These notes would constitute part of the material of "Axiomatic and Transformational Geometry" (MATH 5330). The catalog description in the 2014-15 ETSU Graduate Catalog was: "Axiomatic and finite geometries. Euclidean geometry (synthetic/analytic), transformational geometries, non-Euclidean and projective geometries." The prerequisites are Calculus 2 (MATH 1920), Linear Algebra (2010), and Mathematical Reasoning (MATH 3000). These notes cover an axiomatic approach to plane projective geometry.

Axiomatic and Transformational Geometry was removed from the catalog in 2015. This course was previously titled "Vector Geometry" and the description in the 1988-90 ETSU Graduate Catalogue was: "Projective geometry, affine geometry and affine transformation, Euclidean geometry, non-Euclidean geometries." It seems that "Vector Geometry" was split into "Introduction to Modern Geometry" (MATH 4157/5157) and "Axiomatic and Transformational Geometry" (MATH 5330) sometime in the 1990s. Honestly, the material in these notes is not at the level of a graduate-only course, and is more appropriate for a cross-listed undergraduate-graduate level class... perhaps it can become "Introduction to Modern Geometry" (MATH 4167/5167) in the future.

Copies of the classnotes are on the internet in PDF format as given below. The "Proofs of Theorems" files were prepared in Beamer and they contain proofs of the results from the class notes. The "Printout of Proofs" are printable PDF files of the Beamer slides without the pauses. These notes and supplements have not been classroom tested (and so may have some typographical errors).

• Chapter 1. The Elements of Perspective.
• Chapter 6. The Axiomatic Foundation.
• Chapter 7. The Complete Four-Point and Complete Four-Line.
• Chapter 8. Conics.
• Chapter 9. The Introduction of Coordinates.
• Additional Chapters.

Chapter 1. The Elements of Perspective.

• Section 1.1. Introduction. Section 1.1 notes
• Section 1.2. The Elements of Perspective. Section 1.2 notes
• Section 1.3. Plane Perspective. Section 1.3 notes
• Section 1.4. Plane Constructions. Section 1.4 notes
• Section 1.5. Conclusion. Section 1.5 notes
• Study Guide 1

Chapter 6. The Axiomatic Foundation.

• Section 6.1. Introduction. Section 6.1 notes
• Section 6.2. The Axioms of Incidence and Connection. Section 6.2 notes
• Section 6.3. The Projectivity Axiom.
• Section 6.4. Further Relations among the Fundamental Theorems.
• Section 6.5. Applications of the Theorem of Pappus.
• Section 6.6. Conclusion.

Chapter 7. The Complete Four-Point and Complete Four-Line.

• Section 7.1. Introduction.
• Section 7.2. Definitions and Fundamental Properties.
• Section 7.3. Harmonic Tetrads.
• Section 7.4. Involutory Hexads.
• Section 7.5. Further Results.
• Section 7.6. Conclusion.

Chapter 8. Conics.

• Section 8.1. Introduction.
• Section 8.2. Definitions and Fundamental Properties.
• Section 8.3. The Intersections of a Line and a Conic.
• Section 8.4. Desargues' Conic Theorem.
• Section 8.5. Pascal's Theorem.
• Section 8.6. Polar Theory.
• Section 8.7. Projectives on a Conic.
• Section 8.8. Further Results.
• Section 8.9. Conclusion.

Chapter 9. The Introduction of Coordinates.

• Section 9.1. Introduction.
• Section 9.2. Fields.
• Section 9.3. The Arithmetic of Collinear Points.
• Section 9.4. Projectives in Nonhomogeneous Coordinates.
• Section 9.5. Homogeneous Coordinates on a Line.
• Section 9.6. Nonhomogeneous Coordinates in the Projective Plane.
• Section 9.7. Homogeneous Coordinates in the Projective Plane.
• Section 9.8. Line Coordinates in the Projective Plane.
• Section 9.9. Conclusion.

Additional Chapters.

• Chapter 2. The Extended Euclidean Plane.
• Chapter 3. A Little Linear Algebra.
• Chapter 4. Further Properties of the Extended Plane.
• Chapter 5. Linear Transformations in Π₂.
• Chapter 10. The Introduction of a Metric.
• Chapter 11. Singular Metric Gauges.
• Appendix 1. A Review of Determinant.
• Appendix 2. A Finite NonDesarguesian Geometry.

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