You are here

Vectors and Transformations in Plane Geometry

Philippe Tondeur
Publish or Perish, Inc.
Publication Date: 
Number of Pages: 
[Reviewed by
Underwood Dudley
, on

Let us take a moment or two to sympathize with geometry, who has seen better days. Mathematics started with geometry, tradition telling us that the very first proof of a theorem occurred when Thales proved that the base angles of an isosceles triangle are equal. Euclidean geometry was mathematics for more than a millennium. Newton cast his results in the Principia in geometric form. At one time “geometer” and “mathematician” were synonymous.

Nowadays there are very few people whose job title is geometer and it is quite possible to be a professional mathematician without having the slightest idea of what the theorem of Menelaus says. Geometry has mostly fallen out of the undergraduate curriculum. There are some states whose legislatures, a few decades behind the times, require that those who wish to be certified as worthy to teach mathematics in their states must complete a course in geometry, and this keeps some college courses in geometry going.

It’s too bad there aren’t more. Geometry deserves better. A course in geometry is much better for students than a course in… well, I won’t offend anyone by being specific, but we could all complete the sentence in one way or another. This book would be very good text indeed for the course.

As its title indicates, it approaches geometry through vectors. There’s a lot to be said for the synthetic approach — I’ve taught such a course, greatly to my benefit, though I’m not so sure about the students’ — but the vector approach has the advantage not only of being very pretty but of connecting to modern mathematics. The book contains vectors, groups, transformations, and matrices as well as the theorems of Ceva and Desargues. The nine-point circle appears on the same page as the Cauchy-Schwartz inequality.

There are only 114 pages of text (followed by 15 pages of answers to odd-numbered exercises), so it is just right for a semester’s work. If students demand more, instructors should be able to provide extra material. Philippe Tondeur is a very smart man, so it is no surprise that his book is clear and well-written. It’s well printed too. Anyone looking around for a geometry text should consider it.

Woody Dudley hopes that all who read this will forever refrain from making jokes about how geometers know the angles.

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