MODERN GEOMETRY was written to provide undergraduate and graduate level mathematics education students with an introduction to both Euclidean and non-Euclidean geometries, appropriate to their needs as future junior and senior high school mathematics teachers. MODERN GEOMETRY provides a systematic survey of Euclidean, hyperbolic, transformation, fractal, and projective geometries. This approach is consistent with the recommendations of the National Council of Teachers of Mathematics (NCTM), the International Society for Technology in Education (ISTE), and other professional organizations active in the preparation and continuing professional development of K-12 mathematics teachers.
1. Geometry Through the Ages. Greek Geometry Before Euclid. Euclid and the Elements. Neutral Geometry. Famous Open Problems in Geometry. 2. Topics in Euclidean Geometry. Elementary Constructions. Exploring Relationships Between Objects. Formal Geometric Proof. 3. Other Geometries. The Concept of Parallelism. Points, Lines, and Curves in Poincares Model of Hyperbolic Space. Polygons in Hyberbolic Space. 4. Transformation Geometry. An Analytic Model of the Euclidean Plane. Representing Linear Transformations in 2-space with Matrices. The Direct Isometries: Translations and Rotations. Indirect Isometries: Reflections. Composition and Analysis of Transformations. Other Linear Transformations. 5. Fractal Geometry. Introduction to Self-similarity. Fractal Dimension. Iterated Function Systems. From Order to Chaos. The Mandelbrot Set. 6. Projective Geometry. Elements of Perspective Drawing. Introduction to Projective Geometry. The Cross Ratio. Applications of the Cross Ratio. Matrix Methods for 3-point Perspective Transformations. Applications of Geometry in Remote Sensing. Applications of Geometry in Terrain Rendering. Index.