Higher Geometry

Author: Zhong Ji / Tang Sulan / Ye Muxiu (editors)
Publisher:
Publish Date: 2005-02-01
Features: This book can be used as a textbook for the advanced geometry course in undergraduate mathematics programs at higher education institutions. The purpose of this book is to briefly introduce the basic knowledge, theories, and methods of projective geometry, aiming to help readers develop geometric spatial concepts, understand Klein's transformation group viewpoint, clarify the intrinsic connections and fundamental differences between projective geometry, affine geometry, and Euclidean geometry, improve their ability to solve geometric problems, and lay a solid foundation for further study in modern mathematics. Additionally, the book briefly introduces the preliminary knowledge of n-dimensional projective space and projective spaces over different base fields (such as the field of real numbers, the field of complex numbers, and finite fields), enabling readers to further understand the concept of abstract spaces and serving as a bridge to modern mathematical knowledge. The book consistently adheres to Klein's transformation group viewpoint, with content focusing on various transformations, including 1-dimensional projective transformations, perspective transformations, and involutions, direct transformations, polar transformations, etc. It also establishes the groups of projective transformations, affine transformations, similarity transformations, and orthogonal transformations, respectively. Each group corresponds to a specific geometry, and the relationships between them are revealed through the relationships of transformation groups. In the process of discussing transformations, some important concepts in projective geometry are introduced. The coordinate method is the primary method used in this book. It successively establishes 1-dimensional, 2-dimensional, 3-dimensional projective coordinate systems and coordinate transformations, primarily using homogeneous coordinates. For affine geometry and Euclidean geometry, non-homogeneous coordinates are used instead. The book does not adopt an axiomatic foundation; it introduces a few axioms at the beginning to reveal the basic characteristics of projective planes and also serve as a basis for proving some theorems. Except for the introduction of infinite elements and projective coordinate systems at the beginning, the entire book's discussion is logically rigorous. The cross-ratio is a fundamental projective invariant and holds an important position in projective geometry, hence the book provides a detailed introduction. A second-order curve can have different definitions. The book defines it using polar transformations, primarily to highlight the role of polar transformations. For the various properties of second-order curves, the book selects only the more important ones to list as theorems, while the general ones are presented as examples and exercises. The methods used in this book are primarily algebraic, so the application of various vector operations and the relationships between transformations are fundamental knowledge that must be mastered. On this basis, it becomes easier to solve and prove problems. Since synthetic methods have their own convenient and skillful characteristics, some theorems are proven using both methods for reference. In practical problem-solving, only one method is needed. When learning advanced geometry, readers should not only understand the content of each chapter but also pay attention to the overall theory, including main concepts, main theorems, main methods, and systematic structure. Only in this way can they achieve a deeper understanding of advanced geometry and better grasp and remember the material. The exercises in this book are selected to help readers understand, master the theory, and methods, and some of them are relatively complex. However, with detailed solutions provided, readers will find them easy to comprehend. As for the problems, difficult ones are avoided. Solutions or hints are provided with the problems for reference.