Read e-book online Pyramid Algorithms: A Dynamic Programming Approach to Curves PDF

By Ron Goldman

Pyramid Algorithms offers a different method of figuring out, examining, and computing the most typical polynomial and spline curve and floor schemes utilized in computer-aided geometric layout, utilising a dynamic programming approach in accordance with recursive pyramids.The recursive pyramid method deals the specific good thing about revealing the whole constitution of algorithms, in addition to relationships among them, at a look. This book-the just one outfitted round this approach-is bound to switch how you take into consideration CAGD and how you practice it, and all it calls for is a easy historical past in calculus and linear algebra, and straightforward programming abilities. * Written through one of many world's most outstanding CAGD researchers* Designed to be used as either a qualified reference and a textbook, and addressed to machine scientists, engineers, mathematicians, theoreticians, and scholars alike* contains chapters on Bezier curves and surfaces, B-splines, blossoming, and multi-sided Bezier patches* depends on an simply understood notation, and concludes every one part with either sensible and theoretical routines that improve and intricate upon the dialogue within the textual content* Foreword by way of Professor Helmut Pottmann, Vienna collage of expertise

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Read or Download Pyramid Algorithms: A Dynamic Programming Approach to Curves and Surfaces for Geometric Modeling PDF

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Extra resources for Pyramid Algorithms: A Dynamic Programming Approach to Curves and Surfaces for Geometric Modeling

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Conclude that the affine transformations on affine space are equivalent to the mass-preserving linear transformation on Grassmann space. 4. Let L be a linear transformation on Grassmann space. a. Show that if L is nonsingular, then L induces a unique, well-defined transformation L* on projective space so that the following diagram commutes: Grassmann space L ---"- Grassmann space Projection Projection Projective space L* Projective space b. - Projective space A transformation L* on projective space induced in this fashion by a linear transformation L on Grassmann space is called a projective transfor- mation.

J2c~,0] = [ q , . . , c ~ , 0 ] , since in projective space we are dealing with equivalence classes of points in Grassmann space. A point in projective space that corresponds to a point in affine space has a nonzero final coordinate. Thus, just as in Grassmann space, we can recover the rectangular coordinates (c 1..... Cn) of an affine point from the homogeneous coordinates [mc 1..... m c n , m ] ot the corresponding projective point by dividing the 1 30 C H A PT E R 1 Introduction: Foundations first n homogeneous coordinates ( m c 1.

We shall see in subsequent chapters that Lagrange polynomials, as well as B6zier and B-spline curves and surfaces, are defined in precisely this fashion. 15) are said to be translation invariant. 1 Ambient Spaces 25 face by a vector v, we need only translate each control point by v. This result follows from the fact that the blending functions form a partition of unity, since for such blending functions n n n P(t) + v = ~,Bk(t)P k + EBk(t)v - ~,Bk(t)(P k + v) k=O k=O k=O f(s,t) + v - EijBij(s,t)Pij + EijBij(s,t)v - E,ijBij(s,t)(Pij + v) .

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