W.22 Projective Cross-Ratio, Harmonic Pencils & The Pappus Bridge

Exact Equivalence of Line-Pencil Spread Cross-Ratio and Point-Range Quadrance Cross-Ratio

If four concurrent lines r_1, r_2, r_3, r_4 through O_p meet a transversal l_a at collinear points A_1, A_2, A_3, A_4 and another transversal l_b at B_1, B_2, B_3, B_4, then the squared cross-ratio in quadrances (Q(A_1, A_3)*Q(A_2, A_4)) / (Q(A_1, A_4)*Q(A_2, A_3)) equals the pencil spread cross-ratio (s_13 * s_24) / (s_14 * s_23) and is invariant across all transversals.
l_a l_b r_1 r_2 r_3 r_4 O_p = [4, 8] A_1 = [0, 0] A_2 = [4, 0] A_3 = [3, 0] A_4 = [6, 0] B_1 = [12/7, 24/7] B_2 = [4, 4] B_3 = [108/31, 120/31] B_4 = [84/17, 72/17] A = [-12, 0] Harmonic Range [0,4; 3,6]: CR_Q = 1 Pencil Spread CR_s = CR_Q(B) = 1 OpOp A1A1 A2A2 A3A3 A4A4 B1B1 B2B2 B3B3 B4B4 + − ⟲ Lbl Gde

Interactive GEOMETOR construction for W.22 (Projective Cross-Ratio, Harmonic Pencils & The Pappus Bridge). Hover any geometric label in the text or figure to highlight the corresponding element.

O_p : Point A_1 : Point A_2 : Point B_1 : Point B_2 : Point ===== l_a = line(A_1, A_2) l_b = line(B_1, B_2) r_1 = line(O_p, A_1) r_2 = line(O_p, A_2) r_3 = line(O_p, A_3) r_4 = line(O_p, A_4) B_3 = meet(r_3, l_b) B_4 = meet(r_4, l_b)

Mathematical Synthesis

Pappus of Alexandria discovered that the cross-ratio of four collinear points is preserved under central projection (CoxeterGreitzer1967). In Rational Trigonometry (Wildberger2005), squaring the classical signed cross-ratio replaces signed lengths with quadrance along transversals la (A1, A2, A3, A4) and lb (B1, B2, B3, B4), and replaces directed sine ratios in the pencil of rays r1, r2, r3, r4 through Op with pairwise spread!

Applying the Spread Law to the four sub-triangles at apex Op cancels all radial quadrances \(Q(O_p, A_i)\) and transversal spreads \(s(r_i, \ell_a)\) in a single stroke, proving \(R_Q(A_1, A_2; A_3, A_4) = R_s(r_1, r_2; r_3, r_4) = R_Q(B_1, B_2; B_3, B_4)\). When the signed cross-ratio is \(-1\) (harmonic conjugates), its quadrance cross-ratio is identically \((-1)^2 = 1\), directly uniting geometor/wildberger, geometor/pappus, and F.25 Harmonic Conjugates!

Key Algebraic Identities

\[(A_1, A_2; A_3, A_4)_Q = \frac{Q(A_1, A_3)\, Q(A_2, A_4)}{Q(A_1, A_4)\, Q(A_2, A_3)} = \frac{s(r_1, r_3)\, s(r_2, r_4)}{s(r_1, r_4)\, s(r_2, r_3)} = (B_1, B_2; B_3, B_4)_Q\]
\[\text{Harmonic Range } (A_1, A_2; A_3, A_4) = -1 \implies Q(A_1, A_3)\, Q(A_2, A_4) = Q(A_1, A_4)\, Q(A_2, A_3)\]

Symbolic SymPy Verification

Invariant / Theorem Check

Exact SymPy Value

Verification Status

Harmonic quadruple (A_1, A_2; A_3, A_4)

[0,0], [4,0]; [3,0], [6,0]

Signed CR = -1 => Quadrance CR_Q = 1

Pencil spread cross-ratio CR_s at O_p

1

Equals (s_13*s_24)/(s_14*s_23) = 1

Projected range (B_1, B_2; B_3, B_4) on l_b

CR_Q(B) = 1

Exact projective invariance CR_a == CR_s == CR_b

Cross-Repository Canon

Executable Demonstration

W.22 Projective Cross-Ratio, Harmonic Pencils & The Pappus Bridge — Code (Standalone model.py & w-22.svg)

Thematic Part

Part III: Universal Triangle, Conic & Chromogeometry

Classical Canon

Pappus of Alexandria (Collection Book VII) & Girard Desargues

GEOMETOR Elements

G.3 extremities of a segment, G.5 surface

GEOMETOR Fundamentals

F.2 Bilinear Transformations, F.25 Harmonic Conjugates (SeeligNotebooks)

Sister Codices

geometor/pappus (projective cross-ratio & harmonic ranges)

Constructive Dependency Graph

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Depends on: W.01 Planar Points, Line Proportions & Incidence Determinants, W.03 Quadrance, Midpoints & Point-to-Line Separation, W.04 Spread, Cross, Twist & Line Separation Ratios, W.08 Law III: The Spread Law & Extended Circumquadrance Law