"相交线与二次贝塞尔三角形"
[英]Intersect Line vs Quadratic Bezier Triangle
问题描述
I'm trying to find the intersection between a line segment and a quadratic bezier triangle for my OpenCL real time raytracer.
我正在尝试为我的 OpenCL 实时光线追踪器找到线段和二次贝塞尔三角形之间的交点。
This question Detect&find intersection ray vs. cubic bezier triangle<\/a> talks about finding the collision point between a ray and a cubic bezier triangle and the main recomendations are to try subdivision, or tensor product bezier patches.这个问题Detect&find intersection ray vs.cubic bezier triangle<\/a>讨论了寻找射线和三次贝塞尔三角形之间的碰撞点，主要建议是尝试细分或张量积贝塞尔补丁。
Here's the equation for a quadratic bezier triangle:这是二次贝塞尔三角形的方程：
Super stumped on this one!超级难倒这个！
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2 个解决方案
解决方案1
0 已采纳 2014-03-26 05:35:41
Line segment has parametric equation P = P0+(P1-P0)*v=P0+d*v , where v is parameter [0..1] To find intersection with brute force, you have to solve system of three quadratic equations like 线段具有参数方程式P = P0+(P1-P0)*v=P0+d*v ，其中v是参数[0..1]要找到具有蛮力的交点，必须求解三个二次方程组，例如
x0+dx*v=AxS^2 + 2*Dx*S*T + 2*Ex*S*U + Bx*T^2 + 2*Fx*T*U + Cx*U^2
This system has 8 possible solutions, and intersection(s) exists only if resulted v,s,t lie in range 0..1 simultaneously. 该系统有8个可能的解，并且仅当结果v，s，t同时位于范围0..1中时才存在交集。 It's not easy to solve this system (possible approaches - Gröbner basis , numerical methods, and solution process might be numerically unstable). 解决这个系统并不容易（可能的方法-Gröbner基础 ，数值方法和求解过程在数值上可能不稳定）。 That is why subdivision method is sometimes recommended. 这就是为什么有时建议使用细分方法的原因。
解决方案2
0 2022-01-29 17:56:15
when your bezier patch has more than 3 sides, ray-intersection is no longer analytic (even if the splines on the side are only quadratic) and a precise enough iterative approach is significantly slower.当你的贝塞尔面片有超过 3 条边时，光线交点不再是解析的（即使边上的样条线只是二次的），并且足够精确的迭代方法会明显变慢。 Bezier patches are pathetic in terms of performance due to inherent recursion, you may want to do FFT for fourier-patches instead.由于固有的递归，Bezier 补丁在性能方面是可悲的，您可能希望对傅立叶补丁进行 FFT。 Shadertoy.com [iq] has various "twisted cube intersection"s (that are not iterated by sphere-tracking==raymarching, which is efficient, but also causes MANY low-precision-cases-artifacts, where ever convergence of iteration is too slow). Shadertoy.com [iq] 有各种“扭曲的立方体交点”（不通过 sphere-tracking==raymarching 进行迭代，这是有效的，但也会导致许多低精度情况下的伪影，其中迭代的收敛太慢的）。
yes, triangular-bezierpatch intersections has only analytic quadratic complexity (with quadratic beziers on the borders), but it involves some pre-computation (unless it is already in barycentric coordinates), that significantly lower the resulting precision (barycentric adds 1 division globally, and sums up over all domains)是的，三角形-贝塞尔补丁交点只有解析二次复杂度（边界上有二次贝塞尔曲线），但它涉及一些预计算（除非它已经在重心坐标中），这会显着降低结果精度（重心在全球范围内增加 1 个除法，并总结所有领域）
This was solved (poorly, as in it has 1 error case and too low precision) on shadertoy in opengl in 2019: https:\/\/www.shadertoy.com\/view\/XsjSDt<\/a> and i optimized it slightly (fixed precision issues, added camera and culling): https:\/\/www.shadertoy.com\/view\/ttjSzw<\/a> + https:\/\/www.shadertoy.com\/view\/wlSyzd<\/a> also see, https:\/\/www.reddit.com\/r\/askmath\/comments\/sfn8mk\/analytic_intersection_ray\/<\/a>这在 2019 年的 opengl 中的 shadertoy 上得到了解决（很差，因为它有 1 个错误情况和太低的精度）： https<\/a> :\/\/www.shadertoy.com\/view\/XsjSDt 我稍微优化了它（修复了精度问题，添加了相机和剔除）： https<\/a> :\/\/www.shadertoy.com\/view\/ttjSzw + https:\/\/www.shadertoy.com\/view\/wlSyzd<\/a>另见， https:\/\/www.reddit.com\/r\/askmath\/comments\/ sfn8mk\/analytic_intersection_ray\/<\/a>
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