Euclidean geometry

Euclidean geometry, also called "flat" or "parabolic" geometry, is named after the Greek mathematician Euclid. Euclid's text Elements is an early systematic treatment of this kind of geometry, based on axioms (or postulates). This is the kind of geometry familiar to most people, since it is the kind usually taught in high school.

Euclidean geometry is distinguished from other geometries by the parallel postulate, which is usually phrased as follows: Through a point not on a given straight line, one and only one line can be drawn that never meets the given line. In particular, this postulate separates Euclidean geometry from hyperbolic geometry, where many parallel lines could be drawn through the point, and from elliptic and projective geometry, where no parallel lines exist. (Euclidean geometry does, however, share the parallel postulate with some geometries, such as certain finite geometries[?].)

Since Euclid's time, other mathematicians have laid out more thorough axiomatic systems for Euclidean geometry, such as David Hilbert and George Birkhoff[?].

Modern Concept of Euclidean Geometry

Today Euclidean geometry is usually constructed rather than axiomatized, by means of analytic geometry. A rectangular coordinate system maps each point in Euclidean space with a unique list of n real numbers (x₁,...,x_n), so we can define it to be the set of all such lists (Rⁿ). We also define a metric (distance function) d by

<math>d(x,y)^2=(x_1-y_1)^2+\ldots+(x_n-y_n)^2</math>

which you might recognise as an application of the Pythagorean Theorem (see also Euclidean distance). This turns Rⁿ into a metric space. Maps that preserve the distance between all pairs of points are called isometries, and include reflections, rotations, translations, and compositions thereof. In matrix notation any of these have the form

<math>x'=\mathbf{A}x+b</math>

where A is an orthogonal matrix and b is a column vector. Isometries are taken as the congruences of Euclidean geometry - that is, we only consider properties preserved by them. That way we do not have to worry about the precise origin or axes, but still consider distances, angles, and so forth.