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Bézier curve

About: Bézier curve is a research topic. Over the lifetime, 1557 publications have been published within this topic receiving 24404 citations. The topic is also known as: Bezier curve.

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25 Aug 1995
TL;DR: This chapter discusses the construction of B-spline Curves and Surfaces using Bezier Curves, as well as five Fundamental Geometric Algorithms, and their application to Curve Interpolation.
Abstract: One Curve and Surface Basics.- 1.1 Implicit and Parametric Forms.- 1.2 Power Basis Form of a Curve.- 1.3 Bezier Curves.- 1.4 Rational Bezier Curves.- 1.5 Tensor Product Surfaces.- Exercises.- Two B-Spline Basis Functions.- 2.1 Introduction.- 2.2 Definition and Properties of B-spline Basis Functions.- 2.3 Derivatives of B-spline Basis Functions.- 2.4 Further Properties of the Basis Functions.- 2.5 Computational Algorithms.- Exercises.- Three B-spline Curves and Surfaces.- 3.1 Introduction.- 3.2 The Definition and Properties of B-spline Curves.- 3.3 The Derivatives of a B-spline Curve.- 3.4 Definition and Properties of B-spline Surfaces.- 3.5 Derivatives of a B-spline Surface.- Exercises.- Four Rational B-spline Curves and Surfaces.- 4.1 Introduction.- 4.2 Definition and Properties of NURBS Curves.- 4.3 Derivatives of a NURBS Curve.- 4.4 Definition and Properties of NURBS Surfaces.- 4.5 Derivatives of a NURBS Surface.- Exercises.- Five Fundamental Geometric Algorithms.- 5.1 Introduction.- 5.2 Knot Insertion.- 5.3 Knot Refinement.- 5.4 Knot Removal.- 5.5 Degree Elevation.- 5.6 Degree Reduction.- Exercises.- Six Advanced Geometric Algorithms.- 6.1 Point Inversion and Projection for Curves and Surfaces.- 6.2 Surface Tangent Vector Inversion.- 6.3 Transformations and Projections of Curves and Surfaces.- 6.4 Reparameterization of NURBS Curves and Surfaces.- 6.5 Curve and Surface Reversal.- 6.6 Conversion Between B-spline and Piecewise Power Basis Forms.- Exercises.- Seven Conics and Circles.- 7.1 Introduction.- 7.2 Various Forms for Representing Conics.- 7.3 The Quadratic Rational Bezier Arc.- 7.4 Infinite Control Points.- 7.5 Construction of Circles.- 7.6 Construction of Conies.- 7.7 Conic Type Classification and Form Conversion.- 7.8 Higher Order Circles.- Exercises.- Eight Construction of Common Surfaces.- 8.1 Introduction.- 8.2 Bilinear Surfaces.- 8.3 The General Cylinder.- 8.4 The Ruled Surface.- 8.5 The Surface of Revolution.- 8.6 Nonuniform Scaling of Surfaces.- 8.7 A Three-sided Spherical Surface.- Nine Curve and Surface Fitting.- 9.1 Introduction.- 9.2 Global Interpolation.- 9.2.1 Global Curve Interpolation to Point Data.- 9.2.2 Global Curve Interpolation with End Derivatives Specified.- 9.2.3 Cubic Spline Curve Interpolation.- 9.2.4 Global Curve Interpolation with First Derivatives Specified.- 9.2.5 Global Surface Interpolation.- 9.3 Local Interpolation.- 9.3.1 Local Curve Interpolation Preliminaries.- 9.3.2 Local Parabolic Curve Interpolation.- 9.3.3 Local Rational Quadratic Curve Interpolation.- 9.3.4 Local Cubic Curve Interpolation.- 9.3.5 Local Bicubic Surface Interpolation.- 9.4 Global Approximation.- 9.4.1 Least Squares Curve Approximation.- 9.4.2 Weighted and Constrained Least Squares Curve Fitting.- 9.4.3 Least Squares Surface Approximation.- 9.4.4 Approximation to Within a Specified Accuracy.- 9.5 Local Approximation.- 9.5.1 Local Rational Quadratic Curve Approximation.- 9.5.2 Local Nonrational Cubic Curve Approximation.- Exercises.- Ten Advanced Surface Construction Techniques.- 10.1 Introduction.- 10.2 Swung Surfaces.- 10.3 Skinned Surfaces.- 10.4 Swept Surfaces.- 10.5 Interpolation of a Bidirectional Curve Network.- 10.6 Coons Surfaces.- Eleven Shape Modification Tools.- 11.1 Introduction.- 11.2 Control Point Repositioning.- 11.3 Weight Modification.- 11.3.1 Modification of One Curve Weight.- 11.3.2 Modification of Two Neighboring Curve Weights.- 11.3.3 Modification of One Surface Weight.- 11.4 Shape Operators.- 11.4.1 Warping.- 11.4.2 Flattening.- 11.4.3 Bending.- 11.5 Constraint-based Curve and Surface Shaping.- 11.5.1 Constraint-based Curve Modification.- 11.5.2 Constraint-based Surface Modification.- Twelve Standards and Data Exchange.- 12.1 Introduction.- 12.2 Knot Vectors.- 12.3 Nurbs Within the Standards.- 12.3.1 IGES.- 12.3.2 STEP.- 12.3.3 PHIGS.- 12.4 Data Exchange to and from a NURBS System.- Thirteen B-spline Programming Concepts.- 13.1 Introduction.- 13.2 Data Types and Portability.- 13.3 Data Structures.- 13.4 Memory Allocation.- 13.5 Error Control.- 13.6 Utility Routines.- 13.7 Arithmetic Routines.- 13.8 Example Programs.- 13.9 Additional Structures.- 13.10 System Structure.- References.

