Ask HN: Why are quadratic Bezier curves described so obtusely?
Laziness? really?<p>I think a lot of important and frequently applied elements of math are, like Bezier splines, something that can be understood in different ways, in different contexts.  e.g. what is a matrix?  You can tell a story about systems of equations, or about tensors or maybe bilinear forms, or about coordinate system transforms, etc.  All of these are valid, and good understanding comes from being able to hold multiple views of the same underlying thing simultaneously.<p>You have embraced one way of interpreting a quadratic Bezier curve, as a curve characterized by some constant second derivative (2 degrees of freedom or DOF), an initial velocity (2 DOF), and an initial position (2 DOF).  Someone from computer-aided design (CAD) might specify the curve in terms of the start point (2 DOF), end point (2 DOF), and control point (2 DOF). Someone implementing all this in code might evaluate the basis functions, or might use Casteljau&#x27;s method to do the lerp of the lerps, as you noted.  All of these end up being different views of the same math object, and all of them are sensible.<p>In particular, what is most &quot;physical&quot; may be subjective.  An important use of splines was CAD, where a human is saying: I need something to go from here, to here, in a smooth way that I can nudge with this control point.  That is how we interact with cubic Bezier splines in drawing programs, and those control points for specifying the spline are nice and tangible, if not physical.  A curve of constant jerk is physical too, I guess, but doesn&#x27;t align as well with how the expressivity of cubic splines frequently gets invoked.<p>There is definitely elitism in how some people describe their use of math (behold my sophistication); it&#x27;s a bummer.  More charitably, you could understand it as the author respecting the time of the reader assumed to already understand something about the topic.  But don&#x27;t let that discourage you.