Hidden intersections

Closing Quilez's hidden intersections: circles as a complex kei, mirrors, invariants. Theory, exercises, storyboards and an interactive explorer.

The idea in five lines

1. A line that "misses" a circle still meets it, at two complex conjugate points (Inigo Quilez's point).
2. Sweep a family of parallel lines: the real curve and its phantom branches form one locus, glued at the tangency points.
3. Treat circles and lines as points of P³(C) with a quadratic form Q; reflecting (inverting) one circle in another is then a linear reflection.
4. Mirroring, C ⋆ L, makes the circles a (complex) kei, and the mirrors generate PO(4,C). A circle meets its mirror image exactly where it meets the mirror (when ⟨C,L⟩ ≠ 0).
5. A single invariant, κ = ⟨C,L⟩²/(Q(C)Q(L)), sorts everything: κ = 1 is tangency, κ = cos²θ for crossing circles, and κ < 0 is possible.

The two Shorts

• Original: Inigo Quilez, Hidden intersections: https://www.youtube.com/shorts/cc-Z_igk4Lo
• Reply: Carlo Perassi, reply Short: https://youtube.com/shorts/IfTOfr-aFRE

All credit for the starting question and the construction goes to Inigo Quilez. This repository does not contain any of his video, audio or frames. video/quilez_transcript.tex describes his Short in text, with a link to the original.

Input × mirror: the full classification

pdf/classification.pdf works through every combination of an input (point, line, circle) and an operator (the mirror). Each parameter can be real, purely imaginary or complex, and the paper also covers how the two objects meet: secant, tangent, disjoint, nested, concentric, orthogonal, through a point, parallel, isotropic.

• 2025 raw parameter pairs collapse to 7 real classes and 35 cells (plus one rule: points cannot be mirrors). Each cell then splits into 88 leaves by the type of common points: two real points, a real double point, a conjugate pair, one real and one non-real, a non-real double point, or two non-real points that are not conjugate.
• A single real number, τ = ⟨X, X̄⟩/|Q(X)|, sorts single objects. Over ℂ, almost everything is one orbit, and κ is the only invariant of a pair.
• Every exclusion is proved, and every leaf has an exact example. verify/classification_verify.py runs 305 checks (about 2.5 minutes).

Sources: classification/ (LaTeX, generated tables tab_*.tex, data, helper modules, review log) and verify/classification_verify.py.

Interactive explorer

https://carlok.github.io/hidden-intersections/: choose a curve (circle, ellipse, parabola, cubic, quartic, lemniscate, Cassini oval) and a line direction, then drag to rotate the complex tangency locus. It is a single static app/index.html with no dependencies, so you can also open it locally.

Repository map

Path	Contents
theory/	theory.tex (the main paper), writeup.tex (phantom loci), section_inv.tex (inversive notes), alt_quandle_note.tex (kei/quandle note), figures figs/, alt_figs/ and their generators
classification/	classification.tex: exhaustive input × mirror classification; classification_build.py turns classification_data.json into the tab_*.tex tables; core/gen/explore helper modules; review log
exercises/	examples16.tex: 16 worked examples (real, complex and mixed), with figures and generators
video/	storyboards (v4, alt v2, third: .tex, .py, narration _script.txt), a text transcript of Quilez's Short, and the TTS / frame / render scripts
verify/	every verification script with its log; check_logs.py fails if any log reports a failure
app/	the standalone interactive explorer (index.html)
pdf/	compiled PDFs: theory, writeup, alt note, examples, classification, storyboards, transcript
docs/	README images
.github/workflows/	ci.yml (verify on every push and PR), pages.yml (verify, rebuild PDFs, deploy app/ + pdf/ to Pages on push to main)

Main reading order: pdf/theory.pdf, then pdf/examples16.pdf, pdf/classification.pdf, pdf/writeup.pdf and pdf/alt_quandle_note.pdf.

Build

python3 -m venv .venv && . .venv/bin/activate
pip install -r requirements.txt
make verify   # all verification scripts (theory_verify.py ~1 min, classification_verify.py ~2.5 min); logs land in verify/
make pdf      # needs tectonic (https://tectonic-typesetting.github.io)
make site     # assembles _site/ exactly as GitHub Pages serves it

Notes:

• The storyboard PDFs in pdf/ are committed rather than rebuilt: their .tex sources include rendered frame dumps (frames_v4/, alt_frames_v2/, alt3_frames/), which are git-ignored. You can regenerate them with the frame scripts in video/, but some earlier frame stages and all audio/video renders are not in the repository.
• The narration scripts (tts_gen.py, alt3_tts.py) need edge-tts and network access; the renders need ffmpeg.

Publishing (GitHub Pages)

After the repository is pushed, set Settings → Pages → Source to GitHub Actions. Every push to main then runs the verification, rebuilds the PDFs and publishes the site at https://carlok.github.io/hidden-intersections/. The suggested About text and topics are in repo_meta.json.

License

• Code (*.py, app/, Makefile, workflows): MIT.
• Text, LaTeX sources, figures and PDFs: CC BY 4.0.
• Inigo Quilez's Short is his own work and is not covered by either license.

Cite this work with CITATION.cff.