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## Über dieses Buch

1 Aims and Features of This Book The contents of t. his book were originally planned t. o be included in a book en­ titled Geometric lIIodeling and CAD/CAM to be written by M. Hosaka and F. Kimura, but since the draft. of my part of the book was finished much earlier than Kimura's, we decided to publish this part separately at first. In it, geometrically oriented basic methods and tools used for analysis and synthesis of curves and surfaces used in CAD/CAM, various expressions and manipulations of free-form surface patches and their connection, interference as well as their qualit. y eval­ uation are treated. They are important elements and procedures of geometric models. And construction and utilization of geometric models which include free-form surfaces are explained in the application examples, in which the meth­ ods and the techniques described in this book were used. In the succeeding book which Kimura is to write, advanced topics such as data structures of geometric models, non-manifold models, geometric inference as well as tolerance problems and product models, process planning and so on are to be included. Conse­ quently, the title of this book is changed to Modeling of Curves and Surfaces in CAD/CAM. Features of this book are the following. Though there are excellent text books in the same field such as G. Farin's Curves and Surfaces for CAD /CAM[l] and C. M.

## Inhaltsverzeichnis

### 1. Excerpts from Vector and Matrix Theory

Abstract
In treating geometric objects or quantities, we use vector and matrix notations and operations throughout this book to simplify their description and also to make the reader’s geometric understanding easier. We show in this chapter only the essence of vector and matrix operations with their application examples, which will be referred to from various places in this book.
Mamoru Hosaka

### 2. Coordinate Transformations and Displacements

Abstract
In CAD and CAM, objects to be designed and manufactured have to be modelled in the computer. In dealing with such models, geometric relations among the objects have to be described and their calculations systematically performed. Vector and matrix operations are very useful in these tasks. Among their applications, coordinate transformation and the movements of solid bodies are the fundamental tools in treating 3D objects mathematically. We explain them in this chapter. A position vector p of a point is denoted by p (1) to indicate that it is seen from the coordinate system C1. The same is applied for its components:
$${p^{(1)}} = {(x,y,z)^{(1)}}.$$
(2.1)
When this point is to be seen from the coordinate system C2, it must be written in the same from as eq. (2.1):
$${p^{(2)}} = {(x,y,z)^{(2)}}.$$
(2.2)
We have to establish the relation between p(1) and p(2). When a body moves to other location changing its orientation, a point p(1) of the body moves to q(1) seen form the same coordinate system. We have to also establish their relation.
Mamoru Hosaka

### 3. Lines, Planes and Polyhedra

Abstract
In advanced CAD, its software has to process information on the models of 3D objects as though a person treats real ones. In construction and processing of the models, their information must be sufficient and have no contradiction, and their processing methods must be robust, efficient and flexible, because the result is directly connected to real processes of physical objects. From this point of view, a polyhedron is a fundamental object to be modelled in the computer. Its information structure and processing methods are clear, and numerical approximation enters least compared with those of other 3D objects. Polygons and polyhedra or wire-frames are also useful in describing objects with free-form surfaces, because free-form curves and surfaces are defined by their control points (refer to Chap. 9). So we explain in this chapter their basic equations as well as their geometric and topological properties and their interference.
Mamoru Hosaka

Abstract
Conics and quadrics are frequently used in various parts of shapes in engineering products. Though they are only one degree higher than lines and planes, their expressive capability is far greater than the latter. Since their geometric and algebraic properties have been fully investigated theoretically, we can use them with confidence. Knowledge of them is necessary not only for their appropriate use, but also for estimating characteristics of the shape around a point on a free- form surface and understanding various techniques of analyzing surface problems.
Mamoru Hosaka

### 5. Theory of Curves

Abstract
Curves are not only the fundamental geometric elements, but also they give us various information on shapes. In CAD/CAM, free-form curves have to be mathematically defined or extracted so that their properties can be controlled and evaluated by the computer. Accordingly, knowledge of the fundamentals of the differential geometry of curves is required to apply geometry to practical problems and it also gives the concepts and methods needed to understand the theory of surfaces which is explained in the next two chapters [1]. The curves are usually expressed in parametric forms, and arc length of the curve is used for the parameter in theoretical treatments because of its simplicity of expression, but for practical uses the parameter is changed from arc length 5 to a more manageable variable parameter t which monotonically increases with arc length. When treating interference problems of curves or surfaces, the curves expressed in implicit forms are sometimes convenient in numerical calculations or theoretical analysis. Therefore, we introduce a method of converting from a parametric form to an implicit one in the last section.
Mamoru Hosaka

### 6. Basic Theory of Surfaces

Abstract
Quadric surfaces, especially bodies of revolution such as spheres, cylinders, cones, or also ruled surfaces have been widely used in engineering products. This is because their geometric properties are easily specified by their designers and the shapes are easily manufactured by machines. On the other hand, free-form surfaces have only been used in special cases and their design and manufacture have required special talent and skill. But in recent years their uses have increased because of strong demands for high performance and aesthetic quality in engineering products. And computers have been used for their design and manufacture. Accordingly, various mathematical expressions of free-forms for engineering use have been developed.
Mamoru Hosaka

