Eye — geometry from the observer's tangent space

EARTH EYE-CORPUS-01: research notes, not a rendering engine. This corpus studies how an observer sees when rays propagate intrinsically in a three-dimensional Riemannian space, including quotient spaces. It makes no choice of raylib, sokol, graphics API, programming language, or numerical backend.

Start here

1. The point-indexed ray state: a direction lives at a point, not in a universal vector space.
2. Geodesic initial-value equations and parallel transport.
3. Observer-local rays, quotient boundaries, and lighting versus optics.
4. Ray-Marching Thurston Geometries, the principal source, followed by the eight models.
5. Checked worked calculations and typed architecture.

Mathematical state: $\mathrm{RayState}=\sum_{x:M}T_xM$. A ray is an initial-value problem for the geodesic spray on $TM$; its optical meaning requires an additional modeling decision. Camera orientation is an orthonormal frame at the camera's point.

Foundations

Points and tangent fibers · Charts and frames · Metrics and connections · Christoffel transformation · Tensor derivatives · Geodesic initial values · Transport and holonomy · Curvature and Jacobi fields · Exponential and cut locus · Intrinsic versus extrinsic.

Seeing and simulation

Observer rays · Moving the observer · Intersections and occlusion · Quotient teleportation · Lighting and optical assumptions · Distance and acceleration · Caching and reprojection.

The eight geometries

Geometry	Study
$\mathbb E^3$	Euclidean reference
$S^3$	Round sphere
$\mathbb H^3$	Hyperbolic space
$S^2\times\mathbb R$	Spherical product
$\mathbb H^2\times\mathbb R$	Hyperbolic product
Nil	Heisenberg geometry
Sol	Anisotropic solvable geometry
$\widetilde{\mathrm{SL}_2(\mathbb R)}$	Universal cover

Worked calculations

Flat polar coordinates · Sphere, great circle and holonomy · Upper half-plane geodesic and transport · Flat three-torus boundary crossing · Lie-group curvature and exact rays.

Source studies

Main paper · Scott · Novello et al. · Functional Differential Geometry · Computational bibliography · Source status, versions, and licenses.

Source-code surveys

Thurston ray marcher · Mathlib · Emmy / SICMUtils · geometry-central · JuliaManifolds · DiGeo · DifferentialGeometry.lean fork · Egison · DiscreteExteriorCalculus.jl · Nearby topology and geometry.

Architectural notes

Point-indexed types · Mathematical and numerical boundaries · Verification contracts · Reuse and licensing.

Evidence policy

Each note distinguishes inspected source, metadata-only, independently derived, and unverified assertions; coordinates and signs belong to the local note and cannot silently be transferred between models. Source-code file links identify pinned snapshots, not promises of binary compatibility. This is original explanatory documentation, not vendored code or a copy of published material.

Scope: One-off research and exposition. The verification record distinguishes exact calculations and link/source checks from unrun numerical integrators or any independent acceptance process. This repository selects no runtime dependencies or rendering engine.