TEXT BOOK OF TRANSFORMATION GEOMETRY - EOPCW

TEXT BOOK OF TRANSFORMATION

GEOMETRY

Written by Begashaw Moltot (MED+MSC)

2007

Text Book of Transformation Geometry by Begashaw M. For your comments, use -0938836262

Contents

CHAPTER- 1

TRANSFORMATIONS

1.1 Revision on Mappings .............................................................................................4 1.2 Types of Mappings...................................................................................................5 1.3 Composition of Transformations and Their Properties..........................................12 1.4 Identity and Inverse Transformations ....................................................................15 1.5 Fixed Points of Mappings and Involution ..............................................................19 1.6 Collineations and Dilatations .................................................................................21 Problem Set 1.1 ............................................................................................................24 1.7 Definitions and examples of Transformation Groups ...........................................28 1.8 Criteria for Transformation Groups .......................................................................30 Problem Set 1.2 ............................................................................................................33

CHAPTER-2

AFFINE GEOMETRY

2.1 Introduction to Affine Spaces ................................................................................36 2.2 Geometry in Affine Space......................................................................................39 2.3 Lines and Planes in Affine space ...........................................................................44 2.3.1 Lines in Affine Geometry ...................................................................................44 2.3.2 Planes in Affine Space ........................................................................................53 Problem Set 2.1 ............................................................................................................54 2.3.3 Collinearity in Affine Space................................................................................56 2.4 The Classical Theorems .........................................................................................59 Problem Set 2.2 ............................................................................................................66

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Text Book of Transformation Geometry by Begashaw M. For your comments, use -0938836262

CHAPTER-3

ISOMETRIC (ORTHOGONAL) TRANSFORMATIONS

3.1 Introductions ..........................................................................................................68 3.2 Definition and Examples of Isometries..................................................................69 3.3 Properties of Isometric (Orthogonal) Transformations..........................................72 3.4 Fundamental Types of Isometric Transformations ................................................76

3.4.1 Translation...................................................................................................76 3.4.2 Reflection ....................................................................................................82 3.4.3 Rotation .......................................................................................................95 3.3.4 Glide Reflection ........................................................................................109 3. 5 Orientation and Orthogonal Transformations .....................................................116 3.5.1 Orientation of Vectors..............................................................................116 3.5.2 Orientation of Plane Figures .....................................................................121 3.5.3 Orientation Preserving and Orientation Reversing Isometries.................123 3.6 Fixed Points of Isometries....................................................................................130 3.7 Linear and Non-linear Isometries.........................................................................132 3.8 Representations of Orthogonal (Isometric) .........................................................136 Transformation as a Product of Reflections.............................................................136 3.8.1 Product of Reflections on Two Lines........................................................137 Case I: When the two Lines are Intersecting .....................................................137 Case II: When the two Lines are Parallel...........................................................144 3.8.2 Product of Reflections on Three Lines......................................................152 Case-I: When the three lines are concurrent ......................................................152 Case-II: When the three lines are parallel ..........................................................152 Case-III: When the three lines are neither parallel nor concurrent ....................156 3.8.3 The Fundamental Theorems of Isometries................................................161 3.9 Equations of Orthogonal Transformations in Coordinates ..................................165 3.10 Equations of Even and Odd Isometries ..............................................................166 3.10.1 Equations of Even Isometries .................................................................166 3.10.2 Equations of Odd Isometries ...................................................................166 3.11 Test for Type of Isometries ................................................................................171 Review Problems On Chapter-3.................................................................................175

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Text Book of Transformation Geometry by Begashaw M. For your comments, use -0938836262

CHAPTER-4

SIMILARITY TRANSFORMATIONS

4.1 Introduction ..........................................................................................................178 4.2 Properties of Similarity Transformations.............................................................180 4.3 Common Types of Similarity Transformations ...................................................182

4.3.1 Isometries ..................................................................................................182 4.3.2 Homothety (Homothetic Transformations)...............................................183 4.4 Representation of Similarity Transformations .....................................................184 4. 5 Equations of Similarity Transformations in Coordinates ...................................190 4.6 Direct and Opposite Similarities ..........................................................................191 Review Problems on Chapter-4 .................................................................................195

CHAPTER-5

AFFINE TRANSFORMATIONS

5.1 Introduction ..........................................................................................................198 5.2 Basic Properties of Affine Transformations ........................................................200 5.3 Types of Affine Transformations.........................................................................203

5.3.1 Line (Skew)-Reflections ...........................................................................203 5.3.2 Compressions ............................................................................................205 5.3.3 Shears ........................................................................................................206 5.3.4 Sililarities ..................................................................................................208 5.4 Affine Transformations and Linear Mappings.....................................................210 5.5 Matrix Representation of Affine Transformations...............................................211 5.6 Orientation and Affine Transformations (Revisited) ...........................................217 5.7 Area and Affine Transformations ........................................................................220 5.8 Inverse of Affine Transformations.......................................................................225 Review Problems on Chapter-5 .................................................................................227 REFERENCES ...........................................................................................................232

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Text Book of Transformation Geometry by Begashaw M. For your comments, use -0938836262

CHAPTER- 1

TRANSFORMATIONS

1.1 Revision on Mappings

Definition: Let X and Y be nonempty sets. Then, a mapping f from X to Y is a rule which assigns to every element x in X exactly one (unique) value f (x) in Y , here, f (x) is called the image of x under f . The set X is said to be the domain of f and Y is the co-domain of f . The set of all images of f is called range of f . In this definition of mappings, the word unique (exactly one) refers to the idea of well definedness. A rule which assigns to every element in the domain (in X ) some value in the co domain (in Y ) is said to be a mapping if it is well defined. To show well-defined ness, it suffices to show that f (x) y, f (x) z y z . Notation: The mapping f from X to Y is denoted symbolically by f : X Y . Examples 1. Let g : R2 R2 be given by g(x, y) (2x,3y). Show that g is a mapping. Solution: Clearly g is a rule which assigns to each value in R2 a value in R2 . Now, let's show that g is well-defined. Suppose g(x, y) (a,b) g(x, y) (c, d)

g(x, y) (a,b) g(x, y) (c, d ) (2x,3y) (a,b) (2x,3y) (c, d ) 2x a,3y b 2x c,3y d a c b d (a,b) (c, d)

This implies that the image of any point (x, y) in R2 is unique and hence g is well defined and it is a mapping.

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