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. 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 2" (MATH 4167/5167) in the future.

These notes constitute the transformational geometry part of Axiomatic and Transformational Geometry. The finite geometry part is covered in my online finite geometry class notes. The projective geometry part is covered in my online notes projective 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 V. Mappings of the Euclidean Plane.

• Section 41. Mappings. Section 41 notes
• Section 42. Isometries. Section 42 notes
• Section 43. The Theorem of Isometries. Section 43 notes
• Supplement. Proofs of Theorems in Section 43. Beamer file of Section 43 proofs
• Supplement. Printout of the Proofs of Theorems in Section 43. Print file of Section 43 proofs
• Section 44. Algebra and Groups. Section 44 notes
• Supplement. Proofs of Theorems in Section 44. Beamer file of Section 44 proofs
• Supplement. Printout of the Proofs of Theorems in Section 44. Print file of Section 44 proofs
• Section 45. Normal Subgroups and Isomorphisms. Section 45 notes
• Supplement. Proofs of Theorems in Section 45. Beamer file of Section 45 proofs
• Supplement. Printout of the Proofs of Theorems in Section 45. Print file of Section 45 proofs
• Section 46. Groups of Mappings. Section 46 notes
• Supplement. Proofs of Theorems in Section 46. Beamer file of Section 46 proofs
• Supplement. Printout of the Proofs of Theorems in Section 46. Print file of Section 46 proofs
• Section 47. Similarity Transformations and Results. Section 47 notes
• Supplement. Proofs of Theorems in Section 47. Beamer file of Section 47 proofs
• Supplement. Printout of the Proofs of Theorems in Section 47. Print file of Section 47 proofs
• Section 48. Groups of Translations and Rotations. Section 48 notes
• Supplement. Proofs of Theorems in Section 48. Beamer file of Section 48 proofs
• Supplement. Printout of the Proofs of Theorems in Section 48. Print file of Section 48 proofs
• Section 49. Indirect Isometries and Reflections. Section 49 notes
• Supplement. Proofs of Theorems in Section 49. Beamer file of Section 49 proofs
• Supplement. Printout of the Proofs of Theorems in Section 49. Print file of Section 49 proofs
• Section 50. More Isometries and Similitudes. Section 50 notes
• Supplement. Proofs of Theorems in Section 50. Beamer file of Section 50 proofs
• Supplement. Printout of the Proofs of Theorems in Section 50. Print file of Section 50 proofs
• Section 51. Line Reflections and Isometries as Reflections. Section 51 notes
• Supplement. Proofs of Theorems in Section 51. Beamer file of Section 51 proofs
• Supplement. Printout of the Proofs of Theorems in Section 51. Print file of Section 51 proofs

Chapter VI. Mappings of the Inversive Plane. Chapter VI Introduction notes

• Section 52. Möbius Transformations. Section 52 notes
• Supplement. Proofs of Theorems in Section 52. Beamer file of Section 52 proofs
• Supplement. Printout of the Proofs of Theorems in Section 52. Print file of Section 52 proofs
• Section 53. Cross-Ratio. Section 53 notes
• Supplement. Proofs of Theorems in Section 53. Beamer file of Section 53 proofs
• Supplement. Printout of the Proofs of Theorems in Section 53. Print file of Section 53 proofs
• Section 54. Circles and Conformality. Section 54 notes
• Supplement. Proofs of Theorems in Section 54. Beamer file of Section 54 proofs
• Supplement. Printout of the Proofs of Theorems in Section 54. Print file of Section 54 proofs
• Section 55. M-Transformations. Section 55 notes
• Supplement. Proofs of Theorems in Section 55. Beamer file of Section 55 proofs
• Supplement. Printout of the Proofs of Theorems in Section 55. Print file of Section 55 proofs
• Section 56. The Poincaré Model of Hyperbolic Geometry. Section 56 notes
• Section 57. Hyperbolic Triangles and Parallels. Section 57 notes
• Supplement. Proofs of Theorems in Section 57. Beamer file of Section 57 proofs
• Supplement. Printout of the Proofs of Theorems in Section 57. Print file of Section 57 proofs
• Section 58. More Hyperbolic Triangles and Distance. Section 58 notes
• Supplement. Proofs of Theorems in Section 58. Beamer file of Section 58 proofs
• Supplement. Printout of the Proofs of Theorems in Section 58. Print file of Section 58 proofs
• Section 59. Horocycles. Section 59 notes
• Supplement. Proofs of Theorems in Section 59. Beamer file of Section 59 proofs
• Supplement. Printout of the Proofs of Theorems in Section 59. Print file of Section 59 proofs

Chapter VII. The Projective Plane and Projective Spaces.

• Section 60. The Complex Projective Plane. Section 60 notes
• Supplement. Proofs of Theorems in Section 60. Beamer file of Section 60 proofs
• Supplement. Printout of the Proofs of Theorems in Section 60. Print file of Section 60 proofs
• Section 61. A Model for the Projective Plane. Section 61 notes
• Section 62. Collinear Points.
• Section 63. Noncollinear Points.
• Section 64. The Triangle of Reference.
• Section 65. The Projective Space of Three Dimensions.

Additional chapters:

• Chapter 0. Preliminary Notions.
• Chapter I. Vectors.
• Chapter II. Circles.
• Chapter III. Coaxial Systems of Circles.
• Chapter IV. The Representation of Circles by Points in Space of Three Dimensions.
• Chapter VIII. The Projective Geometry of n Dimensions.
• Chapter IX. The Projective Generation of Conics and Quadrics.
• Chapter X. Prelude to Algebraic Geometry.

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