Yeah, interesting stuff. So, the whole debate between closed forms and numerical methods is kind of overplayed. Even if you had a closed form for arc length, you’d still need numerical methods to compute exp and log. What functions actually work best for arc length? That’s the real question.
Also, while some folks are hyped about Pythagorean-Hodograph curves, I think they’re kinda niche. Euler spirals seem more practical, even if you have to compute a special function for them. Numerical solutions tend to be more stable anyway, especially in cases where a closed form might break down, like near straight lines.
You know, it’s kinda interesting to see how RTS games have changed over the years. Like, Dune II set the stage with its mechanics, but then games like Dawn of War and Blood and Magic started doing their own thing. It’s sorta like how FPS games evolved from Doom to stuff like Half-Life and now all these battle royale games. Just shows how game design, player feedback, and tech keep changing things up. Makes you wonder what the next big thing in RTS games will be, especially with AI and procedural generation getting better. Anyway, just some thoughts.
Also, while some folks are hyped about Pythagorean-Hodograph curves, I think they’re kinda niche. Euler spirals seem more practical, even if you have to compute a special function for them. Numerical solutions tend to be more stable anyway, especially in cases where a closed form might break down, like near straight lines.