Posts tagged 'collinearity'

W.01 Planar Points, Line Proportions & Incidence Determinants

Over any field F of characteristic not equal to 2, a planar point is an ordered pair A = [x, y] and a planar line is a homogeneous 3-proportion l = <a : b : c> with (a, b) != (0, 0) representing a*x + b*y + c = 0; three points are collinear and three lines are concurrent if and only if their respective 3x3 coordinate determinants vanish.

W.07 Law II: Triple Quad Formula, Archimedes’ Function & Quadrea

Three planar points A_1, A_2, A_3 with pairwise quadrances Q_1, Q_2, Q_3 are collinear if and only if Archimedes’ symmetric quadratic form A(Q_1, Q_2, Q_3) = (Q_1 + Q_2 + Q_3)^2 - 2(Q_1^2 + Q_2^2 + Q_3^2) vanishes; for any triangle, A(Q_1, Q_2, Q_3) equals 4 times the squared coordinate determinant (16 * Delta^2), defining the triangle’s quadrea.