Sectional-curvature preserving skinning surfaces

doi:10.1016/0167-8396(95)00048-8

Computer Aided Geometric Design

Volume 13, Issue 7, October 1996, Pages 601-619

https://doi.org/10.1016/0167-8396(95)00048-8 Get rights and content

Abstract

In this work we develop a method for constructing sectional-curvature preserving (scp) C²-continuous surfaces, which interpolate point-sets lying on parallel planes. The working function space consists of skinning surfaces, whose skeletal lines and blending functions are polynomial splines of nonuniform degree. The asymptotic behaviour of these surfaces and their sectional curvature, as the segment degrees tend to infinity globally or semilocally, is thoroughly studied. This study provides sufficient geometrical conditions on the given data ensuring that, it the segment degrees increase semilocally then the surface will eventually become scp in the corresponding parameter subdomain. Based on the obtained asymptotic results, an automatic algorithm for constructing scp interpolatory surfaces is developed and numerically tested.

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Cited by (11)

Local T-spline surface skinning with shape preservation
2018, CAD Computer Aided Design
Citation Excerpt :
There are many studies [15–24] on surface fairing, which will be considered in further work.
Surface skinning is a surface generation method that uses a set of given cross-sectional curves, and it is widely used in free-form surface design. In the B-spline surface skinning, the given B-spline curves should be compatible, that is, the curves should have the same degree and knot sequence. While making the curves compatible, lots of control vertices are generated. Although T-spline surface skinning methods have been introduced to reduce the number of control vertices, the T-spline skinning method that was proposed by Nasri et al. (2012) can generate a wiggled surface when the given B-spline curves are not sufficiently compatible. The intermediate cross sections that were introduced for T-spline surface skinning cannot preserve the shape of the given B-spline curves if the adjacent B-spline curves do not have sufficient common knots, and it can cause wiggles on the surface. In this paper, we analyze this issue and suggest a modified method to remove the wiggles on the skinned T-spline surface. Furthermore, we propose an algorithm for shape preservation of the surface. Our approach is verified by suggesting some examples compared to the Nasri et al. (2012)’s method.
G1-smooth branching surface construction from cross sections
2007, CAD Computer Aided Design
This paper proposes a framework for constructing $G^{1}$ surfaces that interpolate data points on parallel cross sections, consisting of simple disjoined and non-nested contours, the number of which may vary from plane to plane. Using appropriately estimated cross tangent vectors at the given points, we split the problem into a sequence of local Hermite problems, each of which can be one of the following three types: “one-to-one”, “one-to-many” or “many-to-many”.
The solution of the “one-to-many” branching problem, where one contour on the $i$ -plane is to be connected to $M_{}$ contours on the $(i + 1)$ -plane, is based on combining skinning with trimming and hole filling. More specifically, we first construct a $C^{1}$ surrounding curve of all $M_{}$ contours on the $(i + 1)$ -plane. Next, we build the so-called surrounding surface that skins the $i$ -plane contour with the $(i + 1)$ -plane surrounding curve, and trim suitably along parts of the surrounding curve that connect contours. The resulting multi-sided hole is covered with quadrilateral Gordon–Coons patches that possess $G^{1}$ continuity. For this purpose, we develop a hole-filling technique that employs shape-preserving guide curves and is able to preserve data symmetries. The “many-to-many” problem is handled by combining the “one-to-many” methodology with a zone-separation technique, that achieves splitting the “many-to-many” problem into two “one-to-many” problems. The methodology, implemented as a C++ Rhino v3.0 plug-in, is illustrated via two synthetic data sets and in the context of two realistic design examples. Finally, the paper concludes with discussing ongoing work towards improving the robustness and the applicability of the method regarding the surrounding curve construction step.
Class A Bézier curves
2006, Computer Aided Geometric Design
The surfacing of multiview 3D drawings via lofting and occlusion reasoning
2017, Proceedings - 30th IEEE Conference on Computer Vision and Pattern Recognition, CVPR 2017
The Surfacing of Multiview 3D Drawings via Lofting and Occlusion Reasoning
2017, arXiv
C1 Hermite shape preserving polynomial splines in R3
2012, 3D Research

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Computer Aided Geometric Design

Abstract

References (21)

Surfaces with rescribed curvature I

Computer Aided Geometric Design

Convexity preserving interpolation and Powell-Sabin elements

Computer Aided Geometric Design

The convexity of Bernstein polynomials over triangles

J. Approx. Theory

A local scheme for bivariate co-monotone interpolation

Computer Aided Geometric Design

Interpolatory convexity-preserving subdivision schemes for curves and surfaces

Computer-Aided Design

On the convexity of functional splines

Computer Aided Geometric Design

Convexity-preserving interpolatory parametric splines of non-uniform polynomial degree

Computer Aided Geometric Design

Convexity-preserving interpolatory subdivision

Computer Aided Geometric Design

Skinning techniques for interactive B-spline surface interpolation

Computer-Aided Design

The direct modification of surface curvatures in car body design

Cited by (11)

Local T-spline surface skinning with shape preservation

G<sup>1</sup>-smooth branching surface construction from cross sections

Class A Bézier curves

The surfacing of multiview 3D drawings via lofting and occlusion reasoning

The Surfacing of Multiview 3D Drawings via Lofting and Occlusion Reasoning

C<sup>1</sup> Hermite shape preserving polynomial splines in R<sup>3</sup>