Geometric transformations for modeling of curves and surfaces

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: self-similar curves in the Euclidean spaces; space-like Bézier curves and surfaces in the MInkowski 3-space; rational curves and surfaces in the 3-sphere; rational ruled surfaces in the Euclidean 3-space. 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 an affine 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. One-to-one 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 tetrahedra is obtained by the use of the quaternion algebra. The second part presents shape curvatures of Frenet curves in any dimension. Shape curvatures are differential-geometric invariants which determine locally a Frenet curve up to a direct similarity. Self-similar 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 space-like Bézier curves and surfaces in the Minkowski 3-space are proved. Rational curves and surfaces in the 3-sphere are obtained. These curves and surfaces are pre-images of Bézier curves and surfaces under stereographic projection. The last fourth part is devoted to nonlinear transformations of the projective and Euclidean spaces. The three-dimensional 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 3-space. A special involution of the six-dimensional 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.