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(OP here) This is taken directly from the paper's introduction, which admittedly uses the more specific terminology of "1-Lipschitz signed distance bounds". T

(OP here) This is taken directly from the paper's introduction, which admittedly uses the more specific terminology of "1-Lipschitz signed distance bounds". The paper cites the original Hart '96 paper on sphere tracing; quoth Hart, "a function is Lipschitz if and only if the magnitude of its derivative remains bounded". https://graphics.stanford.edu/courses/cs348b-20-spring-content/uploads/hart.pdf https://graphics.stanford.edu/courses/cs348b-20-spring-conte... I wonder if there's a terminology schism here between computer graphics and numerical analysis folks.

constantcrying 1y ago
>I wonder if there's a terminology schism here between computer graphics and numerical analysis folks. The first group just pretends every function has a derivative (even when it clearly does not), the other doesn't. The linked Wikipedia article gets it exactly right, I do not know why you would link to something which straight up says your definition is incorrect. There is no point in talking about Lipschitz continuity when assuming that there is a derivative, you assume that it is Lipschitz because it is a weaker assumption. The key reason Lipschitz continuity is interesting because it allows you to talk about functions without a derivative, almost like they have one. It is the actual thing which makes any of this work.
sfpotter 1y ago
The concept of a Lipschitz function comes from mathematical analysis; neither computer graphics nor numerical analysis. It's straightforward to find the definition of a Lipschitz function online, and it is not in terms of its derivative. If a function is differentiable, then your quote applies; but again, it isn't the definition of a Lipschitz function. I'd say this is a little pedantic, save for the fact that your function of interest (an SDF) isn't a differentiable function! It has big, crucially important subset of points (the caustic sets) where it fails to be differentiable.

constantcrying 1y ago

>I wonder if there's a terminology schism here between computer graphics and numerical analysis folks. The first group just pretends every function has a derivative (even when it clearly does not), the other doesn't. The linked Wikipedia article gets it exactly right, I do not know why you would link to something which straight up says your definition is incorrect. There is no point in talking about Lipschitz continuity when assuming that there is a derivative, you assume that it is Lipschitz because it is a weaker assumption. The key reason Lipschitz continuity is interesting because it allows you to talk about functions without a derivative, almost like they have one. It is the actual thing which makes any of this work.

sfpotter 1y ago

The concept of a Lipschitz function comes from mathematical analysis; neither computer graphics nor numerical analysis. It's straightforward to find the definition of a Lipschitz function online, and it is not in terms of its derivative. If a function is differentiable, then your quote applies; but again, it isn't the definition of a Lipschitz function. I'd say this is a little pedantic, save for the fact that your function of interest (an SDF) isn't a differentiable function! It has big, crucially important subset of points (the caustic sets) where it fails to be differentiable.