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Repulsive Surfaces
arXiv - CS - Graphics Pub Date : 2021-07-04 , DOI: arxiv-2107.01664 Chris Yu, Caleb Brakensiek, Henrik Schumacher, Keenan Crane
arXiv - CS - Graphics Pub Date : 2021-07-04 , DOI: arxiv-2107.01664 Chris Yu, Caleb Brakensiek, Henrik Schumacher, Keenan Crane
Functionals that penalize bending or stretching of a surface play a key role
in geometric and scientific computing, but to date have ignored a very basic
requirement: in many situations, surfaces must not pass through themselves or
each other. This paper develops a numerical framework for optimization of
surface geometry while avoiding (self-)collision. The starting point is the
tangent-point energy, which effectively pushes apart pairs of points that are
close in space but distant along the surface. We develop a discretization of
this energy for triangle meshes, and introduce a novel acceleration scheme
based on a fractional Sobolev inner product. In contrast to similar schemes
developed for curves, we avoid the complexity of building a multiresolution
mesh hierarchy by decomposing our preconditioner into two ordinary Poisson
equations, plus forward application of a fractional differential operator. We
further accelerate this scheme via hierarchical approximation, and describe how
to incorporate a variety of constraints (on area, volume, etc.). Finally, we
explore how this machinery might be applied to problems in mathematical
visualization, geometric modeling, and geometry processing.
中文翻译:
排斥面
惩罚曲面弯曲或拉伸的泛函在几何和科学计算中发挥着关键作用,但迄今为止忽略了一个非常基本的要求:在许多情况下,曲面不能通过自身或彼此。本文开发了一个用于优化表面几何形状同时避免(自)碰撞的数值框架。起点是切点能量,它有效地将空间上靠近但沿表面远离的点对分开。我们为三角形网格开发了这种能量的离散化,并引入了一种基于分数 Sobolev 内积的新型加速方案。与为曲线开发的类似方案相比,我们通过将预处理器分解为两个普通泊松方程,避免了构建多分辨率网格层次结构的复杂性,加上分数微分算子的前向应用。我们通过层次逼近进一步加速了这个方案,并描述了如何结合各种约束(面积、体积等)。最后,我们探索如何将这种机制应用于数学可视化、几何建模和几何处理中的问题。
更新日期:2021-07-06
中文翻译:
排斥面
惩罚曲面弯曲或拉伸的泛函在几何和科学计算中发挥着关键作用,但迄今为止忽略了一个非常基本的要求:在许多情况下,曲面不能通过自身或彼此。本文开发了一个用于优化表面几何形状同时避免(自)碰撞的数值框架。起点是切点能量,它有效地将空间上靠近但沿表面远离的点对分开。我们为三角形网格开发了这种能量的离散化,并引入了一种基于分数 Sobolev 内积的新型加速方案。与为曲线开发的类似方案相比,我们通过将预处理器分解为两个普通泊松方程,避免了构建多分辨率网格层次结构的复杂性,加上分数微分算子的前向应用。我们通过层次逼近进一步加速了这个方案,并描述了如何结合各种约束(面积、体积等)。最后,我们探索如何将这种机制应用于数学可视化、几何建模和几何处理中的问题。