Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:  define a discrete Laplace–Beltrami operator for simplicial surfaces (Definition 16). It depends only on the intrinsic geometry of the surface and its edge ights are positive. Our Laplace operator is similar to the ll known finite-elements Laplacian (the so called “cotan formula”) except that it is based on the intrinsic Delaunay triangulation of the simplicial surface. This leads to new definitions of discrete harmonic functions, discrete mean curvature, and discrete minimal surfaces. The definition of the discrete Laplace–Beltrami operator depends on the existence and uniqueness of Delaunay tessellations in piecewise flat surfaces. While the existence is known,  prove the uniqueness. Using Rippa’s Theorem  show that, as claimed, Musin’s harmonic index provides an optimality criterion for Delaunay triangulations, and this can be used to prove that the edge flipping algorithm terminates also in the setting of piecewise flat surfaces.
['Alexander I. Bobenko', 'Boris Springborn']
I am writing a research paper on a idea A Discrete Laplace–Beltrami Operator for Simplicial Surfaces
[{"from": "human", "value": "I am writing a research paper on a idea A Discrete Laplace\u2013Beltrami Operator for Simplicial Surfaces"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:  define a discrete Laplace\u2013Beltrami operator for simplicial surfaces (Definition\u00a016). It depends only on the intrinsic geometry of the surface and its edge ights are positive. Our Laplace operator is similar to the ll known finite-elements Laplacian (the so called \u201ccotan formula\u201d) except that it is based on the intrinsic Delaunay triangulation of the simplicial surface. This leads to new definitions of discrete harmonic functions, discrete mean curvature, and discrete minimal surfaces. The definition of the discrete Laplace\u2013Beltrami operator depends on the existence and uniqueness of Delaunay tessellations in piecewise flat surfaces. While the existence is known,  prove the uniqueness. Using Rippa\u2019s Theorem  show that, as claimed, Musin\u2019s harmonic index provides an optimality criterion for Delaunay triangulations, and this can be used to prove that the edge flipping algorithm terminates also in the setting of piecewise flat surfaces."}]