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

Algebraic Separation of Unoriented Lines Replacing Transcendental Angle

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.

Interactive GEOMETOR construction for W.04 (Spread, Cross, Twist & Line Separation Ratios). Hover any geometric label in the text or figure to highlight the corresponding element.

A : Point B : Point P_2 : Point ===== l_1 = line(A, B) l_2 = line(A, P_2) C = foot(B, l_2) n_bc = segment(B, C)

Mathematical Synthesis

Classical angle requires circular arc-length integration, ray orientation, and \(\pi\) ambiguities across eight possible arcs at an intersection. Norman J. Wildberger replaces angle with spread \(s(\ell_1, \ell_2) \in F\) (Wildberger2005): if l1 and l2 intersect at A, choosing any point B \(\neq A\) on l1 and dropping altitude nbc to its foot C on l2 yields the invariant ratio \(s(\ell_1, \ell_2) = Q(B, C) / Q(A, B)\).

Paired with spread are the cross \(c(\ell_1, \ell_2) = 1 - s(\ell_1, \ell_2) = Q(A, C)/Q(A, B)\) and the twist \(t(\ell_1, \ell_2) = s / c = Q(B, C)/Q(A, C)\), which squares the signed turn \(u(\ell_1, \ell_2)\). Supplementary angles share the exact same line, so supplementary spreads are identical, while orthogonal lines swap spread and cross (\(s \leftrightarrow 1 - s\)).

Key Algebraic Identities

\[s(\ell_1, \ell_2) = \frac{(a_1 b_2 - a_2 b_1)^2}{(a_1^2 + b_1^2)(a_2^2 + b_2^2)} = \frac{Q(B, C)}{Q(A, B)}, \qquad c(\ell_1, \ell_2) = 1 - s(\ell_1, \ell_2) = \frac{(a_1 a_2 + b_1 b_2)^2}{(a_1^2 + b_1^2)(a_2^2 + b_2^2)}\]

\[t(\ell_1, \ell_2) = \frac{s(\ell_1, \ell_2)}{c(\ell_1, \ell_2)} = u(\ell_1, \ell_2)^2, \qquad \ell_3 \perp \ell_2 \implies s(\ell_1, \ell_3) = 1 - s(\ell_1, \ell_2)\]

Symbolic SymPy Verification

Invariant / Theorem Check	Exact SymPy Value	Verification Status
Spread s(l_1, l_2)	1/5	Matches Q(B,C)/Q(A,B) = 1/5
Cross c(l_1, l_2) = 1 - s	4/5	Matches Q(A,C)/Q(A,B) = 4/5
Twist t(l_1, l_2) = s / c	1/4	Matches u(l_1,l_2)^2 = (-1/2)^2
Complementary orthogonal spread	4/5	s(l_1, l_2^perp) = c(l_1, l_2)

Cross-Repository Canon

Executable Demonstration	W.04 Spread, Cross, Twist & Line Separation Ratios — Code (Standalone model.py & w-04.svg)
Thematic Part	Part I: Preliminaries & Foundational Invariants
Euclid’s Elements	I.11, I.13, VI.4 (EuclidElements)
GEOMETOR Elements	G.3 extremities of a segment, G.5 surface
GEOMETOR Fundamentals	F.14 Conway Notation
Sister Codices	geometor/divine (exact algebraic ratios without floating-point trig)

Constructive Dependency Graph

Depends on: W.01 Planar Points, Line Proportions & Incidence Determinants, W.03 Quadrance, Midpoints & Point-to-Line Separation

Required for: W.05 Spread Polynomials S_n(s) & Pentagonal Golden Spreads, W.06 Law I: Pythagoras’ Theorem & Right Triangle Invariants, W.08 Law III: The Spread Law & Extended Circumquadrance Law, W.09 Law IV: The Cross Law & Quadrea-Spread Identity, W.10 Law V: Triple Spread, Triple Cross & Quadruple Spread Formulas, W.12 Vertex Bisectors, Square Vertices & The Four Incenters, W.22 Projective Cross-Ratio, Harmonic Pencils & The Pappus Bridge, W.28 Rational Polar Coordinates, Classical Curves & The Ellipson