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It is not the case that cubic Beziers are 3D. There’s a good chance that the font you’re using to read this (especially if you print it out) defines the outline

It is not the case that cubic Beziers are 3D. There’s a good chance that the font you’re using to read this (especially if you print it out) defines the outline of (2D) letters with a sequence of cubic Beziers. Lots of 2D computer graphics uses cubic Beziers because they ~balance the needs of having lots of DOF for expressivity, while not being so complicated that they’re a pain to evaluate. And the control points that define a quadratic Bezier could be in any dimension, including 3, or even 1, which is good for plotting the basis functions. Where you are shapes how you make sense of things. As you learn to see Bézier curves from different viewpoints, the particular physical viewpoint you have now will coexist with others. It’s always that way learning math. That feeling of being disoriented is something to treasure, btw; you won’t always have it. My hunch is that this video https://youtu.be/aVwxzDHniEw https://youtu.be/aVwxzDHniEw comes from someone who’s path through learning Bézier curves was non-traditional, but they do a beautiful job of explaining and illustrating things. It showed me that I have more to learn too.

xeonmc 4y ago
That's not quite what I meant when I said cubic are generally 3D. I'm saying that flat-span cubic Beziers are a tiny subset of possible general-span cubic Beziers. The degrees of freedom pertain to the _embedding dimensions_ (could be ten dimensions for instance, which gives each of the control points 10 DoF), but the _span_ of them can be no more than two (for quadratic) or three (for cubic), because there are only ever that many vectors involved respectively. The linked video is in fact one of the sources I had in mind when referring to "lerps of lerps" style of description, just stretched to half an hour with some auxiliary elaboration on what lerps-of-lerps means. Of course they are all still useful tidbits, but it is still fundamentally the same approach of defining an elephant as the sum of its legs and trunks, albeit with more details about said body parts, but still not saying what the thing _is_ from a more cohesive standpoint. By no means am I asserting "my way is the only way", rather the intent is to point out that the candidate explanations should offer something more than just disembodied _description_ of parts

xeonmc 4y ago

That's not quite what I meant when I said cubic are generally 3D. I'm saying that flat-span cubic Beziers are a tiny subset of possible general-span cubic Beziers. The degrees of freedom pertain to the _embedding dimensions_ (could be ten dimensions for instance, which gives each of the control points 10 DoF), but the _span_ of them can be no more than two (for quadratic) or three (for cubic), because there are only ever that many vectors involved respectively. The linked video is in fact one of the sources I had in mind when referring to "lerps of lerps" style of description, just stretched to half an hour with some auxiliary elaboration on what lerps-of-lerps means. Of course they are all still useful tidbits, but it is still fundamentally the same approach of defining an elephant as the sum of its legs and trunks, albeit with more details about said body parts, but still not saying what the thing _is_ from a more cohesive standpoint. By no means am I asserting "my way is the only way", rather the intent is to point out that the candidate explanations should offer something more than just disembodied _description_ of parts