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The Klein View of Geometry |
In Theorem E2, we showed that each element t of E(2) can be written in matrix form as t(x) = Ax + b, where A is an orthogonal 2x2 matrix, and bεR2. In Theorem S2, we showed that each element of S(2) has a similar from, with A replaced by a scalar multiple kA.
Both of these groups are clearly subgroups of the group obtained by allowing A to be any
Definitions Affine geometry is the geometry with set R2 and group A(2).
Since E(2) is a subgroup of A(2), this geometry is related to euclidean geometry. Indeed,
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For the moment, we will concentrate on the fundamental theorem, developing only such results as are required for this purpose.
Note that, throughout our proofs, we will use the convention that the position
Theorem A0
Proof
(2) If M is a line parallel to L, then we may take d as a direction
Thus, collinearity is an affine property. It follows that a list of non-collinear
The Fundamental Theorem of Affine Geometry
Much as in inversive geometry, this follows easily from a special case with
The (O,X,Y) Theorem This result allows us to find geometrically defined generators for A(2).
The study of the properties of affine geometry does not require any
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