[javajosh]

beautifully 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.

[ad8e]

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.

[unconed]

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.

[1] https://i.stack.imgur.com/I9wKC.png

[pavlov]

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.”