Fuhua Cheng

Fairing spline curves and surfaces by minimizing energy

Abstract New algorithms for the classical problem of fairing cubic spline curves and bicubic spline surfaces are presented. To fair a cubic spline curve or a bicubic spline surface with abnormal portions, the algorithms (automatically or interactively) identify the ‘bad’ data points and replace them with new points produced by minimizing the strain energy of the new curve or surface. The proposed algorithms are more general than the existing algorithms in that the new algorithms can adjust more than one ‘bad’ data point in each modification step and they include the existing algorithms [Computer-Aided Design 15(5) (1983) 288–293; 28 (1996) 59–66] as special cases. Test results of the new algorithms are included.

B -spline curves and surfaces viewed as digital filters

Abstract In this paper, we show that B-spline curves and surfaces can be viewed as digital filters. Viewing B-spline problems as digital filters allows one to predict some properties of the generated curves and surfaces. We find that even-order B-splines and odd-order B-splines behave differently when used in curve and surface interpolation. Even-order B-splines generate smoother curves and surfaces than do odd-order B-splines.

Energy and B-spline interproximation

In this paper, we study B-spline curve interproximation with different energy forms and parametrization techniques, and present an interproximation scheme for B-spline surfaces. It shows that the energy form has a much bigger impact on the generated curve than the parametrization technique. With the same energy form, different parametrization techniques generate relatively small difference on the corresponding curves. With the same parametrization technique, however, different energy forms make significant difference on the shape and smoothness of the resulting curves. Furthermore, interproximating B-spline curves generated by minimizing approximated energy forms are far from being good approximations to the optimal curves. They tend to generate flatter regions and sharper turns than curves generated by minimizing the exact energy form. The interproximation scheme for surfaces is aimed at generating a smooth surface to interpolate a grid of data which could either be a point or a region. This is achieved by minimizing a strain energy based on squared principal curvatures for bicubic B-spline surfaces. The surface interproximation process is also studied with different energy forms and parametrization techniques. The test results of the surface interproximation process also show the same conclusion as the curve interproximation process.

Estimating subdivision depths for rational curves and surfaces

An algorithm to estimate subdivision depths for rational curves and surfaces is presented. The subdivision depth is not estimated for the given curve/surface directly. The algorithm computes a subdivision depth for the polynomial curve/surface of which the given rational curve/surface is the image under the standard perspective projection. This subdivision depth, however, guarantees the required flatness of the given curve/surface after the subdivision. This work has applications in surface rendering, surface/surface intersection, and mesh generation.

On the Rates of Approximation of Bernstein Type Operators

Asymptotic behavior of two Bernstein-type operators is studied in this paper. In the first case, the rate of convergence of a Bernstein operator for a bounded function f is studied at points x where f(x+) and f(x-) exist. In the second case, the rate of convergence of a Szasz operator for a function f whose derivative is of bounded variation is studied at points x where f(x+) and f(x-) exist. Estimates of the rate of convergence are obtained for both cases and the estimates are the best possible for continuous points.

Removing local irregularities of NURBS surfaces by modifying highlight lines

The highlight line model is a powerful tool in assessing the quality of a surface. A method to remove local irregularities of NURBS surface by modifying its highlight lines is presented. The method is intuitive and suitable for real-time interactive design. It allows a designer to remove local irregularities of a NURBS surface by a simple operation to smooth the highlight lines. Modification of the control points of the surface is achieved by solving a system of linear equations. Test results are included.

Loop subdivision surface based progressive interpolation

A new method for constructing interpolating Loop subdivision surfaces is presented. The new method is an extension of the progressive interpolation technique for B-splines. Given a triangular mesh M, the idea is to iteratively upgrade the vertices of M to generate a new control mesh

A method for determining knots in parametric curve interpolation

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Interproximation : interpolation and approximation using cubic spline curves

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Subdivision Depth Computation for Catmull-Clark Subdivision Surfaces

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