Non-metric geometry {projective geometry}| can require that parallel lines meet at infinity {elliptic parallel axiom}: at a vanishing point on the horizon line. In the projective plane, two lines intersect at one point {elliptic incidence property}, so, for a line and a point not on the line, all lines through the point intersect the line once. In projective space, two planes intersect at one line.
Projective geometry involves constructions with straightedge only (no compass). Straight lines remain straight. Projective geometry is not about circles, angles, and parallels. All conic sections are equivalent.
Projective geometry has no circles, no angles, no measurements (no metric), no parallels, and no between-ness (no intermediacy).
Collinearity is invariant in projective geometry. Therefore, projective geometry has invariant point and line structures (incidence structure) (incidence relation) that make generalized planes. The Fano plane is the simplest 2-dimensional finite incidence structure.
Cross ratio is relation of projective harmonic conjugates. Cross ratios are invariant in projective geometry. Cross ratio is ratio of distance ratios. For four distinct collinear points A, B, C, and D in sequence, the cross ratio is (AC/BC)*(BD/AD) = (medium1/short)*(medium2/long) = (AC/BC) / (AD/BD) = (medium1/short)/(long/medium2) = (AC*BD)/(BC*AD) = (13*24)/(23*14) = (medium1*medium2)/(short*long). For a line, if three points and the cross ratio are known, the fourth point is known.
Projective geometry has surface curve categories.
Projective geometry includes non-Riemannian geometry, single elliptic geometry, double elliptic geometry, and hyperbolic geometry.
Mathematical Sciences>Geometry>Kinds>Projective Geometry
3-Geometry-Kinds-Projective Geometry
Outline of Knowledge Database Home Page
Description of Outline of Knowledge Database
Date Modified: 2022.0224