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 course was removed from the catalog in 2015. The prerequisites are Calculus 2 (MATH 1920), Linear Algebra (2010), and Mathematical Reasoning (MATH 3000).
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.
These notes constitute the projective geometry part of Axiomatic and Transformational Geometry. The transformational geometry part is covered in my online transformational geometry class notes.
The finite geometry part is covered in my online finite geometry class notes. Online notes for axiomatic geometry are also available, though they are primarily meant for use in Introduction to Modern Geometry. In fact, these notes on transformational geometry could be used as supplemental material in Introduction to Modern Geometry.
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 I. Introduction: The Propositions of Incidence.
- Chapter II. Related Ranges and Pencils: Involutions.
- Chapter III. The Conic.
- Chapter IV. Absolute Elements: The Circle: Foci of Conics.
- Chapter V. The Equation of a Line and of a Conic: Algebraic Correspondence on a Conic: The Harmonic Locus and Envelope.
- Chapter VI. Metrical Geometry.
Chapter I. Introduction: The Propositions of Incidence.
- Section 1. Historical Note.
- Section 2. The Projective Method.
- Section 3. Desargues' Theorem.
- Section 4. The Analytic Method.
- Section 5. Analytic Proof of Desargus' Theorem.
- Section 6. Pappus' Theorem.
- Section 7. The Fourth Harmonic Point.
- Section 8. The Complete Quadrangle.
Chapter II. Related Ranges and Pencils: Involutions.
- Section 9. Related Ranges.
- Section 10. The cross Ratio.
- Section 11. The Cross Ration Property of a (1 - 1) Correspondence.
- Section 12. Ranges in Perspective.
- Section 13. Related Ranges on the Same Base; Double Points.
- Section 14. Related Pencels.
- Section 15. Involution on a Line.
- Section 16. Cross Ratio Property of an Involution.
- Section 17. Involution Property of the Complete Quadrangle.
- Section 18. An Algebraic Representation of an Involution.
- Section 19. Pencils in Involution.
Chapter III. The Conic.
- Section 20. Introduction.
- Section 21. Projective Definition of the Conic.
- Section 22. Related Ranges on a Conic.
- Section 23. Involution on a Conic.
- Section 24. The Conic as an Envelope.
- Section 25. Desargues' Theorem.
- Section 26. Pascal's Theorem.
- Section 27. Pole and Polar.
- Section 28. Properties of Two Conics.
- Section 29. Pencil's of Conics.
Chapter IV. Absolute Elements: The Circle: Foci of Conics.
- Section 30. Introduction.
- Section 31. Absolute Elements.
- Section 32. The Circle.
- Section 33. The Conic and the Absolute Points.
- Section 34. Central Properties of Conics; Conjugate Diameters.
- Section 35. Foci and Axes of a Conic.
- Section 36. The Director Circle.
- Section 37. Confocal Conics.
- Section 38. The Auxiliary Circle.
- Section 39. Some Properties of the Parabola.
- Section 40. Some Properties of the Rectangular Hyperbola.
- Section 41. The Hyperbola of Apollonius.
- Section 42. The Frégier Point.
Chapter V. The Equation of a Line and of a Conic: Algebraic Correspondence on a Conic: The Harmonic Locus and Envelope.
- Section 43. The Equation of a Line.
- Section 44. The Equation of a conic.
- Section 45. Tangent, Pole and Polar.
- Section 46. The Line-Equation of a Conic.
- Section 47. Special Forms for the Equation of a Conic.
- Section 48. Correspondence between Points of a Conic.
- Section 49. The symmetrical (2 - 2) Correspondence of Points on a Conic.
- Section 50. The Harmonic Envelope.
- Section 51. A Conic Associated with Three Conics of a Pencil.
Chapter VI. Metrical Geometry.
- Section 52. Introduction.
- Section 53. Projective Definition of Distance and Angle.
- Section 54. The Absolute Conic.
- Section 55. Algebraic Expressions for Distance and Angle.
- Section 56. Real and Complex Points and Lines.
- Section 57. Real and Complex Conics.
- Section 58. Metrical Geometry.
- Section 59. Distance and Angle in Euclidean Geometry.
- Section 60. The Euclidean Equivalents of Simple Projective Elements.
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