New PDF release: Discrete differential geometry: An applied introduction

By Desbrun M., et al.

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Forman 2005] and [Bj¨orner and Welker 1995] and references therein). 4 Calculus ex Geometrica Given the overwhelming geometric nature of the most fundamental and successful calculus of these last few centuries, it seems relevant to approach computations from a geometric standpoint. One of the key insights that percolated down from the theory of differential forms is rather simple and intuitive: one needs to recognize that different physical quantities have different properties, and must be treated accordingly.

The basic building blocks of the parameterization are the kites formed by the endpoints of an edge (vi , v j ) and the incident face circumcircle centers ci jk and c jil for triangles ti jk and t jil respectively (at the boundary t jil is missing). Since all relations are scale invariant it is convenient to introduce the logarithmic radius variables ρt = log rt for t ∈ T . With these definitions ∂S = 2π − ∑ 2 fe (ρk − ρl ) − ∑ 2(π − θe ), ∂ ρk {e∈k}∩E {e∈k}∩E int (9) bdy giving us the desired equivalence of ∇ρ S = 0 and Equation 7.

For example, fluid flux is sometimes called a pseudo-2-form: indeed, given a transverse direction, we know how much flux is going through a piece of surface; it does not depend on the orientation of the surface itself. Vorticity is, however, a true 2-form: given an orientation of the surface, the integration gives us the circulation around that surface boundary induced by the surface orientation. It does not depend on the transverse direction of the surface. But if we have an orientation of the ambient space, we can always associate transverse direction with internal orientation of the submanifold.

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Discrete differential geometry: An applied introduction by Desbrun M., et al.


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