F.2 Bilinear Transformations — Folios

Archival manuscript scans and diplomatic transcriptions from seelig’s notebooks for F.2 (Bilinear Transformations) (SeeligNotebooks).

Folio 0003 (Page 1 of 8): Bilinear Transformation & Derivatives

[Folio 0003]

Folio 0003 (original scan 0003.jpg) from resources/seelig/02_bilinear-transformations.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0003 (0003.jpg, Sheet 1 of 8):

Defines \(\mathbb{B}(1,x) = \frac{1+x}{1-x} = A(x)B(x)\), derives \(\mathbb{B}' = 2B^2\) and \(\mathbb{B}'' = 4B^3\), and plots the hyperbola with vertical asymptote \(x = 1\).

\[\mathbb{B}(x) = \frac{1 + x}{1 - x}, \qquad \mathbb{B}^{-1}(y) = \frac{y - 1}{y + 1}, \qquad \mathbb{B}'(x) = \frac{2}{(1 - x)^2} = 2 B^2\]

Editorial & Symbolic Notes

Archival scan 0003.jpg from Fred W. Seelig’s notebook dossier on F.2 Bilinear Transformations (Page 1 of 8). Companion OpenDocument worksheet(s): Harmonic Conjugates, Apollonian Circles, HC Tables. Classical Euclidean foundations: VI.3.

Folio 0004 (Page 2 of 8): Inverse Transformation & Unit Square Grid

[Folio 0004]

Folio 0004 (original scan 0004.jpg) from resources/seelig/02_bilinear-transformations.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0004 (0004.jpg, Sheet 2 of 8):

Plots \(\mathbb{B}(x)\) and \(\mathbb{B}^{-1}(y) = \frac{y-1}{y+1}\) on a unit-square grid and re-verifies \(\mathbb{B}' = B(1+\mathbb{B}) = 2B^2\).

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

Editorial & Symbolic Notes

Archival scan 0004.jpg from Fred W. Seelig’s notebook dossier on F.2 Bilinear Transformations (Page 2 of 8). Companion OpenDocument worksheet(s): Harmonic Conjugates, Apollonian Circles, HC Tables. Classical Euclidean foundations: VI.3.

Folio 0005 (Page 3 of 8): Second Derivative Forms & C.a.R. Conic Fit (FWS 12/5/2013)

[Folio 0005]

Folio 0005 (original scan 0005.jpg) from resources/seelig/02_bilinear-transformations.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0005 (0005.jpg, Sheet 3 of 8):

Expresses \(\mathbb{B}''\) in two algebraic forms and derives the C.a.R. conic curve equation \(x/4 - y/4 - xy/4 = 1/4\).

\[\mathbb{B}(x) = \frac{1 + x}{1 - x}, \qquad \mathbb{B}^{-1}(y) = \frac{y - 1}{y + 1}, \qquad \mathbb{B}'(x) = \frac{2}{(1 - x)^2} = 2 B^2\]

Editorial & Symbolic Notes

Archival scan 0005.jpg from Fred W. Seelig’s notebook dossier on F.2 Bilinear Transformations (Page 3 of 8). Companion OpenDocument worksheet(s): Harmonic Conjugates, Apollonian Circles, HC Tables. Classical Euclidean foundations: VI.3.

Folio 0006 (Page 4 of 8): Table 1 for B(x) & Inverse Derivatives

[Folio 0006]

Folio 0006 (original scan 0006.jpg) from resources/seelig/02_bilinear-transformations.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0006 (0006.jpg, Sheet 4 of 8):

Tabulates \(x, A, B, \mathbb{B}, \mathbb{B}'\) for \(x \in \{-1, 0, 1/2, -1/3, 3\}\) and differentiates \(\mathbb{B}^{-1}(y)\).

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

Editorial & Symbolic Notes

Archival scan 0006.jpg from Fred W. Seelig’s notebook dossier on F.2 Bilinear Transformations (Page 4 of 8). Companion OpenDocument worksheet(s): Harmonic Conjugates, Apollonian Circles, HC Tables. Classical Euclidean foundations: VI.3.

