Abstract
Aiming to optimize the shape of closed embedded curves within prescribed isotopy classes, we use a gradient-based approach to approximate stationary points of the Möbius energy. The gradients are computed with respect to Sobolev inner products similar to the W^{3/2,2}-inner product. This leads to optimization methods that are significantly more efficient and robust than standard techniques based on L^2-gradients.
Highlights
Let γ : T → Rm be a sufficiently smooth embedding1 of the circle T into Euclidean space
The applicability of self-avoiding energies is heavily limited by their immense cost: Typical discretizations replace the double integrals by double sums which leads to a computational complexity of at least Ω((N · m)2) for evaluating the discrete Möbius energy and its derivative, where N · m is the number of degrees of freedom of the discretized geometry
This issue can be mended by sophisticated kernel compression techniques, see [99]. We focus on another issue that is more related to mathematical optimization, namely the fact that, for N → ∞, the discretized optimization problems become increasingly ill-conditioned
Summary
Let γ : T → Rm be a sufficiently smooth embedding of the circle T into Euclidean space. The term |γ (x) − γ (y)|−2 blows up whenever a selfcontact emerges, lending itself as contact barrier for modeling impermeability of curves and rods. Y) guarantees that the energy is finite for sufficiently smooth embeddings This way, any time-continuous descent method like, e.g., a gradient flow, will necessarily preserve the isotopy class.. This way, any time-continuous descent method like, e.g., a gradient flow, will necessarily preserve the isotopy class.2 Another pleasant feature of the Möbius energy is that its critical points enjoy higher smoothness. This is in agreement with the interpretation of W 3/2,2-gradient descent as a coarse discretization of an ordinary differential equation. Further potential fields of applications for repulsive energies include computer graphics [10,79], packing problems [30,31], the modeling of coiling and kinking of submarine communications cables [24,100], and even solar coronal structures [70]
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