Metric Geometry & Transformations

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Abstractions about spaces, curves, coordinate systems, metrics, and transformations that preserve or compare geometric structure. They include isometries, reflections, inversions, projections, distance bounds, canonical shapes, and geometric constants across Euclidean and generalized spaces.

46 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.

• Abstract index notation — A basis-free tensor notation in which index letters name argument slots and variance rather than numerical components.
• CAT(k) space — A geodesic metric space whose triangles are no thicker than comparison triangles in the constant-curvature model space of curvature k, within the prescribed perimeter range.
• 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.
• Covering number — The minimum number of radius-r balls required to cover a specified subset of a metric or pseudometric space.
• Crofton formula — An integral-geometric identity recovering a curve’s length from the invariant-measure average number of intersections with lines.
• Curve — A one-dimensional path represented, in its broad topological form, as the continuous image of an interval in a space.
• Denjoy's theorem on rotation number — A regularity theorem stating that an orientation-preserving circle diffeomorphism with irrational rotation number and derivative of bounded variation is topologically conjugate to the corresponding irrational rotation.
• Direction (geometry) — An equivalence class of parallel oriented lines, rays, or nonzero vectors that retains orientation while discarding position and usually magnitude.
• Equivalence of metrics — A relation between metrics that captures equality of induced topology, uniformity, Lipschitz structure, or another declared level of geometric behavior.
• Fixed points of isometry groups in Euclidean space — The common-fixed-point set of a Euclidean isometry group, which is either empty or an affine subspace.
• Golden ellipse — An ellipse whose major-to-minor semiaxis ratio equals the golden ratio.
• Hadamard space — A complete geodesic metric space of nonpositive curvature in the CAT(0) sense, equivalently satisfying a strong midpoint convexity inequality.
• 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.
• Injective metric space — A metric space into which every nonexpansive map from a subspace extends nonexpansively over the containing space, equivalently a hyperconvex space.
• Inversion transformation — A conformal coordinate transformation that maps a nonzero point to a reciprocal radial position and, with translations and rotations, extends Poincaré symmetry toward the conformal group.
• Isometry group — The group of all bijective self-maps of a metric space that preserve every distance, with composition encoding the space's exact metric symmetries.
• Kepler–Bouwkamp constant — The positive constant equal to the infinite product of cos(pi divided by k) for integers k from three onward, arising as the limiting radius in nested regular-polygon incircle construction.
• 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.
• Nabla symbol — The symbol ∇, read nabla or del, used to denote the vector differential operator whose combination with scalar and vector fields yields gradient, divergence, curl, and related coordinate-dependent expressions.
• Optical space — A coordinate-bearing mathematical view of an optical system associated with a refractive region such as object, image or intermediate space.
• Orthoptic (geometry) — The locus of points from which two tangents to a given curve meet at a right angle.
• Parabola — A conic curve whose points are equidistant from a fixed focus and directrix, equivalently a nondegenerate quadratic curve of eccentricity one.
• Pi — The dimensionless mathematical constant equal to a Euclidean circle's circumference divided by its diameter, equivalently characterized through analysis, with an irrational transcendental value near 3.14159.
• 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.
• Polar space — An incidence geometry of points and singular subspaces modeled by isotropic subspaces of a form and satisfying the one-or-all axiom.
• Polyhedral space — A metric space assembled by gluing constant-curvature simplices isometrically along compatible faces.
• Positive form — A real differential form of Hodge type (p,p) satisfying a declared positivity cone, with several inequivalent notions in higher bidegree.
• Positively separated sets — Two nonempty subsets of a metric space whose infimum pairwise distance is strictly greater than zero.
• Procrustes transformation — A similarity transformation using translation, rotation, reflection if allowed and uniform scaling while preserving shape.
• Quillen metric — A Hermitian metric on a determinant line of elliptic operators obtained by correcting an L2 metric with a zeta-regularized analytic torsion factor.
• Radon transform — An integral transform mapping a function to its integrals over hyperplanes, with inversion reconstructing the original under suitable conditions.
• Scalar field — A function assigning one scalar quantity to every point of a space or spacetime region, invariant under coordinate changes appropriate to a scalar.
• 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.
• Special conformal transformation — A conformal map obtained by composing inversion, translation, and inversion, represented in Euclidean or Minkowski coordinates by a characteristic fractional transformation.
• Star domain — A set containing a point from which the line segment to every point of the set remains inside the set.
• Statistical manifold — A differentiable family of probability distributions equipped with information-geometric structures, most canonically the Fisher information metric and compatible affine connections.
• Superquadrics — A parameterized family of three-dimensional shapes that generalizes quadrics by replacing squared coordinate terms with adjustable powers, producing rounded, boxlike or pinched forms.
• 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.
• Translation plane — A projective plane containing a line whose elation group acts transitively on the affine points of each line parallel to a fixed direction, yielding an affine translation structure.
• Translation surface — A surface assembled from Euclidean polygons by identifying parallel boundary edges through translations, yielding a flat metric away from finitely many conical singularities.
• Triangle inequality — The distance or norm axiom stating that a direct separation is no greater than the length of any two-step path, d(x,z)≤d(x,y)+d(y,z).
• Two-point tensor — A tensor-like linear map whose indices transform in two different vector spaces, commonly connecting a material reference configuration with a current spatial configuration.
• Ultrametric space — A metric space satisfying the strong triangle inequality, so every triangle is isosceles with its two largest distances equal and balls form a nested hierarchy.
• Uniformly disconnected space — A metric space with one scale-independent constant preventing any two distinct points from being joined by a chain of sufficiently small relative steps.
• Unit sphere — The set of all vectors or points at norm exactly one from an origin in a specified normed space.