TOC & Ch. 0 & Ch. 1 Axiom Table of Contents Ch. 0 Introduction Ch. 1 Axiomatic Systems 1.1.1 Introduction 1.1.2 Examples 1.1.3 History 1.2 A Finite Geometry 1.3 Finite Projective 1.4 Applications

Ch. 2 Neutral Geometry Ch. 2 Table of Contents 2.1.1 Introduction 2.1.2 History 2.1.3 Analytic Models 2.2 Incidence Axioms 2.3 Distance/Ruler Axioms 2.4.1 Plane Separation Axiom 2.4.2 Angle & Measure 2.5.1 Supplement Postulate 2.5.2 SAS Postulate 2.6.1 Parallel Lines 2.6.2 Saccheri Quadrilateral 2.7.1 Euclid Parallel Postulate 2.7.2 Hyperbolic Parallel Postulate 2.7.3 Elliptic Parallel Postulate 2.8 Euclid/Hyperbolic/Elliptic Birkhoff's Axioms Hilbert's Axioms SMSG Axioms

Ch. 3 Transformational Ch. 3 Table of Contents 3.1.1 Introduction 3.1.2 History 3.2.1 Definitions 3.2.2 Analytic Model 3.2.3 Affine Transformation 3.3.1 Isometry 3.3.2 Model/Collinearity 3.3.3 Model/Isometry 3.4.1 Direct Isometry 3.4.2 Model/Direct 3.5.1 Indirect Isometry 3.5.2 Model/Indirect 3.6.1 Similarity Transformation 3.6.2 Model/Similarity 3.7 Other Affine Transformations

Ch. 4 Projective Geometry Ch. 4 Table of Contents 4.1.1 Introduction 4.1.2 Historical 4.2.1 Axioms 4.2.2 Basic Theorems 4.3 Duality 4.4 Desargue's Theorem 4.5.1 Harmonic Sets 4.5.2 Music & Harmonic Sets 4.6.1 Definitions for Projectivity 4.6.2 Fundamental Theorem 4.6.3 Projectivity/Harmonic Sets 4.6.4 Alternate Construction 4.7.1 Conics 4.7.2 Pascal's Theorem 4.7.3 Tangents to Conics

Other Topics Ch. 5 Spherical Geometry Ch. 6 Fractal Geometry Ch. 7 Topology

Appendices Internet Resources Index Geometer's Sketchpad/GeoGebra JavaSketchpad/GeoGebraHTML Video Lectures Logic Review References Acknowledgements

      3.7 Other Affine Transformations of the Euclidean Plane  Printout
Mathematicians are like lovers…Grant a mathematician the least principle, and he will draw from it a consequence which you must grant him also, and from this consequence another.
—Bernard de Fontenelle (1657–1757)

      This section is an exploration section. The reader is asked to investigate various affine transformations that were not covered in previous sections and to conjecture general properties. Hopefully, the reader does not just accept the proposed conjectures, but proves or attempts to prove them. The reader should note that the matrices for isometries and similarities had certain positions which had the same value, |a₁₁| = |a₂₂| and |a₁₂| = |a₂₁|. What happens when these values are not the same?

 Definition. An affine transformation of the Euclidean plane, T, is a mapping that maps each point X of the Euclidean plane to a point T(X) of the Euclidean plane defined by T(X) = AX where det(A) is nonzero and

 where each a_ij is a real number.

May use computer software or calculators to aid in the investigations that follow.

Investigation Exercise 3.108. Suppose  where a is neither 0 nor 1. Given a square PQRS with vertices P(1, 1, 1), Q(–1, 1, 1), R(–1, –1, 1), and S(1, –1, 1).

1. Find the image of the square for different values of a such as –3, –1/2, 1/2, 3, etc.  
2. Describe how the transformation changed the square PQRS.  
3. Are there any invariant points or lines? If yes, what are they?
4. Let  Find the image of the square PQRS with vertices P(3, 4, 1), Q(1, 4, 1), R(1, 2, 1), and S(3, 2, 1), then repeat parts (b) and (c).
5. Let  Find the image of the square PQRS with vertices P(1, 1, 1), Q(–1, 1, 1), R(–1, –1, 1), and S(1, –1, 1) for different values of a not 0 or 1, then repeat parts (b) and (c).
6. These are examples of what is called a shear. Write a general definition of a shear.
7. From your definition derive one or more matrices for a shear. Write any relationships you find between shears and isometries.
8. Write any properties you find for shears.

Investigation Exercise 3.109. Suppose  where a is not –1, 0, or 1. Given a square PQRS with vertices P(1, 0, 1), Q(0, 1, 1), R(–1, 0, 1), and S(0, –1, 1).  

1. Find the image of the square for different values of a such as –3, –1/2, 1/2, 3, etc.  
2. Describe how the transformation changed the square PQRS.  
3. Are there any invariant points or lines? If yes, what are they?
4. Let  Find the image of the square PQRS with vertices P(3, 3, 1), Q(2, 4, 1), R(1, 3, 1), and S(2, 2, 1), then repeat parts (b) and (c). 
5. Let  Find the image of the square PQRS with vertices P(1, 0, 1), Q(0, 1, 1), R(–1, 0, 1), and S(0, –1, 1).  for different values of a not –1, 0, or 1, then repeat parts (b) and (c).
6. These are examples of what is called a strain. Write a general definition of a strain.  
7. From your definition derive one or more matrices for a strain.
8. Write any relationships you find between strains and isometries.
9. Write any properties you find for strains.

Investigation Exercise 3.110. Let  with  and       

1. Verify the computation.
2. Write a theorem that the computation implies.