- Paperback: 156 pages
- Publisher: Birkhäuser; 2004 edition (July 23, 2004)
- Language: English
- ISBN-10: 0817641610
- ISBN-13: 978-0817641610
- Product Dimensions: 6.1 x 0.4 x 9.2 inches
- Shipping Weight: 8.8 ounces (View shipping rates and policies)
- Average Customer Review: 3 customer reviews
- Amazon Best Sellers Rank: #1,534,585 in Books (See Top 100 in Books)
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Lines and Curves: A Practical Geometry Handbook 2004th Edition
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"This book was originally published in Russian in 1978 and translated into English by A. Kundu for a 1980 edition. The present edition is based on that translation. It consists of brief expository sections followed by problems that are non-trivial and will be new to most American readers.... There is no book like this one, and it is well worth buying." ―MAA Reviews
"If only some fo the ideas of this book would slip in teaching at school the pupils would not lament for boring mathematics. And if you [are looking] for a fasicinating, exciting, but by no means trivial approach to the beginnings of the theory of algebraic curves buy this book!" ―Monashefte für Mathematik
"An engaging presentation, meant to attract young talent to the study of elementary geometry, of several topics in plane Euclidean geometry that share a certain `dynamic' quality: geometric loci, many of which are trajectories, being defined in terms of motions, minima and maxima, conic sections" ―Zentralblatt Math
From the Back Cover
"Lines and Curves" is a unique adventure in the world of geometry. Originally written in Russian and used in the Gelfand Correspondence School, this work has since become a classic: unlike standard textbooks that use the subject primarily to introduce axiomatic reasoning through formal geometric proofs, "Lines and Curves" maintains mathematical rigor, but also strikes a balance between creative storytelling and surprising examples of geometric properties. This newly revised and expanded edition includes more than 200 theoretical and practical problems in which formal geometry provides simple and elegant insight, and the book points the reader toward important areas of modern mathematics.
One of the key strengths of the text is its reinterpretation of geometry in the context of motion, whereby curves are realized as trajectories of moving points instead of as stationary configurations in the plane. This novel approach, rooted in physics and kinematics, yields unusually intuitive and straightforward proofs of many otherwise difficult results. The geometrical properties of paths traced by moving points, the sets of points satisfying given geometric constraints, and questions of maxima and minima are all emphasized; therefore, "Lines and Curves" is well positioned for companion use with software packages like "The Geometer’s Sketchpad®," and it can serve as a guidebook for engineers. Its deeper, interdisciplinary treatment is ideal for more theoretical readers, and the development from first principles makes the book accessible to undergraduates, advanced high school students, teachers, and puzzle enthusiasts alike. A wide audience will profit from this clear and diverse examination of the subject.
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The style is engaging and entertaining. I invite anyone to read the Introduction (available free from Amazon) to get a taste of the material. To keep my comments concrete, consider Chapter 2, "The Alphabet": no fewer than six different interpretations of a straight line as a geometric locus are explained (and will consistently be used throughout the rest of the book). The same goes for the circle, for which at least four interpretations are given. Other conics (ellipses, parabolas, hyperbolas) are treated similarly in Chapter 6.
A further remarkable feature is the authors' willingness to employ analytic geometry at crucial places where resorting to purely synthetic methods would be cumbersome and not particularly illuminating. The best illustration of this is the "Theorem on the Squares of the Distances" in Chapter 2 (What is the locus of all the points in the plane whose weighted sum of squares of distances to given fixed points is equal to a constant?) Another instance is to be found already in section 0.2 of the Introduction (read it from the links above!) Exercise for the reader of this review: solve 0.2 using no analytic methods, but rather by modifying the argument of 0.1 and using the fact that the compression/dilation by a factor of b/a of a circle of radius a with respect to a diameter is an ellipse of semiaxis lengths a and b. The latter approach will seem natural enough to a reader who has absorbed the main lessons of "Lines and Curves".
I can only assume that readers of this little gem will want to go further. The book does not have a bibliography, but I can offer the following suggestions: "Geometric Transformations" (volumes I-III) by I.M. Yaglom, H.S.M. Coxeter's "Geometry Revisited", and the hard-to-get but delightful monograph "The Kinematic Method in Geometrical Problems" by Lyubich and Shor.
(Note: While my personal favorite is number theory, "Lines and Curves" still holds a special place in my heart fifteen years after reading Mir Publishers' Spanish translation. English readers should feel very fortunate indeed that this 2004 Birkhäuser translation is available.)
This work takes you right into the details of solving problems relating to lines and curves. The illustrations that helpfully accompany it are useful and relevant and really add to the learning experience.
Topics it covers include: Sets, points, lines, intersections, curves - and that's just getting started!
If you are looking to learn about geometry, or teach a class on the subject, this book is definitely something you would want to consider. It is a great value.