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1988 | Book

Curves and Surfaces in Computer Aided Geometric Design

Author: Professor Fujio Yamaguchi

Publisher: Springer Berlin Heidelberg

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Table of Contents

Frontmatter
0. Mathematical Description of Shape Information
Abstract
Shapes of industrial products can be roughly classified into those that consist of combinations of elementary geometrical surfaces and those that cannot be expressed in terms of elementary surfaces, but vary in a complicated manner. Many examples of the former type are found among parts of machines. Most machine parts have elementary geometrical shapes such as planes and cylinders. This is because, as long as a more complicated shape is not functionally required, simpler shapes are far simpler from the point of view of production. In this book, these shapes are called Type 1 shapes. Meanwhile, the shapes of such objects as automobile bodies, telephone receivers, ship hulls and electric vacuum cleaners contain many curved surfaces that vary freely in a complicated manner. Let us call these Type 2 shapes.
Fujio Yamaguchi
1. Basic Theory of Curves and Surfaces
Abstract
Before a computer can perform processing relating to a shape, a mathematical description of that shape must be provided on the computer’s memory. Such a description should preserve as many of the properties of the actual object shape as possible. From the point of view of computer processing the following properties are particularly important.
Fujio Yamaguchi
2. Lagrange Interpolation
Abstract
It is known that there exists one at most nth-order polynomial that connects the (n+1) points (x 0, f 0), (x 1, f 1),..., (x n, f n) having different abscissas 8).
Fujio Yamaguchi
3. Hermite Interpolation
Abstract
Hermite interpolation is a generalized form of Lagrange interpolation. Whereas Lagrange interpolation interpolates only between values of a function f 0, f 1, ..., f n at different abscissas x 0, x 1, ..., x n , Hermite interpolation also interpolates between higher order derivatives (Fig. 3.1). The following discussion deals with Hermite interpolation of function values and slopes.
Fujio Yamaguchi
4. Spline Interpolation
Abstract
When a smooth curve passing through a specified sequence of points is generated, use of the shape of a curve produced by a long narrow elastic band such as a steel band has long been used in the design of, for example, ships and automobiles. An elastic band used for such a purpose is called a spline. The spline can be made to assume the shape of a smooth curve passing through the specified points by attaching a suitable number of weights, called weights or ducks (Fig. 4.1).
Fujio Yamaguchi
5. The Bernstein Approximation
Abstract
As explained in Chap. 3, curves and surfaces based on Hermite interpolation position vectors of 2 points Q 0and Q 1 and the tangent vectors at those points \( {\dot Q_0} \) and \( {\dot Q_1} \) (Chap. 3):
$$ P\left( t \right) = \left[ {\begin{array}{*{20}{c}} {{t^3}}&{{t^2}}&t&1 \end{array}} \right]\left[ {\begin{array}{*{20}{r}} 2&{ - 2}&1&1 \\ { - 3}&3&{ - 2}&{ - 1} \\ 0&0&1&0 \\ 1&0&0&0 \end{array}} \right]\left[ {\begin{array}{*{20}{l}} {{Q_0}} \\ {{Q_1}} \\ {{{\dot Q}_0}} \\ {{{\dot Q}_1}} \end{array}} \right]. $$
(5.1)
Fujio Yamaguchi
6. The B-Spline Approximation
Abstract
If (n+1) ordered position vectors Q0, Q1, ..., Qn−1, Q n are given (Fig. 6.1), consider the (n−2) linear combinations:
$$ {P_i}(t) = {X_0}(t){Q_{i - 1}} + {X_1}(t){Q_i} + {X_2}(t){Q_{i + 1}} + {X_3}(t){Q_{i + 2}}(i = 1,2,...,n - 2)$$
(6.1)
each formed from four successive points. X0(t), X1(t), X2(t) and X3(t) are polynomials in the parameter t(0≦t≦1). P i (t) is a curve segment expressed in terms of the varying parameter. The condition for two neighboring curve segments P i (t) and Pi+1(t) to be continuous at the point corresponding to t=1 for the first segment and t=0 for the second, that is, for P i (1)= Pi+1(0) to hold for all Q i (j=i−1, i, ..., i+3), is:
$$ \left. {\begin{array}{*{20}{l}} {{X_0}(1) = {X_3}(0) = 0} \\ {{X_1}(1) = {X_0}(0)} \\ {{X_2}(1) = {X_1}(0)} \\ {{X_3}(1) = {X_2}(0)} \end{array}} \right\} ]$$
(6.2)
Fujio Yamaguchi
7. The Rational Polynomial Curves
Abstract
Conic section curves are in a mutual central projection relationship*). Consequently, arbitrary conic section curves can be derived by performing a suitable affine transformation and then a central projection on one conic section curve. For the initial conic section curve, let us use the simplest one to express, the parabola shown in Fig. 7.1:
$$ \left[ {\begin{array}{*{20}{c}} x&y&1 \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {{t^2}}&t&1 \end{array}} \right]. $$
Fujio Yamaguchi
Backmatter
Metadata
Title
Curves and Surfaces in Computer Aided Geometric Design
Author
Professor Fujio Yamaguchi
Copyright Year
1988
Publisher
Springer Berlin Heidelberg
Electronic ISBN
978-3-642-48952-5
Print ISBN
978-3-642-48954-9
DOI
https://doi.org/10.1007/978-3-642-48952-5