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The first systematic effort to use transformations as the foundation of geometry was made by Felix Klein in the 19th century, under the name Erlangen programme. For nearly a century this approach remained confined to mathematics research circles.
Heinrich Guggenheimer (1977) Differential Geometry, Dover, New York, ISBN 0-486-63433-7. Covers the work of Lie, Klein and Cartan. On p. 139 Guggenheimer sums up the field by noting, "A Klein geometry is the theory of geometric invariants of a transitive transformation group (Erlangen program, 1872)".
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 ...
The program proposed a unified system of geometry that has become the accepted modern method. Klein showed how the essential properties of a given geometry could be represented by the group of transformations that preserve those properties. Thus the program's definition of geometry encompassed both Euclidean and non-Euclidean geometry.
2.2.2 First Bianchi identity. ... This is a list of formulas encountered in Riemannian geometry. ... Note that this transformation formula is for the mean curvature ...
Details for the first representation are given in the next section. In the terms of Felix Klein's Erlangen programme, we read off from this that Euclidean geometry, the geometry of the Euclidean group of symmetries, is, therefore, a specialisation of affine geometry. All affine theorems apply.
Affine transformation (Euclidean geometry); Bäcklund transform; Bilinear transform; Box–Muller transform; Burrows–Wheeler transform (data compression); Chirplet transform ...
The first and most important was the creation of analytic geometry, or geometry with coordinates and equations, by René Descartes (1596–1650) and Pierre de Fermat (1601–1665). This was a necessary precursor to the development of calculus and a precise quantitative science of physics .
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