4 ms·

Neat site. Unless I missed it, I didn't see a discussion of blossoms, which I've found to be a very nice tool when dealing with Bezier curves. There's a great

by kxyvr 11y ago

Neat site. Unless I missed it, I didn't see a discussion of blossoms, which I've found to be a very nice tool when dealing with Bezier curves. There's a great paper called, "Blossoming : a connect-the-dots approach to splines" by Ramshaw (http://www.hpl.hp.com/techreports/Compaq-DEC/SRC-RR-19.pdf http://www.hpl.hp.com/techreports/Compaq-DEC/SRC-RR-19.pdf) that discusses these ideas. I've also gained a lot of value out of Goldman's book, "Pyramid Algorithms," (https://books.google.com/books?id=xFuhIl1rb2sC https://books.google.com/books?id=xFuhIl1rb2sC). Finally, it's not a great book, but Prautzsch, Boehm, and Paluszny's book, "Bezier and B-Spline Techniques," (https://www.springer.com/us/book/9783540437611 https://www.springer.com/us/book/9783540437611) has the most compact and correct description of triangle Bernstein-Bezier patches that I've seen, so I refer to it when working in higher dimensions.

TheRealPomax 11y ago
Sure, but let's be fair: blossoms are "Bezier curves are nice but what if we did something completely different", which isn't really in scope =)
- jacobolus 11y ago
  That’s not a fair summary. Blossoms are the same thing, but with a “cleaner” but more abstract mathematical description. They make certain proofs easier, but require a bit more mathematical sophistication for the reader. Personally I like Ramshaw’s later whitepaper (“On multiplying points: the paired algebras of forms and sites”) even better, I found it yielded some neat insights, but it’s been mostly ignored in the literature. Probably because it’s even more abstract than the blossoming paper, and takes significant amounts of concentrated effort to digest for anyone who isn’t a professional mathematician: http://www.hpl.hp.com/techreports/Compaq-DEC/SRC-RR-169.pdf http://www.hpl.hp.com/techreports/Compaq-DEC/SRC-RR-169.pdf
  - TheRealPomax 11y ago
    "completely" was perhaps a poor choice of word, but insofar as Bezier theory is concerned, blossoms do depart from it quite a bit. It's a great topic (and your links are excellent!) but not something that would likely find its way into the Primer any time soon (although that said, it would be nice to add a "further exploration" section with links to material such as this, so I just filed https://github.com/Pomax/BezierInfo-2/issues/26 https://github.com/Pomax/BezierInfo-2/issues/26 and hopefully I can get to that soon)
  - kxyvr 11y ago
    Thanks for the link to Ramshaw's other paper. I'd not seen that before and I think it'll be very helpful for some work that I occasionally toy with. As a side note to the parent comment, things like blossoms, polar forms, and sites aren't really introductory material, but it's something that I've found incredibly useful and a tool that I wish more people new about. Beyond graphics, Bezier surfaces have an extremely rich structure of properties that can be exploited and, for me, blossoms were the key to understanding that. For example, by using blossoms, I can force the intersection between two different triangular Bernstein-Bezier patches to be smooth and differentiable up to p-2 derivatives where p is the polynomial order. And, this can be done with linear constraints. See proposition 21.1 in Ramshaw's 1987 paper. There's a litany of other properties as well. Anyway, this is more of a way to say that Bezier curves have important applications in optimal design and other fields outside of computer graphics, but I think that requires using some of the lesser known properties of which blossoms are one tool. Alright, so most of that was for the parent commentator's benefit, but I'll say thanks again for the link to the other Ramshaw paper. I didn't realize there was a continuation of the original blossoming work and I think the results in that newer paper will be incredibly helpful for me.

by kxyvr 11y ago

TheRealPomax 11y ago

Sure, but let's be fair: blossoms are "Bezier curves are nice but what if we did something completely different", which isn't really in scope =)

jacobolus 11y ago
That’s not a fair summary. Blossoms are the same thing, but with a “cleaner” but more abstract mathematical description. They make certain proofs easier, but require a bit more mathematical sophistication for the reader. Personally I like Ramshaw’s later whitepaper (“On multiplying points: the paired algebras of forms and sites”) even better, I found it yielded some neat insights, but it’s been mostly ignored in the literature. Probably because it’s even more abstract than the blossoming paper, and takes significant amounts of concentrated effort to digest for anyone who isn’t a professional mathematician: http://www.hpl.hp.com/techreports/Compaq-DEC/SRC-RR-169.pdf http://www.hpl.hp.com/techreports/Compaq-DEC/SRC-RR-169.pdf
- TheRealPomax 11y ago
  "completely" was perhaps a poor choice of word, but insofar as Bezier theory is concerned, blossoms do depart from it quite a bit. It's a great topic (and your links are excellent!) but not something that would likely find its way into the Primer any time soon (although that said, it would be nice to add a "further exploration" section with links to material such as this, so I just filed https://github.com/Pomax/BezierInfo-2/issues/26 https://github.com/Pomax/BezierInfo-2/issues/26 and hopefully I can get to that soon)
- kxyvr 11y ago
  Thanks for the link to Ramshaw's other paper. I'd not seen that before and I think it'll be very helpful for some work that I occasionally toy with. As a side note to the parent comment, things like blossoms, polar forms, and sites aren't really introductory material, but it's something that I've found incredibly useful and a tool that I wish more people new about. Beyond graphics, Bezier surfaces have an extremely rich structure of properties that can be exploited and, for me, blossoms were the key to understanding that. For example, by using blossoms, I can force the intersection between two different triangular Bernstein-Bezier patches to be smooth and differentiable up to p-2 derivatives where p is the polynomial order. And, this can be done with linear constraints. See proposition 21.1 in Ramshaw's 1987 paper. There's a litany of other properties as well. Anyway, this is more of a way to say that Bezier curves have important applications in optimal design and other fields outside of computer graphics, but I think that requires using some of the lesser known properties of which blossoms are one tool. Alright, so most of that was for the parent commentator's benefit, but I'll say thanks again for the link to the other Ramshaw paper. I didn't realize there was a continuation of the original blossoming work and I think the results in that newer paper will be incredibly helpful for me.