Victor Gutenmacher, N. B. Vasilyev

Lines and Curves (eBook, PDF)

A Practical Geometry Handbook
Übersetzer: Kundu, A.

• Format: PDF

Produktbeschreibung

"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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Produktdetails

• Verlag: Birkhäuser Boston
• Seitenzahl: 156
• Erscheinungstermin: 14. März 2013
• Englisch
• ISBN-13: 9781475738094
• Artikelnr.: 44194614

Inhaltsangabe

1 Sets of Points.- 2 The Alphabet.- 3 Logical Combinations.- 4 Maximum and Minimum.- 5 Level Curves.- 6 Quadratic Curves.- 7 Rotations and Trajectories.- 8 Drawings, Animation, and the Magic Triangle.- Answers, Hints, Solutions.- A Summary of Results from Analytic Geometry.- B Some Facts from School Geometry.- B.1 Proportional segments.- B.2 Distances, perpendiculars.- B.3 The circle.- B.4 Triangles.- C A Dozen Assignments.- C.1 Name the "letters".- C.2 Transformations and constructions.- C.3 Rotating straight lines.- C.4 Straight lines and linear relations.- C.5 The tangency principle (conditional extremum).- C.6 Partitions.- C.7 Ellipses, hyperbolas, and parabolas.- C.8 Envelopes, infinite unions.- C.9 Tangents to cycloids.- C.1O Equations of curves.- C.11 Geometrical practical work.- C.12 Small investigations.- About Victor Gutenmacher.- About N. B. Vasilyev.

Rezensionen

"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