An excellent exposition! This is the way mathematics should be taught. This is a great explanation of Gaussian curvature and Bezier curves, splines, and patche

An excellent exposition! This is the way mathematics should be taught. This is a great explanation of Gaussian curvature and Bezier curves, splines, and patches, B-Splines, NURBS, and subdivision surfaces. It reads like a story, and nothing is introduced without the reader having an understanding why it's needed, what problem it solves, and how. One thing I wish was mentioned (in the final notes, perhaps): The surfaces formed by dragging lines in space are called Ruled Surfaces, and have many interesting applications in fields ranging from pure math to architecture[1]. If you have ever seen string art in two or three dimensions[2], it's the same thing. Hyperboloid[3] is an example of particular importance, as it allows for building light, structurally sound structures - particularly, towers and domes[4]. You have walked up a Helicoid[5] if you ever went up or down a spiral staircase[6]. This would give you an intuition about the seeming paradox of having a curved surface made out of straight lines. The Mobius Strip[7] is a ruled surface of a special kind: it only has one side (an ant crawling on a Mobius strip would eventually walk over both sides of the piece of paper it was made from, without ever touching the boundary). It has always surprised me just how much complexity one can get out of one simple rule. But that's why math is interesting :) [1] https://en.wikipedia.org/wiki/Ruled_surface https://en.wikipedia.org/wiki/Ruled_surface [2] https://www.guidepatterns.com/wp-content/uploads/2015/01/3D-String-Art.jpg https://www.guidepatterns.com/wp-content/uploads/2015/01/3D-... [3] https://polyhedr.com/hyperboloid-net.html https://polyhedr.com/hyperboloid-net.html [4] https://en.wikipedia.org/wiki/Hyperboloid_structure https://en.wikipedia.org/wiki/Hyperboloid_structure [5] https://en.wikipedia.org/wiki/Helicoid https://en.wikipedia.org/wiki/Helicoid [6] https://www.architectureartdesigns.com/16-elegant-modern-spiral-stairs-design-ideas-that-will-fit-every-home-decor/ https://www.architectureartdesigns.com/16-elegant-modern-spi... [7] https://en.wikipedia.org/wiki/M%C3%B6bius_strip https://en.wikipedia.org/wiki/M%C3%B6bius_strip