关键词：参数表面；环形循环；弯曲能量；梯度下降；定期同伦
摘 要：We focus on an optimization problem on parameterized surfaces of genus one. In particular we trade off the penalty functions for bending a toroidal path and for applying a twist to it and aim to find local minima of this cost function. This analysis forms a key element in demonstrating the different regular homotopy classes of tori. A generalization of this surface optimization, which considers curvature as well as any shearing of its parameter grid, may be used to find the most optimal direct path from an arbitrary closed manifold of genus one into one of the four basic representatives of the four regular homotopy classes of tori.