chore: translate all German text to English across code, docs, and CI
Unified the codebase language to English throughout. German text appeared in code comments, test file headers, CI step names, and several markdown documents. All natural-language text is now English; proper nouns (Institut für Mathematik, Technische Universität Berlin) are unchanged. Files changed: - .gitea/workflows/cpp-tests.yml — CI step names and job comments - code/include/mesh_utils.hpp — inline comment - code/tests/cgal/CMakeLists.txt — section comment block - code/tests/cgal/test_geometry_utils.cpp — full file header + all test comments - doc/math/references.md — geometry-central section - doc/math/validation.md — Section 9 (geometry-central cross-validation) - doc/roadmap/phases.md — Optional geometry-central track (GC-1/2/3) Co-Authored-By: Claude Sonnet 4.6 <noreply@anthropic.com>
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Tarik Moussa
@@ -97,58 +97,57 @@ Java features from `de.varylab.discreteconformal` not yet in C++:
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## ◼ Optional / Hypothetical — geometry-central Cross-Comparison
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> **Status: keine geplante Phase — rein explorativ.**
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> Diese Punkte sind keine Voraussetzung für Phase 8–10. Sie sind
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> interessant, weil geometry-central (Keenan Crane, CMU) auf denselben
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> mathematischen Grundlagen wie conformallab++ aufbaut — insbesondere auf
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> **Springborn 2020** und der direkten Weiterentwicklung durch
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> **Status: no planned phase — purely exploratory.**
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> These items are not prerequisites for Phase 8–10. They are
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> of interest because geometry-central (Keenan Crane, CMU) is built on the same
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> mathematical foundations as conformallab++ — in particular
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> **Springborn 2020** and its direct extension by
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> **Gillespie, Springborn & Crane (SIGGRAPH 2021)**.
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> Der entscheidende Unterschied: geometry-central löst dasselbe Problem
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> (diskrete konforme Äquivalenz) mit **intrinsischen Triangulierungen +
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> Ptolemäischen Flips**, während conformallab++ **Newton auf der
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> Original-Triangulierung** anwendet.
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> The key difference: geometry-central solves the same problem
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> (discrete conformal equivalence) using **intrinsic triangulations +
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> Ptolemaic flips**, while conformallab++ applies **Newton on the
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> original triangulation**.
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```
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GC-1 [optional, jetzt möglich]
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Mathematischer Output-Vergleich
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→ gleiche Testnetze (cathead.obj, brezel.obj, torus_4x4.off) in
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beide Bibliotheken laden
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→ UV-Koordinaten, u-Vektor, Residualnorm vergleichen
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→ Normalisierungskonventionen abgleichen (u-Mittelwert, Skalierung)
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Ziel: unabhängige Kreuz-Validierung der Konvergenzpunkte.
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Aufwand: kleines Python/C++ Vergleichsskript, kein Bibliotheks-Umbau.
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GC-1 [optional, possible now]
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Mathematical output comparison
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→ load the same test meshes (cathead.obj, brezel.obj, torus_4x4.off) into
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both libraries
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→ compare UV coordinates, u-vector, residual norm
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→ align normalisation conventions (u mean, scaling)
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Goal: independent cross-validation of convergence points.
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Effort: small Python/C++ comparison script, no library restructuring.
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GC-2 [optional, sinnvoll nach Phase 8]
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Intrinsic Delaunay Pre-Conditioning
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→ Vor dem Newton-Solver: geometry-central SignpostIntrinsicTriangulation
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auf die Eingabe anwenden
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→ Ptolemäische Flips konditionieren die Hessian-Matrix vor
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→ Hypothese: weniger Newton-Iterationen auf nicht-Delaunay-Eingaben
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→ Implementierbar als optionaler cmake-Flag: -DWITH_GC_PRECOND=ON
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Abhängigkeit: geometry-central als optionale externe Abhängigkeit
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(header-only Teile genügen für den Flip-Algorithmus).
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GC-2 [optional, useful after Phase 8]
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Intrinsic Delaunay pre-conditioning
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→ before the Newton solver: apply geometry-central SignpostIntrinsicTriangulation
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to the input
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→ Ptolemaic flips pre-condition the Hessian matrix
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→ hypothesis: fewer Newton iterations on non-Delaunay inputs
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→ implementable as an optional cmake flag: -DWITH_GC_PRECOND=ON
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Dependency: geometry-central as an optional external dependency
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(header-only parts suffice for the flip algorithm).
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GC-3 [hypothetisch, Phase 10+ Forschung]
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Ptolemäische Flip-basierter Solver als alternativer Backend
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→ Statt Newton: Ptolemäische Flips + penultimate-step Normalisierung
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(Gillespie–Springborn–Crane 2021 Algorithmus)
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→ Vergleich: Konvergenzradius, Robustheit auf pathologischen Netzen,
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numerische Stabilität auf hohen Genus-Flächen
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→ Für conformallab++ interessant, weil der Newton-Ansatz auf
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stark nicht-Delaunay Netzen (z.B. nach Remeshing) instabil
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werden kann.
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Keine Implementierung geplant — Konzeptnotiz für Phase 10-Forschung.
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GC-3 [hypothetical, Phase 10+ research]
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Ptolemaic flip-based solver as an alternative backend
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→ instead of Newton: Ptolemaic flips + penultimate-step normalisation
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(Gillespie–Springborn–Crane 2021 algorithm)
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→ comparison: convergence radius, robustness on pathological meshes,
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numerical stability on high-genus surfaces
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→ relevant for conformallab++ because the Newton approach can become
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unstable on strongly non-Delaunay meshes (e.g. after remeshing).
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No implementation planned — conceptual note for Phase 10 research.
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```
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**Verbindung zur Literatur:**
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Das Springborn 2020-Papier ("Ideal Hyperbolic Polyhedra and Discrete
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Uniformization") ist in conformallab++ als HyperIdeal-Geometriemodus
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bereits implementiert (Phase 2/3). Die Gillespie–Springborn–Crane
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2021-Erweiterung — die geometry-central implementiert — ergänzt dies um
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intrinsische Triangulierungen und macht den Algorithmus robust gegen
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schlechte Eingangs-Triangulierungen. Beide teilen denselben
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mathematischen Kern (diskrete konforme Äquivalenz, Gauss–Bonnet,
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Variationsprinzip von Bobenko–Springborn 2004).
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**Connection to the literature:**
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The Springborn 2020 paper ("Ideal Hyperbolic Polyhedra and Discrete
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Uniformization") is already implemented in conformallab++ as the HyperIdeal
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geometry mode (Phase 2/3). The Gillespie–Springborn–Crane
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2021 extension — implemented in geometry-central — augments this with
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intrinsic triangulations and makes the algorithm robust against
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poor input triangulations. Both share the same
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mathematical core (discrete conformal equivalence, Gauss–Bonnet,
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variational principle of Bobenko–Springborn 2004).
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---
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