Georgi H. Georgiev Geometric Transformations for Modeling of Curves and Surfaces ISBN: 9783844006919 Preis: 48,80 € / 97,60 SFR 

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Onetoone correspondences of this shape space onto itself are studied. Analogously, the shape space of similar tetrahedra is the set of equivalence classes of tetrahedra with respect to the group of space direct similarities. The representation of the shape space of similar tetrahedral is obtained by the use of the quaternion algebra. The second part presents shape curvatures of Frenet curves in any dimension. Shape curvatures are differentialgeometric invariants which determine locally a Frenet curve up to a direct similarity. Selfsimilar curves, i.e. curves with constant shape curvatures, are completely classified. In the third part, Bézier curves and surfaces are considered and studied from different points of view. Formulae for shape curvatures of quadratic and cubic Bézier curves are derived in an explicit form. The matrix representation of cubic Bézier curves is used to express the change of their shape curvatures under an arbitrary affine transformation. Sufficient conditions for spacelike Bézier curves and surfaces in the Minkowski 3space are proved. Rational curves and surfaces in the 3sphere are obtained. These curves and surfaces are preimages of Bézier curves and surfaces under stereographic projection. The last fourth part is devoted to nonlinear transformations of the projective and Euclidean spaces. The threedimensional geometric algebra is used for an investigation of plane quadratic transformations. Rational ruled surfaces of any degree passing through two lines are expressed by their parametric and implicit equations. These rational surfaces are images of planes under special birational transformations of the Euclidean 3space. A special involution of the sixdimensional complex projective space is also examined. The book is intended for graduate students and researchers working in areas as mathematics and computer aided geometric design. 
Publisher´s description: Geometric transformations, curves and surfaces appear in many branches of mathematics. They also play a significant role in modern areas of computer science, such as computer aided geometric design, computer graphics and computer vision. The aim of this book is not only to demonstrate that geometric transformations, curves and surfaces are closely related, but also to give direct constructions for curves and surfaces applied in geometric modelling. Several classes of such curves and surfaces are presented: selfsimilar curves in the Euclidean spaces; spacelike Bézier curves and surfaces in the Minkowski 3space; rational curves and surfaces in the 3sphere; rational ruled surfaces in the Euclidean 3space. The book is divided into four parts. In the first part, shape spaces of similar triangles and tetrahedra are discussed. A direct similarity of the Euclidean space is anffine transformation which preserves the angles and the orientation. The shape space of similar triangles can be considered as the set of equivalence classes of triangles with respect to the group of plane direct similarities.

Quelle: Zentralblatt MATH 1276 1  
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