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Projective Geometry & Duality

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Abstractions about projective lines, bundles, extensions, collineations, poles and polars, quadrics, configurations, and projective curves.

10 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.

  • Collineation — A bijection of projective spaces that preserves collinearity.
  • Hyperboloid — A nondegenerate central quadric whose real canonical equation has either one sheet or two sheets according to the signs of its squared-coordinate terms.
  • Klein configuration — A symmetric incidence configuration of sixty points and sixty planes in projective three-space, with fifteen incidences through every point and on every plane.
  • Pole and polar — A reciprocal point-line correspondence induced by a nondegenerate conic or quadric that reverses incidence.
  • Projective bundle — A fiber bundle or scheme morphism locally modeled on projective space, often obtained by projectivizing a vector bundle.
  • Projective line — The one-dimensional projective space over a field or ring, commonly the one-dimensional subspaces of a two-dimensional vector space and equivalently an affine line completed by points at infinity.
  • Projectively extended real line — The real line completed by one unsigned point at infinity, yielding a topological circle and the real projective line.
  • Projectivization — The construction that maps a nonzero vector space, cone or vector bundle to its space of one-dimensional linear subspaces by quotienting nonzero vectors under scalar equivalence.
  • Unital (geometry) — A 2-(n³+1,n+1,1) block design in which every pair of points lies on exactly one block, with embedded unitals meeting each projective-plane line in one or n+1 points.
  • W-curve — A curve in projective space invariant under a one-parameter subgroup of projective transformations, so its entire path is an orbit and its projective differential invariants remain constant.