Chun-Gang Zhu

NUMERICAL SOLUTION OF BURGERS–FISHER EQUATION BY CUBIC B-SPLINE QUASI-INTERPOLATION

Abstract In this paper, numerical solution of the Burgers’ equation is presented based on the cubic B-spline quasi-interpolation. At first the cubic B-spline quasi-interpolation is introduced. Moreover, the numerical scheme is presented, by using the derivative of the quasi-interpolation to approximate the spatial derivative of the dependent variable and a low order forward difference to approximate the time derivative of the dependent variable. The accuracy of the proposed method is demonstrated by some test problems. The numerical results are found in good agreement with exact solutions. The advantage of the resulting scheme is that the algorithm is very simple, so it is very easy to implement.

Parametric representation of a surface pencil with a common line of curvature

Line of curvature on a surface plays an important role in practical applications. A curve on a surface is a line of curvature if its tangents are always in the direction of the principal curvature. By utilizing the Frenet frame, the surface pencil can be expressed as a linear combination of the components of the local frame. With this parametric representation, we derive the necessary and sufficient condition for the given curve to be the line of curvature on the surface. Moreover, the necessary and sufficient condition for the given curve to satisfy the line of curvature and the geodesic requirements is also analyzed.

An approach for designing a developable surface through a given line of curvature

Developable surface and line of curvature play an important role in geometric design and surface analysis. This paper proposes a new method to construct a developable surface possessing a given curve as the line of curvature of it. We analyze the necessary and sufficient conditions when the resulting developable surface is a cylinder, cone or tangent surface. Finally, we illustrate the convenience and efficiency of this method by some representative examples.

Functional splines with different degrees of smoothness and their applications

Implicit curves and surfaces are extensively used in interpolation, approximation and blending. [Li J, Hoschek J, Hartmann E. G^n^-^1-functional splines for interpolation and approximation of curves, surfaces and solids. Computer Aided Geometric Design 1990;7:209-20] presented a functional method for constructing G^n^-^1 curves and surfaces which are called functional splines. In this paper, functional splines with different degrees of smoothness are presented and applied to some typical problems.

Injectivity of NURBS curves

The injectivity of NURBS curve implies the curve has no self-intersection. In this paper, we propose a geometric condition on the control polygon which guarantees the NURBS curve to be injective for all possible choices of positive weights. The proof is based on the degree elevation algorithm and toric degeneration theory of NURBS curve.

Multivariate spline approximation of the signed distance function

The signed distance function can effectively support many geometry processing tasks such as decimates, smoothing and shape reconstruction since it provides efficient access to distance estimates. In this paper, we present an adaptive method to approximate the signed distance function of a smooth curve by using polynomial splines over type-2 triangulation. The trimmed offsets are also studied.

The number of regular control surfaces of toric patch

Abstract Through a rational map, a toric patch is defined associated to a lattice polygon, which is the convex of a given finite integer lattice points set A . The classical rational Bezier curves, rational triangular and tensor-product patches are special cases of toric patches. One of the geometric meanings of toric patch is that the limiting of the patch is its regular control surface, when all weights tend to infinity. In this paper, we study the number of regular decompositions of A , and the relationship between regular decompositions and the corresponding secondary polytope. What is more, we indicate that the number of regular control surfaces of toric patch associated with A is equal to the number of regular decompositions of A .

Progressive iterative approximation for regularized least square bivariate B-spline surface fitting

Abstract Recently, the use of progressive iterative approximation (PIA) to fit data points has received a deal of attention benefitting from its simplicity, flexibility, and generality. In this paper, we present a novel progressive iterative approximation for regularized least square bivariate B-spline surface fitting (RLSPIA). RLSPIA extends the PIA property of univariate NTP (normalized totally positive) bases to linear dependent non-tensor product bivariate B-spline bases, which leads to a lower order fitting result than common tensor product B-spline surface. During each iteration, the weights for generating fairing updating surface are obtained by solving an energy minimization problem with box constraints iteratively. Furthermore, an accelerating term is introduced to speed up the convergence rate of RLSPIA, which is comparable favourably with the theoretical optimal one. Several examples are provided to illustrate the efficiency and effectiveness of the proposed method.

G1 continuity of four pieces of developable surfaces with Bézier boundaries

Abstract For potential applications in geometric design and manufacturing of material, the G 1 connection of many pieces of developable surfaces is an important issue. In this paper, by using de Casteljau algorithm we study the G 1 connection of four pieces of developable surfaces with Bezier boundary curves. We convert these surfaces to tensor form firstly, then characterize the constrains of the control points of the surfaces need to satisfy when G 1 connecting them. This method can also be extended to the case when the developable surfaces possess Bezier boundary curves with different degrees.

Geometric conditions of non-self-intersecting NURBS surfaces

NURBS surface is very useful in geometric modeling, animation, image morphing and deformation. Constructing non-self-intersecting (injective) NURBS surfaces is an important process in surface and solid modeling. In this paper, the injective conditions of tensor product NURBS surface are studied, based on the geometric positions of control points, which are equivalent to the surface to be non-self-intersecting for all positive weights. Finally, some representative examples are provided.