Isometry
A Bijective Map between two Metric Spaces that preserves distances,i.e.,
where
is the Map and 
is the Distance function.An isometry of the Plane is a linear transformation which preserves length. Isometries include Rotation,Translation, Reflection, Glides, and the Identity Map. If an isometry hasmore than one Fixed Point, it must be either the identity transformation or a reflection. Every isometry of periodtwo (two applications of the transformation preserving lengths in the original configuration) is either a reflection or ahalf turn rotation. Every isometry in the plane is the product of at most three reflections (at most two if there is aFixed Point). Every finite group of isometries has at least one Fixed Point.
See also Distance, Euclidean Motion, Hjelmslev's Theorem, Length (Curve), Reflection,Rotation, Translation
References
Gray, A. ``Isometries of Surfaces.'' §13.2 in Modern Differential Geometry of Curves and Surfaces. Boca Raton, FL: CRC Press, pp. 255-258, 1993.
- Rational Quadrilateral
- Rational Triangle
- Ratio Test
- RATS Sequence
- Ray
- Rayleigh Differential Equation
- Rayleigh Distribution
- Rayleigh-Ritz Variational Technique
- Rayleigh's Formulas
- Rayleigh's Theorem
- Real Analysis
- Real Axis
- Real Function
- Realizer
- Real Line
- Real Matrix
- Real Number
- Real Part
- Real Polynomial
- Real Projective Plane
- Real Quadratic Field
- Rearrangement Theorem
- Reciprocal
- Reciprocal Difference
- Reciprocal Polyhedron