4,552 citations

01 Nov 1987
TL;DR: This paper presents an Explicity Formulation for Cubic Beta-splines, a simple Approximation technique for Uniform Cubic B-spline Surfaces, and discusses its applications in Rendering and Evaluation and simulation.
Abstract: 1 Introduction 2 Preliminaries 3 Hermite and Cubic Spline Interpolation 4 A Simple Approximation Technique - Uniform Cubic B-splines 5 Splines in a More General Setting 6 The One-Sided Basis 7 Divided Differences 8 General B-splines 9 B-spline Properties 10 Bezier Curves 11. Knot Insertion 12 The Oslo Algorithm 13 Parametric vs. Geometric Continuity 14 Uniformly-Shaped Beta-spline Surfaces 15 Geometric Continuity, Reparametrization, and the Chain Rule 16 Continuously-Shaped Beta-splines 17 An Explicity Formulation for Cubic Beta-splines 18 Discretely-Shaped Beta-splines 19 B-spline Representations for Beta-splines 20 Rendering and Evaluation 21 Selected Applications

1,292 citations

01 Aug 1989
TL;DR: A simple, unified, algorithmic approach to change of basis procedures in computer aided geometric design, R.A. Froyland, et al wonderful triangle.
Abstract: Symmetrizing multiaffine polynomials, P.J. Barry norm estimates for inverses of distance matrices, B.J.C. Baxter numerical treatment of surface-surface-intersection and contouring, K.-H. Brakhage modelling closed surfaces - a comparison of existing methods, P. Brunet and A. Vinacua a new characterization of plane elastica, G. Brunnett POLynomials, POLar forms, and interPOL-ation, Casteljau pyramid patches provide potential polynomial paradigms, A.S. Cavaretta and C.A. Micchelli implicitizing rational surfaces with base points by applying perturbations and the factors of zero theorem, E.-W. Chionh and R.C. Goldman wavelets and multiscale interpolation, C.K. Chui and X. Shi a curve intersection algorithm with processing of singular cases - introduction of a clipping technique, M. Daniel best approximation of parametric curves by splines, W.L.F. Degen an approximately G1 cubic surface interpolant, T. DeRose and S. Mann on the G2 continuity of piecewise parametric surfaces, W. Du and F.J.M. Schmitt stationary and non-stationary binary subdivision schemes, N. Dyn and D. Levin MQ-curves are curves in tension, M. Eck offset approximation improvement by control point perturbation, G. Elber and E. Cohen curves and surfaces in geometrical optics, R.T. Farouki and J.-C.A. Chastang evaluation and properties of a derivative of a NURBS curve, M.S. Floater hybrid cubic Bezier triangle patches, T.A. Foley and K. Opitz modelling geological structures using splines, L.A. Froyland, et al wonderful triangle - a simple, unified, algorithmic approach to change of basis procedures in computer aided geometric design, R.N. Goldman an arbitrary mesh network scheme using rational splines, J.A. Gregory and P.K. Yuen Bezier curves and surface patches on quadrics, J. Hoschek monotonicity preserving interplation using C2 rational cubic Bezier curves, M.K. Ismail on piecewise quadratic G2 approximation and interplation, J. Kozak and M. Lokar non-affine blossoms and subdivision for Q-splines, R. Kulkarni. Part contents.

253 citations

Proceedings ArticleDOI
01 Aug 1987
TL;DR: Three-dimensionalscan-conversion algorithms, that scan-convert 3D parametric objects into their discrete voxelmap representation within a Cubic Frame Buffer (CFB), are presented and emply third-order forward difference techniques.
Abstract: Three-dimensional (3D) scan-conversion algorithms, that scan-convert 3D parametric objects into their discrete voxelmap representation within a Cubic Frame Buffer (CFB), are presented. The parametric objects that are studied include Bezier form of cubic parametric curves, bicubic parametric surface patches, and tricubic parametric volumes. The converted objects in discrete 3D space maintain pre-defined application-dependent connectivity and fidelity requirements.The algorithms introduced here emply third-order forward difference techniques. Efficient versions of the algorithms based on first-order decision mechanisms, which employ only integer arithmetic, are also discussed. All algorithms are incremental and use only simple operations inside the inner algorithm loops. They perform scan-conversion with computational complexity which is linear in the number of voxels written to the CFB. All the algorithms have been implemented as part of the CUBE Architecture, which is a voxel-based system for 3D graphics.

202 citations

Journal ArticleDOI
Charles Loop1, James F. Blinn1
01 Jul 2005
TL;DR: A method for resolution independent rendering of paths and bounded regions, defined by quadratic and cubic spline curves, that leverages the parallelism of programmable graphics hardware to achieve high performance is presented.
Abstract: We present a method for resolution independent rendering of paths and bounded regions, defined by quadratic and cubic spline curves, that leverages the parallelism of programmable graphics hardware to achieve high performance. A simple implicit equation for a parametric curve is found in a space that can be thought of as an analog to texture space. The image of a curve's Bezier control points are found in this space and assigned to the control points as texture coordinates. When the triangle(s) corresponding to the Bezier curve control hull are rendered, a pixel shader program evaluates the implicit equation for a pixel's interpolated texture coordinates to determine an inside/outside test for the curve. We extend our technique to handle anti-aliasing of boundaries. We also construct a vector image from mosaics of triangulated Bezier control points and show how to deform such images to create resolution independent texture on three dimensional objects.

201 citations

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