Plane (mathematics)

Basic properties

Key properties of a plane include: any two distinct points in the plane determine a unique straight line contained in the plane; any three points that are not collinear determine a unique plane; the plane contains an infinite collection of points and lines. In Euclidean geometry a plane has no depth and extends infinitely in all directions. It can be described by sets of points points that satisfy certain equations or relations.

Analytic and vector descriptions

In analytic geometry a plane in three-dimensional space can be given by a linear equation of the form ax + by + cz + d = 0, where a, b and c are not all zero; this is a standard way to represent a flat surface algebraically. Alternatively, a plane can be expressed parametrically as the set of points p + su + tv where p is a fixed point in space and u, v are independent direction vectors. These descriptions make planes central in linear algebra, multivariable calculus and computer graphics.

Intersections and relationships

Two distinct planes in the same three-dimensional ambient space either are parallel (no intersection), coincide (all points in common) or intersect along a single straight line. Inside a plane, lines behave according to familiar planar rules from elementary geometry, such as parallelism and angle measures. Many elementary theorems and constructions—triangles, circles, polygons—are defined as figures that lie entirely within a single plane; such shapes are often called plane figures and are typically named using capital letters placed at vertices or by a Greek letter such as a capital pi symbol.

History, contexts and distinctions

The concept of a plane goes back to early Greek geometry, where flat surfaces were treated informally, and it was formalized in Euclid's axioms. Modern mathematics studies planes in several related frameworks: the Euclidean plane (with distances and angles), affine planes (preserving parallelism but not distances), and projective planes (adding 'points at infinity' to simplify intersection laws). Each framework emphasizes different properties useful in pure mathematics, engineering and the visual sciences.

Uses and examples

Planes are used to model flat regions in physics, engineering drawings, architectural plans and geographic maps (local approximations of the Earth's surface). In computer-aided design and graphics, planes serve as clipping surfaces, reference frames and support for constructing objects. As basic building blocks in mathematics, planes underlie the study of curves, surfaces and higher-dimensional analogues and provide an approachable setting to learn about vectors, coordinates and transformations.

• Three noncollinear points determine a plane.
• Two planes intersect in a line unless they are parallel or identical.
• Planes are central in both synthetic and analytic branches of geometry.

For further foundational reading see standard geometry texts and analytic geometry references, which develop these ideas with proofs, examples and applications.

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