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We propose a specialized form of the curved-knot B-spline surface of Hayes [1982] that we call regular curved-knot spline surface. Unlike the original formulation where the knots of the first parametric coordinate can evolve arbitrarily with respect to the second coordinate, our formulation designs the knot functions as special curves that guarantee a monotonic blending of the knots corresponding to opposite surface boundaries. Furthermore, we demonstrate that local derivatives on the boundary can be described as an ordinary B-spline surface. The latter property allows for constructing smooth transitions between B-spline boundaries with different knot vectors. [ABSTRACT FROM AUTHOR]

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: 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]

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]