4 ms·
The geometric intuition is important but missing. Calculus and algebra is not everything. Short version: Imagine a line connecting A to B. Now imagine a poin
by dietrichepp 10y ago
The geometric intuition is important but missing. Calculus and algebra is not everything.
Short version:
Imagine a line connecting A to B. Now imagine a point sliding across that line, starting at A and going to B.
Now consider a line A to B and a second line B to C. A point slides across each line: one from A to B, another from B to C. Now draw a line connecting these two points, and imagine a point sliding across that line.
This moving line starts as A to B, ends as B to C, so the point sliding along it goes from A to C along a curved path. This is a quadratic Bézier curve.
Add another layer and you get cubic.
- blahedo 10y agoThe crucial insight I drew from the OP was that Bézier curves are essentially a weighted average of the endpoints and control points, with the weights shifting an a parametric way. Somehow that had never clicked before. Your comment gives a nice way to elabourate on how the parameterisation works.
- Retric 10y agoWhich gets to their real advantage, they are easy to create by dragging around a few points.
- to3m 10y agoFor whatever it's worth, I found this explanation useful, at least when rendered visually, when first encountering this stuff. But clearly the geometric viewpoint isn't important to everybody, because the author of the post would have mentioned it if so! See also, maybe: http://psychclassics.yorku.ca/Galton/imagery.htm http://psychclassics.yorku.ca/Galton/imagery.htm http://www.nytimes.com/2015/06/23/science/aphantasia-minds-eye-blind.html?_r=0 http://www.nytimes.com/2015/06/23/science/aphantasia-minds-e... https://www.facebook.com/notes/blake-ross/aphantasia-how-it-feels-to-be-blind-in-your-mind/10156834777480504/ https://www.facebook.com/notes/blake-ross/aphantasia-how-it-...
- userbinator 10y agoWikipedia has some good animations of this: https://en.wikipedia.org/wiki/B%C3%A9zier_curve#Constructing_B.C3.A9zier_curves https://en.wikipedia.org/wiki/B%C3%A9zier_curve#Constructing...
- agumonkey 10y agoThis was a big revelation for me. It's just "nested percentages" in mathematical clothing. How to connect simple concepts (I think almost everybody would understand linear interpolation, even in [a, 1-a] form) and get polynomials.