Lines and Curves: A Practical Geometry Handbook

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