Lines and Curves: A Practical Geometry Handbook

Springer Science & Business Media, Jul 23, 2004 - Mathematics - 156 pages

"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 SketchpadŽ," 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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Contents

 Sets of Points 7 A family of lines and motion 11 Construction problems 12 Additional problems 15 The Alphabet 19 A circle and a pair of arcs 21 Squares of distances 23 Distances from straight lines 29
 Addition of rotations 91 A theorem on two circles 99 Velocities and tangents 101 Parametric equations 106 Conclusion 109 Drawings Animation and the Magic Triangle 111 The envelope of a family of lines 114 The magic triangle 115

 The entire alphabet 33 Logical Combinations 35 Intersections and unions 39 The cheese problem 44 Maximum and Minimum 47 Where to put the point? 49 The motorboat problem 51 Level Curves 55 Functions on a plane 57 The graph of a function 58 The map of a function 62 Extrema of functions 64 Quadratic Curves 67 Foci and tangents 70 The focal property of a parabola 72 Curves as the envelopes of straight lines 75 Equations of curves 78 A note about the elimination of radicals 81 The end of our alphabet 82 Algebraic curves 87 Rotations and Trajectories 89
 A tiny ring on a circle 117 Two pedestrians on a circle and the Steiner deltoid 118 How three points move around three symmetric circles 120 The rotation of the WallaceSimson line and the Feuerbach circle 124 Steiners triangle and Morleys triangle 125 Answers Hints Solutions 127 Summary of Results from Analytic Geometry 135 Some Facts from School Geometry 137 B2 Distances perpendiculars 138 B3 The circle 139 A Dozen Assignments 141 C2 Transformations and constructions 142 C4 Straight lines and linear relations 143 C6 Partitions 144 C8 Envelopes infinite unions 145 C10 Equations of curves 146 C12 Small investigations 148 Index 149 About Victor Gutenmacher 153 About N B Vasilyev 155 Copyright

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