A Dynamic Programming Approach to Curves and Surfaces for by Ron Goldman

By Ron Goldman

Pyramid Algorithms offers a different method of figuring out, studying, and computing the commonest polynomial and spline curve and floor schemes utilized in computer-aided geometric layout, utilizing a dynamic programming strategy in accordance with recursive pyramids.
The recursive pyramid method deals the distinctive good thing about revealing the full constitution of algorithms, in addition to relationships among them, at a look. This book-the just one equipped round this approach-is sure to switch how you take into consideration CAGD and how you practice it, and all it calls for is a easy history in calculus and linear algebra, and easy programming skills.
* Written through one of many world's most outstanding CAGD researchers
* Designed to be used as either a certified 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
* is determined by an simply understood notation, and concludes each one part with either sensible and theoretical workouts that increase and intricate upon the dialogue within the text
* Foreword via Professor Helmut Pottmann, Vienna collage of know-how

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

Example text

P,m(t) denote a polynomial curve of degree p + 1 that interpolates the points Po ..... Pp,Pm at the parameters t o ..... tp,t m. p_l,m(t). tm - t p tm - t p 3. n (t) even if we leave the control points fixed. 4. Let P(t) be the Lagrange interpolating polynomial for the control points PO..... Pn and nodes t o ..... t n. Form a new Lagrange interpolating curve Q(t) by replacing each node tk by the node rk = ark + b for some fixed constants a > 0 and b. Show that changing all the nodes in this way has no affect on the shape of the interpolating curve.

We still must address one more preliminary issue before we can proceed to our main theme. We need to decide how we shall represent curves and surfaces inside our ambient spaces. Four types of representations for curves and surfaces are common in computer graphics and geometric design: explicit, implicit, parametric, and procedural. Here we shall look briefly at each of these alternatives and then settle on one particular form to use throughout this text. When you first studied analytic geometry, you used rectangular coordinates and considered equations of the form y = f(x).

T n. n(t) - . n(tk) = Pk k = 0 ..... n. n (t) is a polynomial of degree n. The parameter values to ..... t n at which the interpolation occurs are called nodes, and the points PO ..... 6). n (t) changes even if we leave the control points fixed (see Exercise 3). Exercises 1. n (tk) = Pk. 2. p,m(t) denote a polynomial curve of degree p + 1 that interpolates the points Po ..... Pp,Pm at the parameters t o ..... tp,t m. p_l,m(t). tm - t p tm - t p 3. n (t) even if we leave the control points fixed.

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