Geometric Objects & Transformations

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Abstractions about spatial objects and the transformations acting on them — curves and constructed loci (helix, cissoid, hodograph), coordinate systems and reference frames (log-polar coordinates, number line, unit sphere), and symmetry-preserving maps such as shear mapping, Procrustes transformation and translational symmetry.

24 abstractions in this family — domain-specific abstractions that sit near one another in structural-signature space (k-means over structural-signature embeddings). Each is shown with its short description.

• 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.
• Cissoid — A plane curve constructed from two base curves and a pole by placing, on each ray through the pole, a point whose directed radius combines the two ray intersections.
• Curve — A one-dimensional path represented, in its broad topological form, as the continuous image of an interval in a space.
• Direction (geometry) — An equivalence class of parallel oriented lines, rays, or nonzero vectors that retains orientation while discarding position and usually magnitude.
• Fundamental domain — A representative region for a group action containing one representative from each orbit, whose translates reconstruct the acted-on space up to boundary identifications.
• Helix — A space curve whose tangent maintains a constant angle with a fixed axis, with the circular helix as the constant-radius canonical case.
• Helmert transformation — A similarity transformation between three-dimensional geodetic coordinate frames using translations, rotations, and scale.
• Hodograph — Represent a changing vector by fixing its tail at one origin and tracing the locus of its head, so geometry in vector space exposes magnitude, direction, and derivative relations such as velocity and shear.
• Labyrinth — A unicursal or multicursal spatial pattern of winding passages organized around a difficult route to a center, exit, or goal.
• Line group — A discrete symmetry group of a structure periodic along one spatial axis, combining axial translations or screw operations with compatible point symmetries.
• Local property — A property that holds around each point or on sufficiently small neighborhoods, even if a corresponding global property may fail.
• Log-polar coordinates — A planar coordinate system whose angular coordinate is polar angle and whose radial coordinate is the logarithm of distance from a chosen origin.
• Mathematical diagram — A visual inscription whose spatial marks and conventions represent mathematical objects, relations, transformations, or proofs.
• Number line — A spatial representation of an ordered number system on a directed line, with a chosen origin, unit interval, and monotone coordinate correspondence.
• Polar space — An incidence geometry of points and singular subspaces modeled by isotropic subspaces of a form and satisfying the one-or-all axiom.
• Positive form — A real differential form of Hodge type (p,p) satisfying a declared positivity cone, with several inequivalent notions in higher bidegree.
• Procrustes transformation — A similarity transformation using translation, rotation, reflection if allowed and uniform scaling while preserving shape.
• Proportionality (mathematics) — A relation in which corresponding quantities maintain a constant ratio, or under inverse proportionality a constant product.
• 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.
• Star domain — A set containing a point from which the line segment to every point of the set remains inside the set.
• 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.
• Translational symmetry — Invariance of an object, field, law or equation under every translation in a stated continuous group or under translations in a discrete lattice.
• Traverse (surveying) — A surveying method that determines a sequence of control stations from measured distances and directions along connected lines.
• Unit sphere — The set of all vectors or points at norm exactly one from an origin in a specified normed space.