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Abstract: Coons’ construction generates a surface patch that interpolates four groups of specified boundary curves and the corresponding cross-boundary derivative curves. This constructive method is simple and widely used in computer aided design. However, at the corner points, it requires compatibility of the boundary conditions, which is usually difficult to satisfy in practice. In order to handle the incompatible case where the normal directions respectively indicated by two adjacent boundaries do not agree with each other at the common corner point, we utilize the property of degenerate parametric surfaces that the normal directions can converge to multiple values at degenerate points, and therefore the local degenerate geometry can satisfy conflicting conditions simultaneously. Following this idea, we use a single patch of -degree polynomial spline surface with four degenerate corners to interpolate incompatible boundary conditions, which are represented by -degree polynomial spline curves with continuity. This method is based on symbolic operations and polynomial reparameterizations for polynomial splines, and without introducing any theoretical errors, it achieves continuity on the boundary except for the four corner points. [Copyright &y& Elsevier]

Abstract: This paper proposes a method to construct a B-spline surface that interpolates the specified four groups of boundary derivative curves in the B-spline form. The discontinuity can be bounded by an arbitrary geometric invariant as the tolerance. The method first handles the six types of the compatibility problems by continuity-preserving reparameterization, knot-insertion and local control-point tuning. The transformed boundary conditions are then parametrically compatible, so the Coons strategy can be applied to construct the final interpolant. Not only can it be used in the reliable geometric modeling, but the approach also can be applied to many other algorithms that require compatibility guarantee. [Copyright &y& Elsevier]

A method for generating an interpolation closed G1 arc spline on a given closed point set is presented. For the odd case, i.e. when the number of the given points is odd, this paper disproves the traditional opinion that there is only one closed G1 arc spline interpolating the given points. In fact, the number of the resultant closed G1 arc splines fulfilling the interpolation condition for the odd case is exactly two. We provide an evaluation method based on the arc length as well such that the choice between those two arc splines is made automatically. For the even case, i.e. when the number of the given points is even, the points are automatically moved based on weight functions such that the interpolation condition for generating closed G1 arc splines is satisfied, and that the adjustment is small. And then, the G1 arc spline is constructed such that the radii of the arcs in the spline are close to each other. Examples are given to illustrate the method. [Copyright &y& Elsevier]

The purpose of this paper is to provide yet another solution to a fundamental problem in computer aided geometric design, i.e., constructing a smooth curve satisfying given endpoint (position and tangent) conditions. A new class of curves, called optimized geometric Hermite (OGH) curves, is introduced. An OGH curve is defined by optimizing the magnitudes of the endpoint tangent vectors in the Hermite interpolation process so that the strain energy of the curve is a minimum. An OGH curve is not only mathematically smooth, i.e., with minimum strain energy, but also geometrically smooth, i.e., loop-, cusp- and fold-free if the geometric smoothness conditions and the tangent direction preserving conditions on the tangent angles are satisfied. If the given tangent vectors do not satisfy the tangent angle constraints, one can use a 2-segment or a 3-segment composite optimized geometric Hermite (COH) curve to meet the requirements. Two techniques for constructing 2-segment COH curves and five techniques for constructing 3-segment COH curves are presented. These techniques ensure automatic satisfaction of the tangent angle constraints for each OGH segment and, consequently, mathematical and geometric smoothness of each segment of the curve. The presented OGH and COH curves, combined with symmetry-based extension schemes, cover tangent angles of all possible cases. The new method has been compared with the high-accuracy Hermite interpolation method by de Boor et al. and the Pythagorean-hodograph (PH) curves by Farouki et al. While the other two methods both would generate unpleasant shapes in some cases, the new method generates satisfactory shapes in all the cases. [Copyright &y& Elsevier]

A G 2 interpolation method to a closed mesh was proposed by Shi et al. in [1]. This note points out that their method has two theoretical defects, which may lead to the failure of G 2 continuity. We fix these defects and correct several formula mistakes. [ABSTRACT FROM AUTHOR]

Abstract: This paper focuses on interpolating vertices and normal vectors of a closed quad-dominant mesh 1 [1] A mesh is quad-dominant if it contains quad facets as its majority. The quantity of quad facets is much more than the quantity of triangular and other multi-sided facets. -continuously using regular Coons B-spline surfaces, which are popular in industrial CAD/CAM systems. We first decompose all non-quadrangular facets into quadrilaterals. The tangential and second-order derivative vectors are then estimated on each vertex of the quads. A least-square adjustment algorithm based on the homogeneous form of continuity condition is applied to achieve curvature continuity. Afterwards, the boundary curves, the first- and the second-order cross-boundary derivative curves are constructed fulfilling continuity and compatibility conditions. Coons B-spline patches are finally generated using these curves as boundary conditions. In this paper, the upper bound of the rank of continuity condition matrices is also strictly proved to be , and the method of tangent-vector estimation is improved to avoid petal-shaped patches in interpolating solids of revolution. Several examples demonstrate its feasibility. [ABSTRACT FROM AUTHOR]

Abstract: Dynamic partial differential equation (PDE) parametric curves which can be expressed as a coupled system of two hyperbolic equations are developed. In curve design, dynamic PDE parametric curves can be modified intuitively and are more flexible than ordinary differential equation (ODE) curves. The calculation of dynamic PDE parametric curves must recur to numerical methods and a three-level finite difference scheme is proposed. Approximation and stability properties for the scheme are proved and convergence property is derived. An example of interpolating PDE curves is presented as an application of dynamic PDE parametric curves. [Copyright &y& Elsevier]

Abstract: A new algorithm for reducing control points in lofted surface interpolation to rows of data points is presented in this paper. The key step of surface lofting is to obtain a set of compatible B-spline curves interpolating each row. Given a set of points and their parameterization, a necessary and sufficient condition is proposed to determine the existence of interpolating B-spline curves defined on a given knot vector. Based on this condition, we first properly construct a common knot vector that guarantees the existence of interpolating B-spline curves to each row of points. Then we calculate a set of interpolating B-spline curves defined on the common knot vector by energy minimization. Using this method, fewer control points are employed while maintaining a visually pleasing shape of the lofted surface. Several experimental results demonstrate the usability and quality of the proposed method. [Copyright &y& Elsevier]