Local smooth surface interpolation: a classification

doi:10.1016/0167-8396(90)90030-U

Computer Aided Geometric Design

Volume 7, Issues 1–4, June 1990, Pages 191-195

https://doi.org/10.1016/0167-8396(90)90030-U Get rights and content

Abstract

A classification of algorithms for local smooth surface interpolation with piecewise polynomials is given.

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Polynomial spline interpolation of incompatible boundary conditions with a single degenerate surface
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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 $(2 (p + 2), 2 (p + 2))$ -degree polynomial spline surface with four degenerate corners to interpolate incompatible boundary conditions, which are represented by $p$ -degree polynomial spline curves with $G^{1}$ continuity. This method is based on symbolic operations and polynomial reparameterizations for polynomial splines, and without introducing any theoretical errors, it achieves $G^{1}$ continuity on the boundary except for the four corner points.
RAGS: Rational geometric splines for surfaces of arbitrary topology
2014, Computer Aided Geometric Design
A construction of spline spaces suitable for representing smooth parametric surfaces of arbitrary topological genus and arbitrary order of continuity is proposed. The obtained splines are a direct generalization of bivariate polynomial splines on planar partitions. They are defined as composite functions consisting of rational functions and are parametrized by a single parameter domain, which is a piecewise planar surface, such as a triangulation of a cloud of 3D points. The idea of the construction is to utilize linear rational transformations (or transition maps) to endow the piecewise planar surface with a particular $C^{\infty}$ -differentiable structure appropriate for defining rational splines.
Constructive G 1 connection of multiple freeform pipes in arbitrary poses
2013, Computer Aided Geometric Design
This paper expands the application scope of pipe connection to multiple pipes having freeform cross sections and arbitrary poses. Starting with a typical case — bifurcation, we first connect every two pipe ends of totally three using side patches. Inside them appear two holes. Usually the 6 sides of each hole are short (on the pipe edges) and long (on the side patch edges) alternately. Then in each hole draw three optimized geometric Hermite (OGH) curves connecting two long edges and roughly parallel to a short edge, thus splitting the hole into 4 regions: three 4-sided and a more regular 6-sided. The former are interpolated by three setback patches, while the latter by hole-filling patches using the well-known central split. As illustrated with several practical examples, the above three building blocks can build miscellaneous multiple pipe blending, guaranteed to be $G^{1}$ -continuous and twist compatible.
Constructing G 1 Bézier surfaces over a boundary curve network with T-junctions
2012, CAD Computer Aided Design
A T-junction occurs in a boundary curve network when one boundary curve ends in the middle of another. We show how to construct $G^{1}$ Bézier surfaces over a boundary curve network with T-junctions. By treating the two micro patches which meet at the edge forming the upright of the ‘T’ as a single macro patch, we reduce the problem to one of achieving continuity between this composite patch and the third patch which has the crossbar of the ‘T’ as an edge. Thus we avoid changes to the boundary network, or to any patches except those that meet at the T-junction. Also, we analyze the singularity of the $G^{1}$ continuity system with the T-junction, and give the constraint to make a consistent system using free variables of weight functions. This is the first method of surfacing the T-junction. We present examples and verify continuity by drawing reflection lines and checking angles.
G2B-spline interpolation to a closed mesh
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This paper focuses on interpolating vertices and normal vectors of a closed quad-dominant mesh ¹ $G^{2}$ -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 $G^{2}$ 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 $G^{2}$ 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 $G^{2}$ continuity condition matrices is also strictly proved to be $2 n - 3$ , and the method of tangent-vector estimation is improved to avoid petal-shaped patches in interpolating solids of revolution. Several examples demonstrate its feasibility.
A weighted bivariate blending rational interpolation based on function values
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Citation Excerpt :
Spline interpolation is a useful tool for designing curves and surfaces in computer aided geometric design, hence in the past decade, there has been considerable interest in spline interpolation. For instance, literature [1] gives a method that interpolating the spatial points by a uniform cubic B-spline, and literatures [2,3] realize the smooth interpolation for surfaces. But this kind of spline interpolation has one disadvantage that given the interpolating data, the interpolating function is unique.
This paper presents a new weighted bivariate blending rational spline interpolation based on function values. This spline interpolation has the following advantages: firstly, it can modify the shape of the interpolating surface by changing the parameters under the condition that the values of the interpolating nodes are fixed; secondly, the interpolating function is C¹-continuous for any positive parameters; thirdly, the interpolating function has a simple and explicit mathematical representation; fourthly, the interpolating function only depends on the values of the function being interpolated, so the computation is simple. In addition, this paper discusses some properties of the interpolating function, such as the bases of the interpolating function, the matrix representation, the bounded property, the error between the interpolating function and the function being interpolated.

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^☆: This research was supported by NSF DMS-8701275.

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Local smooth surface interpolation: a classification☆

Abstract

J. Approx. Theory

Computer Aided Design

Computer Aided Geometric Design

Computer Aided Geometric Design

Computer Aided Geometric Design

Computer Aided Geometric Design

Computer Aided Design

Computer Aided Geometric Design

Computer Aided Design

J. Approx. Theory

Computer Aided Geometric Design

Computer Aided Geometric Design

Hermite interpolation using real algebraic surfaces

Proc. 5th ACM Symposium on Computational Geometry

Essai de définition numérique des courbes et des surfaces expérimentales

B-form basics

Design of solids with free-form surfaces

Computer Graphics

Finite element stiffness metrices for the analysis of plate bending

Proc. 1st Conf. Matrix Methods in Struct. Mechanics

Surfaces for computer aided design of space forms