【SIGGRAPH Asia 2012专题*技术论文】雕塑，堆叠，结构和盒：快速定向包围盒优化的旋转群SO（3，R）

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【SIGGRAPH Asia 2012专题*技术论文】
雕塑，堆叠，结构和盒：快速定向包围盒优化的旋转群SO（3，R）Fast oriented bounding box optimization on the rotation group SO(3, R)
Chia-Tche Chang, Bastien Gorissen, Samuel Melchior
本文发现的最小体积方向包围盒（OBB）的N点的有限集合，记为X⊂R3的问题。该问题由在寻找的长方体，即，矩形平行六面体，包封X的最小体积。为2D的情况下，在图1中说明了这一点。每个OBB被定义及其中心X∈R3，其尺寸∈R 3，和其取向R∈SO（3，R）。这种旋转组是在R上的特殊正交群的3度：
SO（3，R）= R∈GL（3，R）| RT R = I= RRT，DET（R）= 1，
其中，一般线性群GL（3，R）是3度，也就是3-3可逆实矩阵的集合。矩阵R的旋转，如在图1中示出的参照帧的前到Eξ。由CH（X）和XC⊂X，分别表示X和它的顶点的集合的凸包。正如N =| X|，让NV =| XC|是在CH（X）的顶点的数量。这个凸包，可以得到其成本高达O（N log N）的预处理步骤。
This article deals with the problem of finding the minimum-volume Oriented Bounding Box (OBB) of a given finite set of N points,denoted by X ⊂ R3. The problem consists in finding the cuboid,that is, rectangular parallelepiped, of minimal volume enclosing X .This is illustrated for the 2D case in Figure 1. Each OBB is defined by its center X ∈ R3, its dimension ∈ R3, and its orientation R ∈ SO(3, R). This rotation group is the special orthogonal group of degree 3 over R:
SO(3, R) = R ∈ GL(3, R) | RT R = I = RRT , det(R) = 1 ,
where GL(3, R) is the general linear group of degree 3, that is,the set of 3-by-3 invertible real matrices. The matrix R rotates the reference frame ex onto eξ as shown in Figure 1. The convex hull of X and the set of its vertices are denoted by CH(X ) and X C ⊂ X , respectively. Just as N = |X |, let NV = |X C| be the number of vertices of CH(X ). This convex hull can be obtained as a preprocessing step with cost O(N log N ).
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