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A Transformation Approach to Tenth Grade Geometry, The Mathematics Teacher, Vol. 65, No. 1 (January 1972), pp. 21-30. Zalman P. Usiskin. The Effects of Teaching Euclidean Geometry via Transformations on Student Achievement and Attitudes in Tenth-Grade Geometry, Journal for Research in Mathematics Education, Vol. 3, No. 4 (Nov., 1972), pp. 249-259.
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Geometric transformations can be distinguished into two types: active or alibi transformations which change the physical position of a set of points relative to a fixed frame of reference or coordinate system (alibi meaning "being somewhere else at the same time"); and passive or alias transformations which leave points fixed but change the ...
Geometry and the Imagination (2nd ed.). Chelsea. ISBN 0-8284-1087-9. John McCleary (2013) Geometry from a Differentiable Viewpoint, Cambridge University Press ISBN 978-0-521-11607-7; Modenov, P. S.; Parkhomenko, A. S. (1965) . Geometric Transformations (2 vols.): Euclidean and Affine Transformations, and Projective Transformations. New York ...
[1] Combining two equal glide reflections gives a pure translation with a translation vector that is twice that of the glide reflection, so the even powers of the glide reflection form a translation group. In the case of glide-reflection symmetry, the symmetry group of an object contains a glide reflection, and hence the group generated by it.
In geometry, inversive geometry is the study of inversion, a transformation of the Euclidean plane that maps circles or lines to other circles or lines and that preserves the angles between crossing curves. Many difficult problems in geometry become much more tractable when an inversion is applied.