Geometry

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22 domain-specific abstractions whose origin domain is Geometry.

• Ambient space (mathematics) — The surrounding mathematical space in which an object is embedded or considered, whose geometry and topology determine extrinsic relations not fixed by the object in isolation.
• Curve — A one-dimensional path represented, in its broad topological form, as the continuous image of an interval in a space.
• Descriptive Geometry — Represent and solve three-dimensional spatial relations through coordinated planar projections whose view directions are deliberately changed to expose true lengths, shapes, incidences, and distances.
• Dihedral Angle — Two intersecting planes or oriented half-planes are compared around their common line, producing an unsigned fold angle or a convention-dependent signed torsion that encodes relative orientation in geometry, molecules, chains, and polyhedra.
• Direction (geometry) — An equivalence class of parallel oriented lines, rays, or nonzero vectors that retains orientation while discarding position and usually magnitude.
• Geometric Transformation — An invertible mapping of a geometric space that preserves the relations declared fundamental by a chosen geometry, thereby organizing transformations and figures by their invariants.
• Golden ellipse — An ellipse whose major-to-minor semiaxis ratio equals the golden ratio.
• Helix — A space curve whose tangent maintains a constant angle with a fixed axis, with the circular helix as the constant-radius canonical case.
• Hilbert metric — A projectively invariant metric on the interior of a bounded convex domain, defined by a logarithmic cross ratio of the two boundary intersections on the line through a point pair.
• Inscribed Square Problem — The open square-peg conjecture that every planar Jordan curve contains four distinct points forming a nondegenerate Euclidean square, without requiring square order to match curve order.
• Motion (geometry) — Transform a metric space by a surjective distance-preserving map, with Euclidean motions further classified by orientation, fixed-point structure, and decomposition into translations, rotations, reflections, and glide operations.
• Point reflection — An affine transformation that sends each point x to 2c−x about a fixed center c, preserving distances and reversing every displacement vector.
• Prismatic surface — Generate a polyhedral ruled surface by translating every point of a polygonal-chain directrix along one fixed direction, producing parallel planar strips joined along parallel generators.
• Procrustes transformation — A similarity transformation using translation, rotation, reflection if allowed and uniform scaling while preserving shape.
• Pseudo-Euclidean Space — A finite-dimensional real vector or affine space equipped with a nondegenerate symmetric bilinear form of mixed signature, admitting positive, negative, and nonzero null directions.
• Quadrisecant — A line meeting a spatial curve or related geometric set in four distinct points, a maximal generic multi-secant whose existence and ordering carry knot- and algebraic-geometric information.
• Shear mapping — An affine transformation that displaces points parallel to a fixed direction by an amount proportional to signed distance from a fixed parallel hyperplane, preserving volume and collinearity while generally changing lengths and angles.
• Simplex — The convex hull of n+1 affinely independent points in n-dimensional space, generalizing a point, segment, triangle and tetrahedron as the simplest full-dimensional polytope.
• Star domain — A set containing a point from which the line segment to every point of the set remains inside the set.
• Star polyhedron — A nonconvex polyhedron with systematic star-like self-intersection or alternating salient and reentrant geometry.
• Three-dimensional space — A geometric space in which a point requires three independent coordinates, most commonly three-dimensional Euclidean space but also including general three-manifolds.
• Uniform polyhedron — A polyhedron with regular polygonal faces and a symmetry group transitive on vertices, including convex, star and self-intersecting examples.