Steiner conic

A conic generated by the intersections of corresponding rays in two projectively related pencils of lines. The correspondence pairs each line through one fixed point with a line through another.

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Projective pencil: A family of lines through one fixed point, considered with projective relations among its members. Each of the two generating families is a pencil centered at a fixed point.
Projective geometry: The geometry of properties preserved by projective transformations, especially incidence and cross-ratio. Steiner's construction is defined by projective relations rather than metric measurements.
Pascal's theorem: A theorem stating that opposite sides of a hexagon inscribed in a conic meet at three collinear points. It provides a projective incidence theorem for points on the conic generated by the pencils.
Jakob Steiner: A Swiss mathematician whose work helped establish synthetic projective geometry. The conic bears Steiner's name and reflects his synthetic approach to geometry.
Projectivity: A bijective transformation of a projective space that preserves incidence and cross-ratio. A projectivity pairs rays in the two pencils to produce the conic.
Central projection: A projection that maps points or lines through rays from a fixed center. Projecting one pencil onto another supplies a basic model for pairing their rays.
Brianchon's theorem: A theorem stating that the three diagonals of a hexagon circumscribed about a conic are concurrent. Its tangent-based counterpart complements point-based constructions of conics.
Jean-Victor Poncelet: A French mathematician who developed foundational methods in projective geometry. Poncelet's work helped shape the projective framework in which Steiner conics are studied.
Conic section: A curve obtained by intersecting a plane with a cone, equivalently a nondegenerate plane curve of degree two. The locus generated by the paired rays is a conic.
Incidence geometry: The study of which geometric objects meet or contain one another. The conic arises from the incidence of each pair of corresponding rays.