Tondeur, Vectors and Transformations in Plane Geometry
x + 135 pages.   Clothbound.   1993

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Contents:.

1. Vectors in the Plane
     1.1 Definition
     1.2 Addition of vectors
     1.4 Formal calculations
     1.5 Equation of a line
     1.6 Parallelograms
     1.7 Centroid of a triangle
     1.8 Centroid of a finite point set
     1.9 Centroid of the zeros of a complex polynomial
    1.10 Centroid of mass-points
    1.11 Barycentric coordinates
    1.12 Theorems of Ceva and Menelaus
    1.13 Theorems of Desargues and Pappus

2. Translations, Dilations, Groups and Symmetries
     2.1 Translations
     2.2 Central dilations
     2.3 Central reflections
     2.4 Dilations
     2.5 Groups of transformations
     2.6 Abstract groups
     2.7 Symmetries of a rectangle
     2.8 Symmetries of a square
     2.9 Symmetries of an equilateral triangle
    2.10 Dihedral groups

3. Scalar Product
     3.1 Definition and elementary properties
     3.2 Orthogonality
     3.3 Circles
     3.4 Cauchy-Schwarz inequality
     3.5 Projection
     3.6 Angles
     3.7 Equation of a line

4. Isometries
     4.1 Definition and examples
     4.2 Fixed points of isometries
     4.3 Reflections
     4.4 Central reflections
     4.5 Isometries with a unique fixed point
     4.6 Products of involutions
     4.7 Translations
     4.8 Rotations
     4.9 Glide reflections
    4.10 Classification of isometries
    4.11 Finite groups of isometries

5. Linear Maps and Matrices
     5.1 The matrix of a linear map
     5.2 Composition of maps and matrix multiplication
     5.3 Matrices for linear isometries
     5.4 Change of basis
     5.5 The trace of a linear map
     5.6 The determinant of a linear map
     5.7 The crystallographic restriction

Answers to Odd-Numbered Exercises. Bibliography

From the Preface:

The first goal of this book is to explain the geometry of the plane by vector methods, in contrast to a synthetic approach. The vector approach is simple and direct. It represents a general method, while the synthetic approach has to many students the aspect of consisting of a multitude of flashes of insight. Each approach has its own charm.

The second goal is to introduce the student to the concept of transformation. ... It gives concrete examples leading to an appreciation of the theory of groups, but does not require any previous knowledge of group theory.

The text has as prerequisites only high-school geometry and algebra. The typical student in this course at the University of Illinois is currently a Junior, frequently taking simultaneously a standard course in abstract algebra. For many undergraduate mathematics majors, this is the only geometry course they will take.
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