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## Inhaltsverzeichnis

### 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

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