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Published December 10, 2007 | public
Journal Article

Accurate, high-order representation of complex three-dimensional surfaces via Fourier continuation analysis

Abstract

We present a new method for construction of high-order parametrizations of surfaces: starting from point clouds, the method we propose can be used to produce full surface parametrizations (by sets of local charts, each one representing a large surface patch – which, typically, contains thousands of the points in the original point-cloud) for complex surfaces of scientific and engineering relevance. The proposed approach accurately renders both smooth and non-smooth portions of a surface: it yields super-algebraically convergent Fourier series approximations to a given surface up to and including all points of geometric singularity, such as corners, edges, conical points, etc. In view of their C^∞ smoothness (except at true geometric singularities) and their properties of high-order approximation, the surfaces produced by this method are suitable for use in conjunction with high-order numerical methods for boundary value problems in domains with complex boundaries, including PDE solvers, integral equation solvers, etc. Our approach is based on a very simple concept: use of Fourier analysis to continue smooth portions of a piecewise smooth function into new functions which, defined on larger domains, are both smooth and periodic. The "continuation functions" arising from a function f converge super-algebraically to f in its domain of definition as discretizations are refined. We demonstrate the capabilities of the proposed approach for a number of surfaces of engineering relevance.

Additional Information

© 2007 Elsevier. Received 11 December 2006, Revised 24 August 2007, Accepted 28 August 2007, Available online 12 September 2007. This work was supported in part by the Air Force Office of Scientific Research, the National Science Foundation and the National Aeronautics and Space Administration.

Additional details

Created:
August 22, 2023
Modified:
October 19, 2023