A multisided generalization of Bézier surfaces
ACM Transactions on Graphics (TOG)
Composite overlapping meshes for the solution of partial differential equations
Journal of Computational Physics
Journal of Computational and Applied Mathematics - Orthogonal polynomials and numerical methods
Locally corrected multidimensional quadrature rules for singular functions
SIAM Journal on Scientific Computing
Rational trigonometric approximations using Fourier series partial sums
Journal of Scientific Computing
Modeling surfaces of arbitrary topology using manifolds
SIGGRAPH '95 Proceedings of the 22nd annual conference on Computer graphics and interactive techniques
Curvature continuous spline surfaces over irregular meshes
Computer Aided Geometric Design
A spectral embedding method applied to the advection-diffusion equation
Journal of Computational Physics
Exponentially Accurate Approximations to Piece-Wise Smooth Periodic Functions
Journal of Scientific Computing
On the Gibbs Phenomenon and Its Resolution
SIAM Review
Degenerate Be´zier patches with continuous curvature
Computer Aided Geometric Design
Numerical solution of the Helmholtz equation in 2D and 3D using a high-order Nystro¨m discretization
Journal of Computational Physics
Modelings surfaces from meshes of arbitrary topology
Computer Aided Geometric Design
Journal of Computational Physics
Numerical Recipes in FORTRAN: The Art of Scientific Computing
Numerical Recipes in FORTRAN: The Art of Scientific Computing
C2 free-form surfaces of degree (3,5)
Computer Aided Geometric Design
A comparison of numerical algorithms for Fourier extension of the first, second, and third kinds
Journal of Computational Physics
Constrained fitting in reverse engineering
Computer Aided Geometric Design
A simple manifold-based construction of surfaces of arbitrary smoothness
ACM SIGGRAPH 2004 Papers
On the Fourier Extension of Nonperiodic Functions
SIAM Journal on Numerical Analysis
Journal of Computational Physics
Multi-domain Fourier-continuation/WENO hybrid solver for conservation laws
Journal of Computational Physics
A Fast Algorithm for Fourier Continuation
SIAM Journal on Scientific Computing
Approximation error in regularized SVD-based Fourier continuations
Applied Numerical Mathematics
The chain collocation method: A spectrally accurate calculus of forms
Journal of Computational Physics
On the resolution power of Fourier extensions for oscillatory functions
Journal of Computational and Applied Mathematics
Spatially Dispersionless, Unconditionally Stable FC---AD Solvers for Variable-Coefficient PDEs
Journal of Scientific Computing
Hi-index | 31.47 |
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.