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双三次Bezier曲面算法

作者:互联网

双三次Bezier曲面

定义

\[p(u,v)= \left[\begin{matrix} u^3 & u^2 & u & 1 \end{matrix}\right] \left[\begin{matrix} -1 & 3 & -3 & 1 \\ 3 & -6 & 3 & 0 \\ -3 & 3 & 0 & 0 \\ 1 & 0 & 0 & 0 \end{matrix}\right] \left[\begin{matrix} P_{0,0} &P_{0,1} & P_{0,2} & P_{0,3}\\ P_{1,0} & P_{1,1} & P_{1,2} & P_{1,3} \\ P_{2,0} & P_{2,1} & P_{2,2} & P_{2,3} \\ P_{3,0} & P_{3,1} &P_{3,2} & P_{3,3} \end{matrix}\right] \left[\begin{matrix} -1 & 3 & -3 & 1 \\ 3 & -6 & 3 & 0 \\ -3 & 3 & 0 & 0 \\ 1 & 0 & 0 & 0 \end{matrix}\right] \left[\begin{matrix} u^3\\ u^2 \\ u \\ 1 \end{matrix}\right] \]

\[P(u,v)=UMPM^TV^T \]

递归细分

曲面片的法向量

\[p_u=\frac{\partial p(u,v)}{\partial u}, p_v=\frac{\partial p(u,v)}{\partial v} \]

细分点在面上的法向量为:

\[N=\frac{\partial p(u,v)}{\partial v}\times \frac{\partial p(u,v)}{\partial v} \]

\[ p^`_u(u,v)= \left[\begin{matrix} 3u^2 & 2u & 1 &0 \end{matrix}\right]MPM^TV^T \]

\[ p^`_v(u,v)=UMPM^T \left[\begin{matrix} 3u^2 & 2u & 1 &0 \end{matrix}\right]^T \]

\[N=p^`_u(u,v)\times p^`_v(u,v) \]

标签:begin,right,partial,matrix,Bezier,算法,曲面,end,left
来源: https://www.cnblogs.com/brilliantM/p/14782187.html