I haven’t ever seen anyone try to implement such a UI in production software though, only my own little experimental doodles. for interoperating 1:1 with systems where the main drawing primitive is a cubic Bézier curve), it works a bit better from a UI perspective to specify the (x, y) locations of on-curve points at t = in each segment and then convert to Bézier basis by multiplying by the appropriate matrix. Personally, I think even for using parametric cubic polynomial segments (i.e. I’ve never been the biggest fan of UIs for manipulating either one. Some computer animation software uses subdivision surfaces instead. Lots of CAD software uses b-splines, specifically “NURBS”, as the main curve primitive. This means that the arclength can be precisely computed in closed form, as can mathematically precise offset curves, which makes them great for defining CAM cut paths. the hypotenuse √(x²+y²) is itself a polynomial. These are curves such that the derivative in the X and Y direction are represented by 'Pythagorean' polynomials, i.e. Īnother type of curve that is very interesting for some applications is a Pythagorean-hodograph curve. He not only proposes a pretty neat type of curve to use, which affords some nice UIs, but also does a great job detailing the history of splines, including other types of smooth “minimum energy” curves, IKARUS splines, Hobby splines (used by METAFONT), etc. Look up Raph Levien’s Ph.D thesis about “spiro” curves (a.k.a. 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. 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.Īlright, 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. There's a litany of other properties as well. See proposition 21.1 in Ramshaw's 1987 paper. And, this can be done with linear constraints. 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. 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. I'd not seen that before and I think it'll be very helpful for some work that I occasionally toy with.Īs 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. Thanks for the link to Ramshaw's other paper.

Bezier curve wiki