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- Hacker News
- Illustrations in the article are indeed precious teaching materialby larodi
- Is it bad that I don't understand this after reading it? I'm not even sure why "lie" he's referring to about bezier curves.by rosstex
- The thing is, the problematic curves often violate the rules put forward as best practices:
- place nodes at extrema (top/bottom, left/right)
- place nodes at points of inflection (middle of an _S_ curve)
- where possible/appropriate observe the "rule of 30" and place off-curve nodes not quote one-third of the way towards the other on-curve node
If one expected (or enforced) those rules, does the problem/math become easier?
by WillAdams - This is very fun and well done, great interactive illustrations, and I love the deep dive into curvature.
I have to wonder - if you want smooth joins, why not use uniform B-splines instead of Bézier curves? With B-splines you don’t have to set constraints or compute curvature, it’s built in. What are the reasons to prefer Bézier?
There’s an old 1985 tech report by Tony DeRose (formerly of Pixar Research) that categorized the different ways to smoothly join curves - "Geometric Continuity: A Parametrization Independent Measure of Continuity for Computer Aided Geometric Design.”
That paper has a simple constraint formula for matching curvature between cubic Béziers, one that has a couple degrees of freedom. It’s the “G2 (curvature continuity)” formula mentioned on Wikipedia here: https://en.wikipedia.org/wiki/Composite_B%C3%A9zier_curve#Sm.... It’d be interesting to know whether that somehow works out to be mathematically equivalent to this post’s technique or not.
by dahart - As someone who has spent a lot of my life editing bezier curves, this is really fantastic.
The one sticking point is instability in the middle of s curves which seems difficult. We're getting better control for every fully convex or non-convex path in exchange for chaos at the boundary between the two. I wonder if incorporating some other error term of the curvature in those regions might help, perhaps guaranteeing that the curvature derivatives are close to continuous whenever the curvatures themselves get too small?
One other thought is that you could use a saturating function of the tangent circle radius so that 0 curvature points don't have to have their control points go to infinity, perhaps making the saturation point a function of the distance to the nearby nodes to keep the scale reasonable.
by moultano - >"The linear interpolations (aka lerps) can be summarized into a single compact formula, e.g. for 4 control points (A,B,C,D):
γ(t)=A⋅(1−t)^3+B⋅3(1−t)^2⋅t+C⋅3(1−t)t^2+D⋅t^3
The rule is simple: descending powers of (1−t), ascending powers of t, with coefficients taken from the n'th row of Pascal's triangle.
I've never seen the connection between the equation for Bezier Curves (more specifically the linear equations of curves with N control points aka "binding points" / "points of stability" / "fixed points" / "immovable points", etc.) and Pascal's Triangle before!
Brilliant!
Great article, too!
- What an amazing resource.
I’ve been building a vector editor that by default draws shapes with smooth curvature and shows the comb. [1] In addition to the four point types mentioned in the article, you get a new “curve” point type. I’ve also been making a font editor with the same drawing capability. [2]
Both are free static web apps that use local storage and have import/export capability for SVG files (and OTF files for the font editor).
by adamschwartz - As another comment mentioned, Raph Levien has a few words to say on the topic of improving Bezier curves.
Simplifying Bézier paths (2023) https://raphlinus.github.io/curves/2023/04/18/bezpath-simpli...
Parallel curves of cubic Béziers (2022) https://raphlinus.github.io/curves/2022/09/09/parallel-bezie...
Fitting cubic Bézier curves (2021) https://raphlinus.github.io/curves/2021/03/11/bezier-fitting...
by lioeters