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Geometric Transformations of the Euclidean Plane

  • Judith N. Cederberg
Part of the Undergraduate Texts in Mathematics book series (UTM)

Abstract

The presentation of non-Euclidean geometry in Chapter 2 was synthetic, that is, figures were studied directly and without use of their algebraic representations. This reflects the manner in which both Euclidean and non-Euclidean geometries were originally developed. However, in the 17th century, French mathematicians Pierre de Fermat (1601–1665) and René Descartes (1596–1650) began using algebraic representations of figures. They realized that by assigning to each point in the plane an ordered pair of real numbers, algebraic techniques could be employed in the study of Euclidean geometry. This study of figures in terms of their algebraic representations by equations is known as analytic geometry.

Keywords

Equilateral Triangle Prove Theorem Affine Transformation Euclidean Plane Invariant Point 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Suggestions for Further Reading

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Suggestions for Viewing

  1. Adventures in Perception (1973, 22 min). An especially effective presentation of the work of M. C. Escher. Produced by Hans Van Gelder, Film Producktie, N. V., The Netherlands. Available from Phoenix/B.F.A. Films, 468 Park Ave. S., New York, NY 10016 (800) 221–1274.Google Scholar
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  3. Isometries (1971; 26 min). Demonstrates that every plane isometry is a translation, rotation, reflection, or glide reflection and that each is the product of at most three reflections. Produced by the College Geometry Project at the University of Minnesota. Available from International Film Bureau, 332 South Michigan Ave., Chicago, IL 60604.Google Scholar
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  6. by reflections. Produced by the College Geometry Project at the University of Minnesota, Available from International Film Bureau, 332 South Michigan Ave., Chicago, IL 60604.Google Scholar
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Suggested Software

  1. The Geometry Center (http://www.geom.umn.edu/) is a great source of downloadable geometry software. In particular, you may want the following:Google Scholar
  2. Geomview—A 3D object viewer.Google Scholar
  3. Kali—A 2D symmetry pattern editor.Google Scholar
  4. Kaleido Tile—Creates tilings of the sphere, plane, or hyperbolic space.Google Scholar
  5. KaleidoMania!—A tool for dynamically creating symmetric designs and exploring the mathematics of symmetry. Available from Key Curriculum Press (http://www.keypress.com/).Google Scholar

Copyright information

© Springer Science+Business Media New York 2001

Authors and Affiliations

  • Judith N. Cederberg
    • 1
  1. 1.Department of MathematicsNorthfieldUSA

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