Folio 0007 (Page 5 of 8): Table 2 for B^{-1}(y) & Conic Fit

[Folio 0007]

Folio 0007 (original scan 0007.jpg) from resources/seelig/02_bilinear-transformations.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0007 (0007.jpg, Sheet 5 of 8):

Tabulates \(y, A, B, \mathbb{B}^{-1}, {\mathbb{B}^{-1}}'\) for \(y \in \{0, 1, 2, 3, 4\}\) and records the C.a.R. conic equation \(-x/4 + y/4 - xy/4 = 1/4\).

\[\mathbb{B}(x) = \frac{1 + x}{1 - x}, \qquad \mathbb{B}^{-1}(y) = \frac{y - 1}{y + 1}, \qquad \mathbb{B}'(x) = \frac{2}{(1 - x)^2} = 2 B^2\]

Editorial & Symbolic Notes

Archival scan 0007.jpg from Fred W. Seelig’s notebook dossier on F.2 Bilinear Transformations (Page 5 of 8). Companion OpenDocument worksheet(s): Harmonic Conjugates, Apollonian Circles, HC Tables. Classical Euclidean foundations: VI.3.

Folio 0008 (Page 6 of 8): Bilinear Transformation — Scaling Properties

[Folio 0008]

Folio 0008 (original scan 0008.jpg) from resources/seelig/02_bilinear-transformations.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0008 (0008.jpg, Sheet 6 of 8):

Sets up the scaling relation \(\mathbb{B}(R, \eta x) = \mathbb{B}(1, x)\).

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

Editorial & Symbolic Notes

Archival scan 0008.jpg from Fred W. Seelig’s notebook dossier on F.2 Bilinear Transformations (Page 6 of 8). Companion OpenDocument worksheet(s): Harmonic Conjugates, Apollonian Circles, HC Tables. Classical Euclidean foundations: VI.3.

Folio 0009 (Page 7 of 8): Apollonian Proportion (FWS 2/7/2014)

[Folio 0009]

Folio 0009 (original scan 0009.jpg) from resources/seelig/02_bilinear-transformations.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0009 (0009.jpg, Sheet 7 of 8):

Applies \(k = \frac{1+\kappa}{1-\kappa}\) to harmonic points \(A, C, B, D\) with \(B=0, A=-d\), deriving \(C = -d/(k+1), D = d/(k-1), M' = d/(k^2-1)\), and \(R = dk/(k^2-1)\).

\[\mathbb{B}(x) = \frac{1 + x}{1 - x}, \qquad \mathbb{B}^{-1}(y) = \frac{y - 1}{y + 1}, \qquad \mathbb{B}'(x) = \frac{2}{(1 - x)^2} = 2 B^2\]

Editorial & Symbolic Notes

Archival scan 0009.jpg from Fred W. Seelig’s notebook dossier on F.2 Bilinear Transformations (Page 7 of 8). Companion OpenDocument worksheet(s): Harmonic Conjugates, Apollonian Circles, HC Tables. Classical Euclidean foundations: VI.3.

Folio 0010 (Page 8 of 8): Apollonian Circles for k = 2, 3, 4

[Folio 0010]

Folio 0010 (original scan 0010.jpg) from resources/seelig/02_bilinear-transformations.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0010 (0010.jpg, Sheet 8 of 8):

Tabulates and sketches the nested Apollonian circles with \((M', R) = (\frac{1}{3}d, \frac{2}{3}d), (\frac{1}{8}d, \frac{3}{8}d), (\frac{1}{15}d, \frac{4}{15}d)\).

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

Editorial & Symbolic Notes

Archival scan 0010.jpg from Fred W. Seelig’s notebook dossier on F.2 Bilinear Transformations (Page 8 of 8). Companion OpenDocument worksheet(s): Harmonic Conjugates, Apollonian Circles, HC Tables. Classical Euclidean foundations: VI.3.