Showing papers in "Computer Aided Geometric Design in 1989"

TL;DR: This paper reviews the work of de Casteljau, de Boor, and Ramshaw in an attempt to clarify the basic principles that underly the polar approach and proposes a consistent system of nomenclature as a possible standard.

TL;DR: This note corrects errors found in some equations of [Sarraga '87] after publication and reformulated in more suitable forms.

TL;DR: It is shown that if a curve has a proper parameterization then the necessary and sufficient condition for the existence of cusps is given by the vanishing of the first derivative vector and a simple algorithm is presented to compute the Proper parameterization of a polynomial curve.

TL;DR: A new approach to the theory of B-splines is presented, based on the principle that polynomials of degree p are equivalent to symmetric p-affine maps, and the basic algorithms for the theory are derived with proofs that are shorter than the currently existing ones.

TL;DR: The frames introduced by Klok can be computed without solving any differential equations, and this paper shows that this ability to computed frames without solving differential equations is feasible.

TL;DR: Conditions of geometric continuity at a vertex where a number of patches meet are investigated and applications to the modelling of closed surfaces and the fitting or triangular patches into a geometric continuous patch complex are sketched.

TL;DR: A polygonal patch is defined to fill an n -sided hole within a C 2 parametric continuous rectangular patch complex to form a continuous patch complex.

TL;DR: Fat arcs in a new bouding region for planar Bezier curves demonstrate that, in the limit, fat arcs for the two halves of a subdivided curve total 1/8 the area of the fat arc of the undivided curve.

TL;DR: An effective method for approximate conversion of spline surfaces using mainly geometric continuity conditions, parameter transformations and nonlinear optimization techniques is introduced.

TL;DR: An efficient algorithm for computing the Bezier points of a generalized cubic β-spline curve is presented and the connection with multiple knot insertion is shown.

TL;DR: This work describes a data structure suitable for representing arbitrary algebraic space curves and gives an algorithm for constructing such a representation for the space curve formed by two intersecting algebraic surfaces.

TL;DR: For data satisfying certain generalized convexity conditions, the existence and uniqueness of a piecewise quadratic curvature continuous parametric spline interpolant in R2 is proved and the numerical construction of the interpolant can be carried out efficiently.

TL;DR: It is demonstrated that a B-spline subdivision patch can be fitted into a general n-sided area of a bicubic surface with at least tangent plane continuity on the boundary.

TL;DR: This paper presents an algorithm to approximate non-uniform rational B-spline curves with non- uniform integral B- Spline curves of lower degree and forms the basis of a similar rational B -spline surface approximation procedure.

TL;DR: The mathematical basis of the entire algorithm is given in the form of a concise procedure, and the crucial steps of patch/patch intersection and region classification are presented in detail from the perspective of an effective implementation.

TL;DR: It is shown that the equations of curves and surfaces in the power basis may be affected by a very important loss of significant digits on the polynomials coefficient; this is due to the required conversion matrices which are ill-conditioned.

TL;DR: A method for finding all zeroes of aspline function using the interval Newton method, which uses an explicit bound on the derivative of a spline function and operates on the whole spline, without breaking it down into polynomial pieces.

TL;DR: In this article, a new classification of geometric continuity is presented, based on the simultaneous application of the concept of "contact of order r" to curves (or surfaces) and to their tangent surfaces and generalized tangent surface; this implies projective invariance.

TL;DR: A sufficient linear convexity condition for bivariate spline (piecewise polynomial) functions over triangulations implies that certain surface subdivision algorithms necessarily produce surfaces which have the wrong shape.

TL;DR: Weighted v-spline curves are used here with a new technique that forms a parametric surface interpolant to a given rectangular set of control points that can be interactively changed by manipulating the control points or by modifying the interval or point tension parameters.

TL;DR: The objective of this paper is to provide an efficient algorithm for the approximation of nonuniform rational B-spline surface patches with non-uniform integral B- Spline patches of lower degree.

TL;DR: The fundamental proposition of blossoming, which expresses the B-spline control points as the blossom values of certain polynomial segments of the spline, is shown to be a simple consequence of the de Boor-Fix formula.

TL;DR: The piecewise C 0 continuous parametric polynomial approximation composed of linear, quadratic and cubic segments, finds application in problems of solid geometric modeling, in surface or data contouring, and in representing offset curves needed for NC tool path generation.

TL;DR: The de Boor-Fix functionals are used to find B-spline representations for certain non-uniform C m interpolating functions with coefficients given in terms of interpolated data.

TL;DR: An application of techniques for free-form shape generation to ab initio design of a yacht hull surface is presented, based on some rough model data of the 12-Meter yacht RAINBOW which in 1934 successfully defended the America's Cup for the United States.

TL;DR: A symbolic algorithm is presented which decomposes a general algebraic set in n dimensions into its irreducible components and it is proposed that this may be a useful tool in CAGD systems since the degrees of the resulting irredUCible components may be smaller than the degree of the originalgebraic set.

TL;DR: A polynomial basis is presented for the quadratic and cubic curves whose convex hulls contain the curve, but with a size of about 77% and 42% of the respective Bezier convex Hulls, and how to subdivide these curves.

TL;DR: A method of surface definition and smoothing using a set of three-dimensional skeletal-lines, which represent the essential features of a surface and can be individually smoothed using the first divided difference, is given.

TL;DR: Group-theoretic methods are used to obtain a system of partial differential equations for the most general curve form invariant under a given group of transformations, which yields a complete classification of both the Euclidean invariant non-adaptive curves, and a well defined sub-class of the Euclid's invariant adaptive curves in two dimensions.

TL;DR: A geometric proof of a Greville-like formula for γ-spline functions is given and it is shown that this principle can be carried over to rationalγ- and γτ-splines functions.