New Bézier curves for vector graphics
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New Bézier curves for vector graphics
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Re: New Bézier curves for vector graphics
#2beautifully made. However I think one's intuition for where the control points go and what they do are very clear if you've ever taken calculus. The line segments control the direction and magnitude of the tangent to the curve at that position. If you understand that, then it makes the motivating problem go away, and all the challenge problems are easy to solve.
Re: New Bézier curves for vector graphics
#3beautifully made. However I think one's intuition for where the control points go and what they do are very clear if you've ever taken calculus. The line segments control the direction and magnitude of the tangent to the curve at that position. If you understand that, then it makes the motivating problem go away, and all the challenge problems are easy to solve.
Interactive curves are primarily meant to be used by designers. They are a tool. When the tool is not intuitive, it’s not really a solution to say “designers should just take calculus and it makes the problem go away.”
Re: New Bézier curves for vector graphics
#4> Béziers can't represent circles. If you try to approximate one by hand, it'll look lopsided.
Many graphics algorithms (vector operations: scale, translate, rotate; raster operations like rasterisation and interpolation) specialise for straight lines (and hence also triangles), Bézier curves, circles, etc separately, rather than forcing everything to be a Bézier curve.
Re: New Bézier curves for vector graphics
#5beautifully made. However I think one's intuition for where the control points go and what they do are very clear if you've ever taken calculus. The line segments control the direction and magnitude of the tangent to the curve at that position. If you understand that, then it makes the motivating problem go away, and all the challenge problems are easy to solve.
It's not as easy to visualize the tangent magnitude as you are suggesting, because the time parametrization makes the calculation fail - the curve doesn't travel forward at a constant rate. This is best seen with the second example: If the left handle is fully extended and the right handle is 0, the Bézier curve looks almost exactly the same as when the handles are reversed. Here's a picture: https://i.imgur.com/WkanN1G.png
The handle varies from 0 to full-strength, but the magnitude of the tangent vector stays constant. This means the handle doesn't decide the magnitude. Tracing the path in your head to visualize the changing tangent vectors would mean visualizing a competition between t^2, (1-t)^2t, and (1-t)^3, which I find difficult, even with some calculus knowledge.
Re: New Bézier curves for vector graphics
#6The guy from the mach engine has also talked about issues dealing with Bézier curves and came up with a interesting set a primitives:
Triangles, quadratic curves and semi-circles.
Mostly this is make it easier to do vector graphics on a gpu, which is why I'm interested in it, but I think it might be easier to understand.
Re: New Bézier curves for vector graphics
#7beautifully made. However I think one's intuition for where the control points go and what they do are very clear if you've ever taken calculus. The line segments control the direction and magnitude of the tangent to the curve at that position. If you understand that, then it makes the motivating problem go away, and all the challenge problems are easy to solve.
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Re: New Bézier curves for vector graphics
#8Obligatory links for anyone that is interested in learning more about Bézier curves:
https://pomax.github.io/bezierinfo/
Re: New Bézier curves for vector graphics
#9Do you have any publicly available code on github?
Re: New Bézier curves for vector graphics
#10beautifully made. However I think one's intuition for where the control points go and what they do are very clear if you've ever taken calculus. The line segments control the direction and magnitude of the tangent to the curve at that position. If you understand that, then it makes the motivating problem go away, and all the challenge problems are easy to solve.
This isn't true, and is obvious when you try to split a bezier without distorting its shape.
As a bezier is just repeated interpolations, the tangents and their lengths can be derived from the intermediate interpolations [1]. This means that if you split a bezier near an existing control point, the new point will have wildly unbalanced tangents on both sides, and yet, connect into the exact same cubic curve.
So both your intuition and your confidence in it are... wrong.
