Posts tagged 'cross'

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.

W.09 Law IV: The Cross Law & Quadrea-Spread Identity

In any triangle A_1 A_2 A_3 with non-null sides Q_1, Q_2 meeting with spread s_3 and cross c_3 = 1 - s_3 at vertex A_3, the quadrances satisfy (Q_1 + Q_2 - Q_3)^2 = 4 * Q_1 * Q_2 * (1 - s_3) = 4 * Q_1 * Q_2 * c_3, equivalently A(Q_1, Q_2, Q_3) = 4 * Q_1 * Q_2 * s_3.