Euclidean plane

The two-dimensional Euclidean space of points, equipped with the familiar notions of distance, angle, and parallelism. Its geometry satisfies Euclid’s parallel postulate.

Euclidean distance: The distance between points defined by the Pythagorean formula in Cartesian coordinates. It supplies the plane’s standard notion of how far apart two points are.
Pythagorean theorem: A theorem stating that a right triangle’s hypotenuse squared equals the sum of its legs’ squares. It yields the distance formula for points in perpendicular coordinates.
Hyperbolic plane: A two-dimensional geometry of constant negative curvature in which multiple parallels pass through an external point. Its parallel behavior differs from the Euclidean plane’s unique parallel.
Compass-and-straightedge construction: A geometric construction using only an unmarked straightedge and a compass. Classical constructions are performed directly in the Euclidean plane.
Euclidean geometry: The geometry of spaces governed by Euclid’s postulates, including the parallel postulate. The plane is the basic two-dimensional setting for this geometry.
Affine line: A one-dimensional affine subspace, extending without bound in both directions. Straight lines in the plane are affine lines, with parallelism defined by their directions.
Elliptic geometry: A non-Euclidean geometry in which no parallel lines exist. It replaces the plane’s parallel postulate with a geometry where lines meet.
Affine transformation: A map that preserves points, straight lines, and parallelism. It describes many plane transformations used in graphics and geometry.
Cartesian coordinate system: A system that locates points using ordered numerical coordinates along perpendicular axes. Coordinates make points and geometric relationships in the plane computable.
Dot product: An operation on vectors whose value encodes their lengths and the angle between them. It provides an algebraic way to measure angles and perpendicularity.

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