三角網格體積計算

Paper read - EFFICIENT FEATURE EXTRACTION FOR 2D/3D OBJECTS IN MESH REPRESENTATION

http://chenlab.ece.cornell.edu/Publication/Cha/icip01_Cha.pdf

可以參見：How to calculate the volume of a 3D mesh object the surface of which is made up triangles

ACB為三角形網格上面的一個三角形面片。對應的OACB四面體的體積為：

\[|V_{OACB}| = |\frac{1}{6}(-x_3 y_2 z_1 + x_2 y_3 z_1 + x_3 y_1 z_2 - x_1y_3z_2 - x_2y_1z_3 + x_1y_2z_3)| \]

有向體積的符號為，\(\vec{OA}\cdot N_{ACB}\)的符號。在實際應用中，直接計算下式即可：

\[V_i' = \frac{1}{6}(-x_{i3} y_{i2} z_{i1} + x_{i2} y_{i3} z_{i1} + x_{i3} y_{i1} z_{i2} - x_{i1}y_{i3}z_{i2} - x_{i2}y_{i1}z_{i3} + x_{i1}y_{i2}z_{i3}) \]

那么三角網格的體積如下：

\[V_{total}' = \sum_i V_i' \]