# Curves and surfaces in computer aided geometric design - Yamaguchi F.

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6.15 Curve Generation by Geometrical Processing................................320

6.16 Interpolation of a Sequence of Points with a Â-Spline Curve...............325

6.17 Matrix Expression of Â-Spline Curves ..................................... 327

6.18 Expression of the Functions C0 0(t), C0A(t), Cl 0(t) and Clfl(f)

by Â-Spline Functions...................................................... 329

6.19 General Â-Spline Surfaces.................................................334

References....................................................................... 336

7. The Rational Polynomial Curves............................................337

7.1 Derivation of Parametric Conic Section Curves ............................ 337

7.2 Classification of Conic Section Curves.................................... 340

7.3 Parabolas................................................................. 342

7.4 Circular Arc Formulas..................................................... 343

7.5 Cubic/Cubic Rational Polynomial Curves.................................... 346

7.6 T-Conic Curves............................................................ 347

References....................................................................... 350

Appendix A: Vector Expression of Simple Geometrical Relations.............. 351

Appendix B: Proofs of Formulas Relating to Â-Spline Functions..............356

Appendix Ñ: Effect of Multiple Knots on Â-Spline Functions....................... 360

Bibliography..................................................................... 366

Subject Index ................................................................... 375

On the Symbols Used in This Book

Among the symbols used in these books, several which require caution are discussed below.

(1) t is the standard symbol for a parameter representing a curve, and è and w for parameters representing a surface. In some places the length of a curve is used as a parameter in the discussion; it is s.

(2) Derivatives with respect to parameters other than s, such as t, are denoted by dots (•); derivatives with respect to s are denoted by primes ('). For example,

dt ’ ds

t, è and w can be regarded as time parameters in the motion of mass points. Since in physics it is customary for derivatives with respect to time to be denoted by dots, this convention has been followed in the present book.

(3) An ordinary coordinate vector is denoted by a slanted letter, for example P. A homogeneous coordinate vector is denoted by a vertical letter, such as P.

(4) In mathematical representations of curves and surfaces, a vector that is given for the purpose of definition is denoted by Q, and a defined curve or surface vector by P. For example in

P(t) = ß0>0 (t) Q0 + Hol(t) Qt + Huo (t) Q0 + H, ä (0 Qt,

the position vectors Q0, Ql and the tangent vectors Q0, are given, and the curve P(t) is defined.

0. Mathematical Description of Shape Information

0.1 Description and Transmission of Shape Information

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.

A designer draws his concept of a shape on paper and proceeds with the design while checking it against the shape that he has drawn. Sometimes during the design work it becomes necessary to build a physical model of the shape. In such a case, blueprints are prepared and given to a model builder. The final step in design is to prepare a set of blueprints on which are written all of the information needed to produce the item that has been designed. The designer must write all of the information needed to produce a 3-dimensional shape on 2-dimensional paper. In the case of a Type 1 shape, these 2-dimensional drawings are called three orthogonal views; in the case of a Type 2 shape they are frequently called curve diagrams.

It has always been difficult to express the considerable amount of information needed to describe a 3-dimensional shape on a limited number of 2-dimensional drawings. In the case of Type 1 shapes the task has been simplified by defining a number of conventions for drawing preparation which the designer can learn and, as he accumulates experience, learn to transmit information effectively to other people on 2-dimensional diagrams. Since Type 1 shapes are geometrically regular and familiar in everyday life, it can be expected that a person who reads the diagram will be able to infer the correct shape from what is on the paper, so that the transmission of shape information proceeds relatively smoothly. Consequently, as far as Type 1 shapes are concerned, if it can be assumed that people will look at the diagrams, this method is very effective for describing and transmitting a considerable amount of 3-dimensional shape information on a limited number of drawings. And since the drawings are in standard formats, they are easy to be manipulated.

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