Posts in category 'Foundational Quadratic Invariants'

W.03 Quadrance, Midpoints & Point-to-Line Separation

The quadrance between A_1 = [x_1, y_1] and A_2 = [x_2, y_2] is the quadratic form Q(A_1, A_2) = (x_2 - x_1)^2 + (y_2 - y_1)^2; midpoint M divides the quadrance by 4, the locus Q(X, A_1) = Q(X, A_2) is the linear perpendicular bisector, and the quadrance from A_0 to a non-null line l = <a : b : c> is (a*x_0 + b*y_0 + c)^2 / (a^2 + b^2).

W.04 Spread, Cross, Twist & Line Separation Ratios

For two non-null lines l_1 = <a_1 : b_1 : c_1> and l_2 = <a_2 : b_2 : c_2>, the spread s(l_1, l_2) = (a_1*b_2 - a_2*b_1)^2 / ((a_1^2 + b_1^2)*(a_2^2 + b_2^2)) depends only on the unoriented lines, vanishes iff l_1 || l_2, equals 1 iff l_1 perp l_2, and equals Q(B, C) / Q(A, B) for any altitude BC dropped from B on l_1 to C on l_2.