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

           3.2.1 Preliminary Definitions and Assumptions  Printout
Knowing is not enough; we must apply.
Willing is not enough; we must do.
—Johann von Goethe (1749–1842)

Definition. A mapping (or function  f  from A to B) of a set A into a set B is a rule that pairs each element of A with exactly one element of a subset of B. The set A is called the domain, and the set of all elements of B (a subset of B) that are paired with an element from A is called the range.

Definition. A mapping  f  from A to B is onto B if for any  b  in B there is at least one  a  in A such that f(a) = b.

Definition. A mapping  f  from A to B is one-to-one if each element of the range of  f  is the image of exactly one element from A.

Definition. A transformation is a one-to-one mapping of a set A onto a set B.

Definition. A transformation of a plane is a transformation that maps points of the plane onto points in the plane.

Definition. A nonempty set G is said to form a group under a binary operation, *, if it satisfies the following conditions:

1. If A and B are in G, then A*B is in G. (The set is closed under the operation, closure.)
2. There exists an element I in G such that for every element A in G, I*A = A*I = A. (The set has an identity.)
3. For every element A in G, there is an element B in G such that A*B = B*A = I, denoted A^–1. (Every element has an inverse.)
4. If A, B, and C are in G, then (A*B)*C = A*(B*C). (associativity)

Theorem 3.0. The set of transformations of a plane is a group under composition.

Proof. The result follows from the following:
       The composition of two transformations of a plane is a transformation (Exercise 3.4).
       The inverse of a transformation is a transformation (Exercise 3.5).
       The identity function is a transformation and composition of functions is associative (Exercise 3.5).//

Exercise 3.2. Which of the following mappings are transformations? Justify.

a.        such that .

b.       such that .

c.        such that .

d.       such that .

e.       Let P be a point in a plane S. Define  by f(P) = P and for any point , f(Q) is the midpoint of .

Exercise 3.3.  Let  and  be transformations defined respectively by f(x,y) = (x  4, y + 1) and g(x,y) = (x + 2, y + 3).

a.       Find the composition .

b.      Find the composition .

c.       Find the inverse of f,  f ¹.

d.      Find the inverse of g,  g ¹.

Exercise 3.4. Prove the composition of two transformations of a plane is a transformation of the plane.

Exercise 3.5. (a) Prove the identity function is a transformation. (b) Prove the inverse of a transformation of a plane is a transformation of the plane. (c) Prove the composition of functions is associative.