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

4.2 Investigate Perspectivity and Projectivity
If I have seen further than others, it is because I have stood on the shoulders of giants.
—Isaac Newton (1642–1727)

• A set of points incident with a line is called a pencil of points (or range of points) and the line is called the axis.
  Example. A set of points {A, B, C, D} on a line l is called a pencil of points with axis l.
   

• A one-to-one mapping between two pencils of points is called a perspectivity if the lines incident with the corresponding points of the two pencils are concurrent. The point where the lines intersect is called the center of the perspectivity. A perspectivity is denoted
  
   where O is the center of perspectivity.
   
• A one-to-one mapping between two pencils of points is called a projectivity if the mapping is a composition of finitely many perspectivities. A projectivity is denoted
  
  .
  
  Examples. The left-hand figure is a perspectivity between two pencils of points with axes p and p',
  
  ,and the right-hand figure is a projectivity,
  
   consisting of the product of two perspectivities,
  
   and
  

  The points in red may be dragged to different locations to explore the perspectivity and projectivity.
  Sorry, this page requires a Java-compatible web browser.      Sorry, this page requires a Java-compatible web browser.
  To reset to the original settings, type the letter "R" on the keyboard.

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