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Conics: A Unified Approach to the Curves of Apollonius
Conics are curves that are obtained by slicing a cone with a plane. They include ellipses, parabolas and hyperbolas, which have many applications in mathematics, physics and engineering. However, the traditional way of studying conics separately can obscure their common features and properties. In this article, we will review a book that offers a different perspective on conics, using projective geometry, complex numbers and other tools to reveal their hidden beauty and elegance.
The book is called Conics, written by Keith Kendig and published by the Mathematical Association of America in 2005. It is part of the Dolciani Mathematical Expositions series, which aims to present mathematical topics in an accessible and engaging way for a broad audience. The book is written in the form of dialogues among three characters: Philosopher, Teacher and Student. Philosopher is a curious outsider who wants to understand conics in a unified way, Teacher is a knowledgeable mathematician who guides Philosopher through various concepts and proofs, and Student is an enthusiastic learner who asks questions and provides examples.
The book covers a wide range of topics related to conics, such as their geometric definition, their algebraic representation, their classification and properties, their relation to circles and spheres, their applications to celestial mechanics and electromagnetism, and their generalizations to higher dimensions. The book also explores some historical aspects of conics, such as the works of Apollonius, Pascal, Newton and others. The book uses many illustrations and diagrams to help the reader visualize the concepts and arguments. The book also includes exercises and hints for further reading at the end of each chapter.
The main theme of the book is to show that conics can be viewed as projections of circles on a plane. This idea allows one to use complex numbers and projective geometry to simplify the treatment of conics and to reveal their symmetries and invariants. For example, one can show that any conic can be transformed into a circle by a suitable change of coordinates, that any two conics are projectively equivalent (i.e., they can be mapped to each other by a projective transformation), that any five points on a conic determine it uniquely (the so-called five-point theorem), that any four conics have ten common points (the so-called ten-point theorem), and so on. The book also shows how these results can be extended to higher dimensions using homogeneous coordinates and matrices.
The book is suitable for readers who have some background in calculus, linear algebra and geometry. It does not assume any prior knowledge of complex numbers or projective geometry, but it introduces them gradually and explains them clearly. The book is also suitable for readers who are interested in the applications of conics to physics and engineering, as it provides many examples and problems that illustrate their relevance. The book is not intended to be a comprehensive or rigorous textbook on conics, but rather a stimulating and enjoyable introduction that invites the reader to explore further.
In conclusion, Conics is a book that offers a fresh and fascinating look at one of the oldest topics in mathematics. It shows how conics can be understood in a unified way using modern tools and techniques. It also shows how conics have influenced many branches of mathematics and science throughout history. It is a book that will appeal to anyone who loves geometry and wants to discover its hidden treasures. ec8f644aee