Euclidean plane: Difference between revisions
imported>Paul Wormer (New page: {{subpages}} The '''Euclidean plane''' is the plane that is the object of study in Euclidean geometry (high-school geometry). The plane and the geometry are named after the ancient-Gre...) |
imported>Richard Pinch m (link) |
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# ρ(''P'',''Q'') ≤ ρ(''P'',''R'') + ρ(''R'',''Q'') (triangular inequality). | # ρ(''P'',''Q'') ≤ ρ(''P'',''R'') + ρ(''R'',''Q'') (triangular inequality). | ||
As is known from Euclidean geometry lines can be drawn between points and different geometric figures (triangles, squares, etc.) can be constructed. A geometric figure can be translated and rotated without change of shape. Such a map is called a ''rigid motion'' of the figure. The totality of rigid motions form a [[group]] of infinite order, the [[Euclidean group]] in two dimensions, often written as ''E''(2). | As is known from Euclidean geometry lines can be drawn between points and different geometric figures (triangles, squares, etc.) can be constructed. A geometric figure can be translated and rotated without change of shape. Such a map is called a ''[[rigid motion]]'' of the figure. The totality of rigid motions form a [[group]] of infinite order, the [[Euclidean group]] in two dimensions, often written as ''E''(2). | ||
Formally, the Euclidean plane is a 2-dimensional [[affine space]] with [[inner product]]. | Formally, the Euclidean plane is a 2-dimensional [[affine space]] with [[inner product]]. | ||
Revision as of 15:47, 25 November 2008
The Euclidean plane is the plane that is the object of study in Euclidean geometry (high-school geometry). The plane and the geometry are named after the ancient-Greek mathematician Euclid.
The Euclidean plane is a collection of points P, Q, R, ... between which a distance ρ is defined, with the properties,
- ρ(P,Q) ≥ 0 and ρ(P,Q) = 0 if and only if P = Q
- ρ(P,Q) = ρ(Q,P)
- ρ(P,Q) ≤ ρ(P,R) + ρ(R,Q) (triangular inequality).
As is known from Euclidean geometry lines can be drawn between points and different geometric figures (triangles, squares, etc.) can be constructed. A geometric figure can be translated and rotated without change of shape. Such a map is called a rigid motion of the figure. The totality of rigid motions form a group of infinite order, the Euclidean group in two dimensions, often written as E(2).
Formally, the Euclidean plane is a 2-dimensional affine space with inner product.