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.

https://www.springer.com/productFlyer_978-3-540-61545-3.pdf?SGWID=0-0-1297-1491853-0

The NURBS Book

The hdograph of a plane parametric curve r(t) = (x(t), y(t)f is the locus described by the first parametric derivative r' (t) = (x ' (t), y ' (t)) of that curve. A polynomial parametric curve is said to have a Pythagorean hodograph if there exists a polynomial a(t) such that x t 2 (t) + y " (t) " Pythagorean triple. " Although Pythagorean-hodograph curves have fewer degrees of freedom than general polynomial curves of the same degree, they exhibit remarkably attractive properties for practical use. For example, their arc length is expressible as a polynomial function of the parameter, and their offsets are rational curves. We present a sufficient-and-necessary algebraic characterization of the Pythagorean-hodograph property, analyze its geometric implications in terms of Bernstein-Bezier forms, and survey the useful attributes it entails in various applications. ~ ' (t) , i.e., (x' (t), y ' (t) , a (t)) form a 1. Introduction The representation of curves and surfaces in a form amenable to efficient, systematic computation is a basic issue in computer-aided design. Those representations Topyright 1990 by International Business Machines Corporation. Copying in printed form for private use is permitted without payment of royalty provided that (1) each reproduction is done without alteration and (2) the Joz~rnal reference and IBM copyright notice are included on the first page. The title and abstract, but no other portions, of this paper may be copied or distributed royalty free without further permission by computer-based and other information-service systems. Permission to republish any other 736 portion of this paper must be obtained from the Editor. that have won widespread acceptance in practical use are almost exclusively parametric formulations, based on (piecewise) polynomial functions (see [ 11 and references therein). Plane curve segments, for example, are usually defined in a form equivalent to " x(t) = aktk, k=O n y (t) = 1 bktk for t E [0, 11. k=O Such segments may be pieced together with various orders of continuity to form spline curves for smooth data interpolation; they are easily rendered by uniformly incrementing t and evaluating the polynomials (1); and algorithmic procedures are available for computing their intersections (see [ 2 ]). An immediate shortcoming of the form (1)-its inability to accommodate conic loci other than the parabola [3]-may be remedied by allowing the rational form r(t) = (X(t)/W(t), Y(t)/ W (t)) , where X(t), Y(t), and W(t) are polynomials. …

Pythagorean hodographs

Layered manufacturing (LM) is emerging as a new manufacturing technology that can enhance the scope of manufacturing. One of the essential tasks in LM is process planning. This paper defines, conceptualizes and reviews the literature in this emerging area. The paper concludes with future projections on the possible directions of research in this area.

A review of process planning techniques in layered manufacturing

https://www.cc.gatech.edu/~jarek/papers/Offsets.pdf

Offsetting operations in solid modelling

A fundamental NC-machining problem is the clearing of areas within specified boundaries from material. The paper presents an efficient algorithm for generating largely optimal tool paths to solve this pocket-machining problem for multiply connected planar pocket areas bounded by curvilinear boundaries. Using certain concepts of computational geometry, i.e. Voronoi diagrams and monotonic pouches, offsets of a pocket boundary are efficiently generated. Further, it is explained how the tool path can be optimized with respect to several criteria arising from technological requirements. The concepts presented have been implemented, and they form the basis of the pocketing package lark .

Pocket Machining Based on Contour-Parallel Tool Paths Generated by Means of Proximity Maps

An offset capability for planar curves and profiles in a solid modeler leads to the solution of some practical design problems.

Offsets of Two-Dimensional Profiles

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Offsets of Two-Dimensional Profiles