math - Building bicubic coons patch from four boundary curves

Question

I want to create s coons patch surface from four boundary curves s1(u), s2(u) q1(v), q2(v)

I know that equations are the following (added screenshots from a presentation):

There are a few parts of the equations that are not fully understand and i did not find any good explanation:

1. In s1(u,v) what is the meaning of p1v(u) and p2v(u)? same goes for q1u(v) and q2u(v) in s2(u,v).

2. In the A matrix, what is the meaning of A00(u,v)..A11(u,v). What would be the value of these parameters if p1,p2 are only functions of u (and not v) and q1,q2 are only function of v.

I would appreciate any help on this issue.

Answer 1

基本上，他们使用上标来表示导数或偏导数。

当我们构建双立方浣熊补丁时，我们将使用Cubic Hermite splines。这些都需要位置和切线作为输入。因此，对于第一条边，P ₁ (u) 是曲线的参数化，P ₁ ^v (u) 是跨边的切线。表面中的任何点实际上都有两个切线，另一个是沿边缘的 P ₁ ^{u (u)，即 dP} ₁ (u) / du。

对于 A 矩阵，这需要补丁的四个角点和导数。对于点 A ₀₀，它使用两个一阶导数 ∂S / ∂u = A ₀₀ ^u和 ∂S / ∂v = A ₀₀ ^u和二阶导数之一 ∂<sup>2S / ∂u ∂v = A ₀₀ ^uv . （我在这里用 S 表示定义表面的函数）。由于沿边缘的曲线的导数必须与拐角处的导数相匹配，因此我们有一些进一步的条件

• A ₀₀ ^u = ∂S / ∂u = P ₁ ^u (0) = Q ₁ ^u (0) = dP ₁ / du (0)
• A ₀₀ ^v = ∂S / ∂v = P ₁ ^v (0) = Q ₁ ^v (0) = dQ ₁ / dv (0)
• A ₀₀ ^uv = ∂<sup>2S / ∂u∂v = ∂P ₁ ^v /∂u (0) = ∂Q ₁ ^u /∂u (0)