A natural analogue of a quadrangle in projective geometry. In projective geometry, distinguished order and betweenness on the lines are nonconcepts hence one does not talk about segments but only lines through two points. For 4 points, it is also usually useful to consider lines joining any two points, rather than joining adjacent for some cyclic ordering.
In projective geometry, a complete quadrangle (often referred to only as a quadrangle in this context) consists of 4 points (vertices) which are coplanar but neither 3 of which are colinear and all the 6 lines (sides) incident with pairs of points.
It is determined and sometimes identified with the 4 points only, but the terminology on sides is understood. Notice that each 3 points being not on the same line is equivalent for this 4 points to form a projective frame.
Two sides which do not meet in the same point are called opposite. Each complete quadrangle has 3 pairs of opposite sides. Intersection of each pair is called a diagonal point of the complete quadrangle and each line through two diagonal points a diagonal line of the complete quadrangle. This terminology is rather distinct from the terminology on diagonals of quadrangles in affine geometry. Fano’s axiom states that the three diagonal points are not on the same line.
Last revised on April 8, 2025 at 09:41:44. See the history of this page for a list of all contributions to it.