Geodesic
Cauchy problems for discrete affine minimal surfaces
Archivum mathematicum, Tome 48 (2012) no. 1, pp. 1-14 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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In this paper we discuss planar quadrilateral (PQ) nets as discrete models for convex affine surfaces. As a main result, we prove a necessary and sufficient condition for a PQ net to admit a Lelieuvre co-normal vector field. Particular attention is given to the class of surfaces with discrete harmonic co-normals, which we call discrete affine minimal surfaces, and the subclass of surfaces with co-planar discrete harmonic co-normals, which we call discrete improper affine spheres. Within this classes, we show how to solve discrete Cauchy problems analogous to the Cauchy problems for smooth analytic improper affine spheres and smooth analytic affine minimal surfaces.
In this paper we discuss planar quadrilateral (PQ) nets as discrete models for convex affine surfaces. As a main result, we prove a necessary and sufficient condition for a PQ net to admit a Lelieuvre co-normal vector field. Particular attention is given to the class of surfaces with discrete harmonic co-normals, which we call discrete affine minimal surfaces, and the subclass of surfaces with co-planar discrete harmonic co-normals, which we call discrete improper affine spheres. Within this classes, we show how to solve discrete Cauchy problems analogous to the Cauchy problems for smooth analytic improper affine spheres and smooth analytic affine minimal surfaces.
DOI : 10.5817/AM2012-1-1
Classification : 39A12, 52C99, 53A15
Keywords: discrete differential geometry; discrete affine minimal surfaces; discrete conjugate nets; PQ meshes 
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Craizer, Marcos; Lewiner, Thomas; Teixeira, Ralph. Cauchy problems for discrete affine minimal surfaces. Archivum mathematicum, Tome 48 (2012) no. 1, pp. 1-14. doi: 10.5817/AM2012-1-1

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