F.2 Bilinear Transformations

Projective Involutions, Derivative Identities, and the Apollonian Proportion

The canonical bilinear transformation B(x) = (1 + x)/(1 - x) and its inverse B^{-1}(y) = (y - 1)/(y + 1) link the internal division ratio kappa = x/r of a segment AB to the Apollonian distance ratio k = (1 + kappa)/(1 - kappa).
l l' / S A / / S C / / S B / / S D / A (0.000, 0.000) C (2.500, 0.000) B (4.000, 0.000) D (10.000, 0.000) S (3.200, 4.200) A' (0.640, 0.840) C' (2.750, 1.500) B' (3.680, 1.810) D' (6.220, 2.680) AA CC BB DD SS A'A' C'C' B'B' D'D' + − ⟲ Lbl Gde

Interactive GEOMETOR construction for F.2 (Bilinear Transformations). Hover any geometric label in the text or figure to highlight the corresponding element.

A : Point B : Point C : Point S : Point ===== D = harmonic_conjugate(A, B, C) l = line(A, B) l_prime = projective_image(S, l)

Mathematical Synthesis

Dated December 5, 2013 and February 7, 2014 (SeeligNotebooks), Seelig’s study of the canonical bilinear transformation \(\mathbb{B}(x) = \frac{1+x}{1-x} = A(x)B(x)\) (with \(A(x) = 1+x\) and \(B(x) = (1-x)^{-1}\)) establishes its differential identities \(\mathbb{B}' = 2B^2 = B(1+\mathbb{B})\) and \(\mathbb{B}'' = 4B^3\), together with the rectangular hyperbola conic equation \(x/4 - y/4 - xy/4 = 1/4\) used for plotting in C.a.R.

In Folios 7–8, Seelig connects \(k = \mathbb{B}(\kappa) = \frac{1+\kappa}{1-\kappa}\) directly to the Apollonian harmonic conjugate division on segment A B of length \(d\), deriving the center \(M' = \frac{d}{k^2-1}\) and radius \(R = \frac{dk}{k^2-1}\) of the Apollonian circles for \(k = 2, 3, 4\).

Key Algebraic Identities

\[\mathbb{B}(x) = \frac{1 + x}{1 - x}, \qquad \mathbb{B}^{-1}(y) = \frac{y - 1}{y + 1}, \qquad \mathbb{B}'(x) = \frac{2}{(1 - x)^2} = 2 B^2\]
\[k = \frac{|AC|}{|BC|} = \frac{1 + \kappa}{1 - \kappa} \iff \kappa = \frac{k - 1}{k + 1}, \qquad M' = \frac{d}{k^2 - 1}, \qquad R = \frac{d k}{k^2 - 1}\]

Cross-Repository Canon & Archival Provenance

Seelig Archive Folios

F.2 Bilinear Transformations — Folios (Folios 0003–0010, 8 scanned manuscript pages in resources/seelig/02_bilinear-transformations)

Euclid’s Elements

VI.3 (EuclidElements)

GEOMETOR Elements (G-Index)

G.5 surface

Encyclopedia of Triangle Centers

General projective / metric locus (Kimberling1998)

Seelig Spreadsheets

Harmonic Conjugates, Apollonian Circles, HC Tables (Spreadsheets)

Constructive Dependency Graph

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Required for: F.5 Centers of Similitude, F.7 Ceva and Menelaus, F.12 Coaxial Circles, F.25 Harmonic Conjugates, F.27 Harmonic Mean, F.31 Logic Tests, F.51 Apollonian Circles and the Problem of Apollonius