Representation of quadric primitives by rational polynomials

doi:10.1016/0167-8396(85)90019-6

Computer Aided Geometric Design

Volume 2, Issues 1–3, September 1985, Pages 151-155

Presented at Oberwolfach 13 November 1984

https://doi.org/10.1016/0167-8396(85)90019-6 Get rights and content

Abstract

A geometric construction for rational conic segments and quadric patches is presented here that provides elegent recipes for avoiding singularities and can easily be used to define quadric patches of different types.

References (8)

W. Boehm
On cubics: a survey
Computer Graphics and Image Processing
(1982)
A.R. Forrest
The twisted cubic curve: a CAGD approach
Computer Aided Design
(1980)
A.A. Ball
CONSURF I
Computer Aided Design
(1974)
G. Farin
Algorithms for rational Bézier curves
Computer Aided Design
(1983)

There are more references available in the full text version of this article.

Cited by (11)

A unified geometric model for designing elastic pivots
2008, Precision Engineering
Citation Excerpt :
This is achieved with quadratic rational Bézier curves, which can exactly represent conic sections via the ratio of polynomial functions. Bézier curves have become a common approach for modeling 2D and 3D curves in the field of computer aided design (CAD) [28–30], but they have not been previously used in modeling the shapes of elastic pivots. A thorough explanation of these curves is available in the book authored by Piegl [31].
Elastic pivots are commonly used in precision machines and instruments as bearings for guiding small displacements or rotations. They provide accurate and frictionless motion with little hysteresis. Elastic pivots are frequently designed with either straight beams or notches shaped as conic sections with circular, elliptical, parabolic, or hyperbolic profiles. The compliance of a pivot is typically predicted by analytical expressions derived from implicit equations for the pivot’s profile. This article describes a more general geometric model suited to representing the entire family of conic section shapes using quadratic rational Bézier curves. This general representation of the geometry is suitable for computer aided design and analysis, especially with finite element methods that yield compliances and stress distributions. The utility of this formulation is demonstrated by optimizing an elastic pivot to meet compliance and stress requirements.
Necessary and sufficient conditions for rational quartic representation of conic sections
2007, Journal of Computational and Applied Mathematics
Conic section is one of the geometric elements most commonly used for shape expression and mechanical accessory cartography. A rational quadratic Bézier curve is just a conic section. It cannot represent an elliptic segment whose center angle is not less than $π$ . However, conics represented in rational quartic format when compared to rational quadratic format, enjoy better properties such as being able to represent conics up to $2 π$ (but not including $2 π$ ) without resorting to negative weights and possessing better parameterization. Therefore, it is actually worth studying the necessary and sufficient conditions for the rational quartic Bézier representation of conics. This paper attributes the rational quartic conic sections to two special kinds, that is, degree-reducible and improperly parameterized; on this basis, the necessary and sufficient conditions for the rational quartic Bézier representation of conics are derived. They are divided into two parts: Bézier control points and weights. These conditions can be used to judge whether a rational quartic Bézier curve is a conic section; or for a given conic section, present positions of the control points and values of the weights of the conic section in form of a rational quartic Bézier curve. Many examples are given to show the use of our results.
Weighted radial displacement: a geometric look at Bezier conics and quadrics
2000, Computer Aided Geometric Design
We describe a simple geometric construction for Bézier conics and quadrics, based on a tool called Weighted Radial Displacement (WRD). The shape of a rational Bézier curve or surface is modified via WRD by choosing an arbitrary point $O$ and displacing the control points along radial directions through $O$ , changing simultaneously the weights. To construct a conic through $O$ , take an arbitrary segment representing a curve of degree n=1, degree raise it to n=2 and apply a WRD. Analogously, if a degree-elevated triangle is modified using a WRD, we get a quadric through $O$ . Any quadratic Bézier patch on a nondegenerate quadric, which defines a stereographic projection, can be obtained through this method. We present a practical algorithm to detect such quadratic Bézier patches lying on quadrics. Bézier patches on degenerate quadrics are also derived via a WRD.
The rational cubic Bézier representation of conics
1992, Computer Aided Geometric Design
The rational cubic Bézier curve is a very useful tool in CAGD. It incorporates both conic sections and parametric cubic curves as special cases, so its advantage is that one can deal with curves of these two kinds in one computer procedure. In this paper, the necessary and sufficient conditions for representing conics by the rational cubic Bézier form in proper parametrization are investigated; these conditions can be divided into two parts: one for weights and the other for Bézier vertices.
Curve and surface constructions using rational B-splines
1987, Computer-Aided Design
The sphere as a rational Bézier surface
1986, Computer Aided Geometric Design

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^∗: Presently at the Technische Universität Braunschweig, F.R.G.

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Representation of quadric primitives by rational polynomials

Abstract

Computer Graphics and Image Processing

Computer Aided Design

Computer Aided Design

Computer Aided Design