# Euclidean plane

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*) ∈ ℝ (real line) - 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*) (triangle inequality).

In other words, the Euclidean plane is a metric space.

Well-known subsets of the Euclidean plane are *straight lines*. Any two non-coinciding points *P* and *Q* [ρ(*P*,*Q*) ≠ 0] in the plane lie on a unique straight line, that is to say, *P* and *Q* determine a unique straight line. The length of the straight line segment connecting the points *P* and *Q* is by definition the Euclidean distance rho;(*P*,*Q*). In a Euclidean plane no upper bound exists on the distance between two points: given a fixed point *P* and a "running" point *Q*, the distance ρ(*P*,*Q*) is unbounded, so that *Q* can "run to infinity" without leaving the plane. The straight line segment connecting *P* and *Q* may be extended to a half-line, which is a line segment terminated in *P* and unbounded on the other side. If *P* also runs to infinity (away from *Q*), i.e., the straight segment is unbounded on both sides, the segment becomes a straight line. Hence, lines and half-lines in a Euclidean plane are of infinite length.

Straight lines may be intersecting. Two straight lines that do *not* intersect are called parallel. The point of intersection of two non-parallel straight lines is a single unique point. Angles between intersecting lines can be measured and expressed in degrees. In particular, straight lines may be perpendicular, that is, their angle of intersection is 90°. The maximum number of perpendicular lines that can intersect at a point of a Euclidean plane is equal for all points; this maximum is two for a Euclidean plane. This fact is expressed by the statement that a Euclidean plane is two-dimensional.

A Cartesian coordinate frame can be erected anywhere in the plane, see this article for details and a figure. Briefly, one chooses an arbitrary point *O* (the origin) and let it be the point of intersection of two perpendicular straight lines, one line called the *x*-axis, the other line called the *y*-axis. Given an arbitrary point *P* in the plane, it is possible to construct a half-line from *P* intersecting the *x*-axis perpendicularly at the point *X*: one says that *X* is the perpendicular projection of *P* on the *x*-axis. The non-negative real number *x* := ρ(*O*,*X*) is the absolute value of the *x*-coordinate of *P*. The non-negative number *y* := ρ(*P*,*X*) is the absolute value of the *y*-coordinate of *P*. It is useful and common to give signs to coordinates, see Cartesian coordinates for the definition of positive and negative coordinates. If *Y* is the perpendicular projection of *P* on the *y*-axis, one sees easily that the distance ρ(*Y*,*P*) is equal to *|x|* and ρ(*O*,*Y*) is equal to *|y|*.

This construction by means of a Cartesian frame (also known as system of axes) shows that in a Euclidean plane every point in the plane is given by a pair of numbers (*x*, *y*) defined with respect to one and the same frame. Let *P* = (*x*, *y*) (this is a short-hand notation for *P* having coordinates *x* and *y*) and let *Q* = (*x′*, *y′*). From the Pythagorean theorem (which holds for Euclidean planes) it follows directly that

Suppose the pairs (*x*, *y*) and (*x*", *y*") with *x* ≠ *x*" and *y*≠*y*" would represent *P* (both pairs with respect to the same Cartesian frame). Then the distance

and because axiom 2 states that two points with non-zero distance are necessarily distinct, it follows that both pairs of numbers cannot simultaneously represent the same point *P*. The coordinate pair (*x*, *y*) represents *P* uniquely. In summary,

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.