I’d be curious to think through an equivalent with your midpoint ball arithmetic, but I feel like it would “need” to be parameterized as a ball of origins (easy) and then something else for the cone of directions —- maybe theta/phi clusters or “cluster of points on the unit sphere” (but converting back and forth is more expensive than the gains).
[1] http://graphics.stanford.edu/~boulos/papers/ia.pdf