### 7. Advanced Applications of Theory of Surfaces

Abstract
Special topics on the geometric properties of surfaces are treated in this chapter. They are developed from the fundamental theory of surfaces which has been explained in the previous chapter. Knowledge of these topics is useful for treating free-form surfaces in advanced problems. First we discuss the umbilics and lines of curvature. On a free-form surface, there are points and regions which have the special characters inherent to its shape, and whose locations do not depend on the coordinate system adopted. The lines of curvature make orthogonal nets on a surface, and the pattern they form exhibits inherent features of the surface. The umbilics are singular points, or curves or regions seeing from the lines of curvature. On the free-form surface the umbilics appear more frequently than our expectation. There are other curves on the surface which depend not only on its inherent features, but also on its orientation with respect to its observers or its environments or its surface physical properties. These curves are useful for describing or evaluating the objects from engineering and aesthetic criteria.
Mamoru Hosaka

### 8. Curves Through Given Points, Interpolation and Extrapolation

Abstract
Methods of constructing a curve which passes through all the points of a given sequence are described. In CAD, input data for a curve are values of a point sequence supplied by automatic measuring instruments or values from digitizers used by a person or supplied from other processes. Since those data contain errors of various sorts, sometimes smoothing procedures such as methods of least squares or appropriate filtering procedures are required.
Mamoru Hosaka

### 9. Bézier Curves and Control Points

Abstract
There are various expressions for describing a shape, but the number of practical ones is limited. A polynomial expression in parametric form with control points is most useful in most applications, for it is not only mathematically simple, but also it has favorable properties for engineering design; its expression does not depend on coordinate systems, geometrical properties are easily grasped and manipulated by control points, unwanted waviness usually does not appear in construction and modification of shapes, and long curves and large areas can be designed by connecting short or small segments.
Mamoru Hosaka

### 10. Connection of Bézier Curves and Relation to Spline Polygons

Abstract
In design of free-form shapes, use of low-degree curves is desirable rather than adopting higher-degree ones or other complex expressions, so long as the objective of the design is attained by connecting low-degree curves. Curves of low degree are easy in analysis, synthesis and calculation of their geometry and also in detailed control of designing shapes, though there are some problems with their connections.
Mamoru Hosaka

### 11. Connection of Bézier Patches and Geometry of Spline Polygons and Nets

Abstract
In Sect. 11.2 of this chapter, we treat surfaces which consist of Bézier patches connecting C (n-1)in all directions. Techniques for obtaining B polygons from an S polygon are extended to those for generating nets of Bézier control points from a given spline net. For simplicity of mathematical manipulation and programming, we adopt a tensor product technique for producing Bézier nets. Connection in G (2) and insertion of vertices of an S net are also included in the first section. In the next three sections, we explain the theoretical aspects of the S polygon, which are the bases of the practical methods in the previous chapter.
Mamoru Hosaka

### 12. Rational Bézier and Spline Expressions

Abstract
Bézier expressions for curves and surfaces are easy to understand and have favorable features for use in CAD and CAM. But they cannot exactly express conics and quadrics such as circles and spheres. By extending the expressions from polynomials to rational forms, we can provide them not only with the ability to express conics and quadrics, but also with possibly increased freedom for design [28][29][32].
Mamoru Hosaka

### 13. Non-regular Connections of Four-Sided Patches and Roundings of Corners

Abstract
Usually in design of free-form surfaces, four-sided patches are connected to synthesize required shapes. For this purpose, S nets can be used advantageously because of their simplicity and the ease of shape control with the automatic adjustment of defined continuity along the boundaries of the patches.
Mamoru Hosaka

### 14. Connections of Patches by Blending

Abstract
Generally, when we have to define a surface patch, its degree of freedom is not always equal to that of given external conditions. If the latter is greater than the former, there must be some relations among the given conditions, or some methods must be provided to increase the degree of freedom of the patch to satisfy the given conditions.
Mamoru Hosaka

### 15. Triangular Surface Patches and Their Connection

Abstract
In Chap. 13, we treated non-four-sided patches, each of which was synthesized from four-sided Bézier patches, and described their uses in special regions of shapes. In this chapter, first we explain a triangular Bézier patch defined by linear interpolations in barycentric coordinates, whose expressions in operator form are similar to that of Bézier patches and their natural extension to rational form. Then we treat methods of connection of triangular patches. Since there is no simple equivalent of a spline net as in the four-sided patch connection [10][19], we cannot control a local shape of connected triangular patches easily. But there are number of connection patterns, which are explained [12].
Mamoru Hosaka

### 16. Surface Intersections

Abstract
Interference calculation of surfaces is an important basic procedure in CAD/CAM technology. It appears in various processes such as set operations in solid modelling, surface modelling, cutter path calculation for NC machining, and calculations in robot motion. Efficiency and robustness of its solutions are required especially for those of free-form surfaces, because the degree of intersection curves becomes so high that their analytical methods are generally hopeless and the numerical methods become unstable where the normal vectors of interfering surfaces on their intersecting curves become nearly parallel.
Mamoru Hosaka

### 17. Applications of the Theories in Industry

Abstract
Various theories and methods explained in the previous chapters were originally developed to meet the practical requirements of Japanese industries. In this chapter their application examples are described: one is style design and stamping die design and manufacture in the integrated CAD/CAM system of a large motor car company. The other is also style design, of small audio devices in a large electronic appliance company.
Mamoru Hosaka

### Backmatter

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