- Basic Concepts
- Parallel lines are coplanar infinite straight lines that never intersect
- Parallel planes are planes in same three-dimensional space that never meet
- Parallel curves keep fixed minimum distance and don't touch
- Line segments and vectors are parallel if they have same or opposite direction
- Euclidean Geometry
- Parallel lines are defined as equidistant and in same plane
- Parallelism is primarily property of affine geometries
- Parallelism is not reflexive and fails to be equivalence relation
- Parallelism is transitive but not reflexive in Euclidean geometry
- Non-Euclidean Geometry
- Hyperbolic geometry allows three types of parallel geodesics
- Spherical geometry has no parallel geodesics
- In hyperbolic geometry, limiting parallel lines exist
- In spherical geometry, all great circles intersect
- Notation
- Parallel symbol is ∥
- Unicode codepoints: ∥ for parallel, ∦ for not parallel
- "Equal and parallel to" represented by U+22D5 (⋕)