双三次 Bezier 曲面

双三次 Bezier 曲面的定义

Bezier 曲面是由 Bezier 曲线扩展得到，它是两组正交的 Bezier 曲线控制点构造空间网格生成的曲面

$$\begin{array}{c}\mathbf{p}(u,v)=\sum _{i=0}^3\sum _{j=0}^3{P}_{i,j}{B}_{i,3}(u)B_{j,3}(v),(u,v)\in [0,1]\times [0,1]\end{array}\text{\hspace{1em}?}$$

其中， $P_{ij}$ 是 16 个控制点，$B_{i,3}(u)$ 和 $B_{j,3}(v)$ 是 Bernstein 基函数。

对 (1) 式展开，有
$$\begin{array}{c}\mathbf{p}(u,v)=\left[\begin{array}{cccc}{B}_{0,3}(u)& B_{1,3}(u)& B_{2,3}(u)& B_{3,3}(u)\end{array}\right]\left[\begin{array}{cccc}{P}_{0,0}& P_{0,1}& P_{0,2}& P_{0,3}\\ P_{1,0}& P_{1,1}& P_{1,2}& P_{1,3}\\ P_{2,0}& P_{2,1}& P_{2,2}& P_{2,3}\\ P_{3,0}& P_{3,1}& P_{3,2}& P_{3,3}\end{array}\right]\left[\begin{array}{c}{B}_{0,3}(v)\\ B_{1,3}(v)\\ B_{2,3}(v)\\ B_{3,3}(v)\end{array}\right]\end{array}\text{\hspace{1em}?}$$

$$B_{0,3}(u)，B_{1,3}(u)，B_{2,3}(u)，B_{3,3}(u)，B_{0,3}(v)，B_{1,3}(v)，B_{2,3}(v)，B_{3,3}(v)$$是三次 Bernstein 基函数，

$$\begin{array}{c}?\begin{array}{l}{B}_{0,3}(u)=?u^3+3u^2?3u+1\\ B_{1,3}(u)=3u^2?6u^2+3u\\ B_{2,3}(u)=?3u^3+3u^2\\ B_{3,3}(u)=u^3\end{array}\end{array}\text{\hspace{1em}?}$$
$$\begin{array}{c}?\begin{array}{l}{B}_{0,3}(v)=?v^3+3v^2?3v+1\\ B_{1,3}(v)=3v^2?6v^2+3v\\ B_{2,3}(v)=?3v^3+3v^2\\ B_{3,3}(v)=v^3\end{array}\end{array}\text{\hspace{1em}?}$$

$$\begin{array}{c}p(u,v)=\left[\begin{array}{cccc}{u}^3& u^2& u& 1\end{array}\right]\left[\begin{array}{cccc}?1& 3& ?3& 1\\ 3& ?6& 3& 0\\ ?3& 3& 0& 0\\ 1& 0& 0& 0\end{array}\right]\left[\begin{array}{cccc}{P}_{0,0}& P_{0,1}& P_{0,2}& P_{0,3}\\ P_{1,0}& P_{1,1}& P_{1,2}& P_{1,3}\\ P_{2,0}& P_{2,1}& P_{2,2}& P_{2,3}\\ P_{3,0}& P_{3,1}& P_{3,2}& P_{3,3}\end{array}\right]\left[\begin{array}{cccc}?1& 3& ?3& 1\\ 3& ?6& 3& 0\\ ?3& 3& 0& 0\\ 1& 0& 0& 0\end{array}\right]\left[\begin{array}{c}{v}^3\\ v^2\\ v\\ 1\end{array}\right]\end{array}\text{\hspace{1em}?}$$

令

$$\mathbf{U}=\left[\begin{array}{cccc}{u}^3& u^2& u& 1\end{array}\right],\mathbf{V}=\left[\begin{array}{cccc}{v}^3& v^2& v& 1\end{array}\right]$$

$$\mathbf{M}=\left[\begin{array}{cccc}?1& 3& ?3& 1\\ 3& ?6& 3& 0\\ ?3& 3& 0& 0\\ 1& 0& 0& 0\end{array}\right],\mathbf{P}=\left[\begin{array}{cccc}{P}_{0,0}& P_{0,1}& P_{0,2}& P_{0,3}\\ P_{1,0}& P_{1,1}& P_{1,2}& P_{1,3}\\ P_{2,0}& P_{2,1}& P_{2,2}& P_{2,3}\\ P_{3,0}& P_{3,1}& P_{3,2}& P_{3,3}\end{array}\right]$$

$$\begin{array}{c}\mathbf{p}(u,v)=\mathbf{U}\mathbf{M}\mathbf{P}{\mathbf{M}}^T{\mathbf{V}}^T\end{array}\text{\hspace{1em}?}$$

$\mathbf{M}$ 是对称矩阵，即 $M^T=M$.

生成曲面时，可以先固定 u 变化 v 得到一簇 Bezier 曲线，然后固定 v，变化 u 得到另一簇 Bezier 曲线，两卒曲线交织生成双三次 Bezier 曲面。

这里的程序算法的示意说明

// 读取控制点

void ReadControlPoint(Point3 P[4][4]) {

	double M[4][4];
	M[0][0] = -1;M[0][1] =  3;M[0][2] = -3;M[0][3] = 1;
	M[1][0] =  3;M[1][1] = -6;M[1][2] =  3;M[1][3] = 0;
	M[2][0] = -3;M[2][1] =  3;M[2][2] =  0;M[2][3] = 0;
	M[3][0] =  1;M[3][1] =  0;M[3][2] =  0;M[3][3] = 0;
	
	Point3 mControlPoint[4][4];
	for (int i=0; i<4; i++) {
		for (int j=0; j<4; j++) {
			mControlPoint[i][j] = P[i][j]	
		}
	}
	// 数字矩阵左乘 三维点矩阵 M * mControlPoint
	// 计算转置矩阵 M^T
	// 数字矩阵右乘 三维点矩阵 mControlPoint * M

	double setp = 0.1;
	double u3, u2, u1, u0, v3, v2, v1, v0;
	// 定义 Point Pt;
	for (double u=0; u<=1; u+=step) {
		for (double v=0; v<=1; v+=step) {
			u3 = u*u*u; v3 = v*v*v;
			u2 = u*u; v2 = v*v;
			u1 = u;	v1 = v;
			u0 = 1; v0 = 1;
			Pt = ....
			// 根据公式计算出 pt
			// 斜二投影
			//if (v == 0) 特殊处理
			// 绘制点
			
		}
	}
	for (double v=0; v<=1; v+=step) {
		for (double u=0; u<=1; u+=step) {
			u3 = u*u*u; v3 = v*v*v;
			u2 = u*u; v2 = v*v;
			u1 = u;	v1 = v;
			u0 = 1; v0 = 1;
			Pt = ....
			// 根据公式计算出 pt
			// 斜二投影
			
			//if (u == 0) 特殊处理
			// 绘制点
			
		}
	}

	
}

// 计算斜二投影

Point2 ObliqueProjection(Point3 P3) {
	Ponit2 P2;
	P2.x = P3.x - P3.z * sqrt(2.0) / 2.0;
	P2.y = P3.y - P3.z * sqrt(2.0) / 2.0;
	return P2;
}

参考 《计算几何算法与实现》孔令德