diff --git a/6-Documentation/famm/MOBIUS_APOLLONIUS_CHORD_PARTITION_GATE.md b/6-Documentation/famm/MOBIUS_APOLLONIUS_CHORD_PARTITION_GATE.md index 10560f31..9adfd569 100644 --- a/6-Documentation/famm/MOBIUS_APOLLONIUS_CHORD_PARTITION_GATE.md +++ b/6-Documentation/famm/MOBIUS_APOLLONIUS_CHORD_PARTITION_GATE.md @@ -4,6 +4,15 @@ Add the uploaded visualization topic — Möbius transforms, Circles of Apollonius, spirals, angles, and trigonometric chord partitions — as a conformal-geometry witness gate in the FAMM/BraidStorm/Universal Shortcut Center stack. +Primary visual source: + +```text +Möbius Transform and Circles of Apollonius - Spirals Angles Spirals Trigonometric Partitions. +YouTube video: https://www.youtube.com/watch?v=ndjz5tVPywM +Local uploaded artifact: Möbius Transform and Circles of Apollonius - Spirals Angles Spirals Trigonometric Partitions (720p, h264).mp4 +Accessed / integrated: 2026-05-18. +``` + The useful project shape is: ```text @@ -39,6 +48,8 @@ T'(z)=\frac{ad-bc}{(cz+d)^2} so local angles are preserved away from the pole. +References: Ahlfors, *Complex Analysis*; Needham, *Visual Complex Analysis*; Beardon, *The Geometry of Discrete Groups*. + ## Apollonius circle witness A Circle of Apollonius is the locus: @@ -61,6 +72,8 @@ fixed source pair (a,b) Under a Möbius transformation, the Apollonius family is transported into another generalized-circle family, while cross-ratio and angle structure provide the guard conditions. +References: Coxeter and Greitzer, *Geometry Revisited*; Needham, *Visual Complex Analysis*. + ## Cross-ratio invariant The primary exact witness is the cross ratio: @@ -95,6 +108,8 @@ Pass condition: R_{\mathrm{cr}}\le \Theta_{\mathrm{tol}} ``` +References: Ahlfors, *Complex Analysis*; Beardon, *The Geometry of Discrete Groups*. + ## Chord / unit-circle partition witness On the unit circle, a chord between two points with angular separation `theta` has length: @@ -118,6 +133,8 @@ angle partition → conformal transport through Möbius map ``` +Reference: Coxeter and Greitzer, *Geometry Revisited*. + ## Spiral connection Möbius transforms can turn simple circle/line pencils into visually spiral-like families under parameterized motion or composition. Project use: @@ -258,6 +275,53 @@ Disallowed claim: A GeoGebra visualization alone proves a new Riemann, spiral, or partition theorem. ``` +## References + +### Visual source + +```bibtex +@online{youtube_mobius_apollonius_spirals_2026, + title = {Möbius Transform and Circles of Apollonius - Spirals Angles Spirals Trigonometric Partitions}, + organization = {YouTube}, + url = {https://www.youtube.com/watch?v=ndjz5tVPywM}, + urldate = {2026-05-18}, + note = {User-supplied video source; local uploaded artifact title: Möbius Transform and Circles of Apollonius - Spirals Angles Spirals Trigonometric Partitions (720p, h264).mp4} +} +``` + +### Mathematical references + +```bibtex +@book{ahlfors1979complex, + title = {Complex Analysis: An Introduction to the Theory of Analytic Functions of One Complex Variable}, + author = {Ahlfors, Lars V.}, + edition = {3}, + publisher = {McGraw-Hill}, + year = {1979} +} + +@book{needham1997visual, + title = {Visual Complex Analysis}, + author = {Needham, Tristan}, + publisher = {Oxford University Press}, + year = {1997} +} + +@book{beardon1983geometry, + title = {The Geometry of Discrete Groups}, + author = {Beardon, Alan F.}, + publisher = {Springer}, + year = {1983} +} + +@book{coxeter1967geometry, + title = {Geometry Revisited}, + author = {Coxeter, H. S. M. and Greitzer, S. L.}, + publisher = {Mathematical Association of America}, + year = {1967} +} +``` + ## Project sentence The Möbius-Apollonius gate turns complex-plane geometry into a receipt-bearing conformal transport layer: Möbius maps move circles, lines, Apollonius ratio loci, and chord partitions while cross-ratio, angle, and chord-length witnesses tell FAMM whether the projected geometry stayed lawful or became a scarred conformal shadow.