Posts in category 'Algebraic & Coordinate Systems'

F.3 Biquadratic Forms in Heron’s Formula

Under the linear substitution X = a/2, W = (b + c)/2, and Y = (c - b)/2, Heron’s area formula reduces to the biquadratic product K = sqrt((W^2 - X^2)(X^2 - Y^2)).

F.14 Conway Notation

In John H. Conway’s triangle notation, setting S = 2K (twice the area of triangle ABC), S_omega = (a^2 + b^2 + c^2)/2, and S_A = (b^2 + c^2 - a^2)/2, S_B = (c^2 + a^2 - b^2)/2, S_C = (a^2 + b^2 - c^2)/2 reduces trigonometric and barycentric identities to polynomial expressions satisfying S_B S_C + S_C S_A + S_A S_B = S^2.

F.31 Logic Tests

Given collinear points A, B, C, the three Boolean inequality tests T_1 = (|CA| < |AB|), T_2 = (|CB| < |AB|), and T_3 = (|CA| < |CB|) uniquely partition the line AB into Region A (-1 < k < 0), interior segment AB (0 < k < infty), and Region B (-infty < k < -1), determining the sign of k = +/- |AC| / |BC|.

F.47 The UVW Coordinate Triangle

Placing the altitude foot H_C at the origin (0, 0) with A = (-U, 0), B = (W, 0), and C = (0, V) parameterizes any triangle by three numbers (U, V, W) with a = sqrt(V^2 + W^2), b = sqrt(U^2 + V^2), c = U + W, and reduces the cevian traces D, E, F through P(p, q) to explicit 2x2 rational matrix formulas.

F.52 Homogeneous Barycentric Coordinates

Every triangle center X(n) has homogeneous barycentric coordinates (u : v : w) normalizable to absolute weights lambda_A + lambda_B + lambda_C = 1 such that X = lambda_A A + lambda_B B + lambda_C C; for the nine-point center N = X(5), lambda_A = 1/4 + (b^2 - c^2)(a^2 - (b^2 + c^2)) / (32 K^2).

F.53 Geometric Half-Angle Trigonometry

In a unit semicircle of radius R = 1 and diameter AB = 2, a point P at central angle theta has altitude sin(theta), base projections 1 + cos(theta) and 1 - cos(theta), and chord |AP| = 2 cos(theta/2) = sqrt(2 + 2 cos(theta)), giving cos^2(theta/2) = (1 + cos(theta))/2 and sin^2(theta/2) = (1 - cos(theta))/2.

F.54 HP-35s Triangle Program and Coordinate Scripts

In the canonical frame A = (0, 0), B = (M, 0), C = (I, J), the forward and inverse barycentric-Cartesian transformations BARY 2 CAR and CAR 2 BARY are given by P = M U + I V, Q = J V and V = Q / J, U = (P - I V) / M, T = 1 - (U + V).