SGA Globe Rendering with Interval Arithmetic

Abstract

We present a Spherical Geometric Algebra (SGA) engine for rendering planetary globes using interval arithmetic bisection instead of traditional ray-marching. This provides mathematically rigorous intersection guarantees at every pixel — no missed surfaces, no depth artifacts.

1. Introduction

Traditional globe rendering uses ray-marching with fixed step sizes or sphere-tracing. These are heuristic approximations that can miss thin features or produce depth errors near the horizon.

Our approach uses interval arithmetic for root-finding in the conformal geometric algebra G(4,1):

f(p) = 0  →  root finding via interval bisection

2. Architecture

2.1 Unified CSG Expression AST

csg_expr.rs → CSGExpr enum (Union, Intersection, Difference, Sphere, Plane, etc.)

Compiles to three targets: - GPU bytecode — postfix stack machine for shader - CPU evaluator — interval arithmetic for collision/raycast - Collision — precise surface queries

2.2 Interval Arithmetic Bisection

Standard ray-marching:

while (steps < max) { p += step * dir; if (sdf(p) < eps) hit; }

Our approach:

1. Coarse sign-change detection (32 samples along ray)
2. Interval bisection on sign-change interval (20 iterations)
3. Exact hit point with rigorous bounds

Guarantees: no missed surfaces, no duplicate hits, correct ordering.

3. Implicit Terrain SDF

d(p) = |p| - (R_earth + height(lat, lon))

Elevation scaling: 2000× (max ~3900m). Camera must pitch downward (-0.5 rad) since horizontal rays follow Earth tangent.

4. Artifact Tracing

Interval bisection in trace_artifacts() reveals: - Floating-point edge cases - CSG operator correctness - Gradient/normal consistency

5. Dual-Number Autodiff (Reverted)

Attempted dual numbers for normals → reverted. WGSL select() doesn't support custom structs. Using 6-eval finite differences instead.

6. Performance