Computable Coordinate System Objects: Theory and Applications

Coordinate systems are the "measuring sticks" of the physical world, forming the cornerstone of scientific computability. This paper presents a unified theoretical framework based on computable coordinate system objects, revolutionizing curvature computation through the dual-frame normalization method. We reduce the traditional complexity of Christoffel symbol calculations (O(n⁶)) to elegant Lie group operations (O(n³)).

Core Innovations:

Intrinsic Gradient Operator (Connection Operator): G_μ = C⁻¹ · ∂c/∂u^μ where intrinsic frame c captures rotation (curvature) and embedding frame C encodes metric, correctly separating both effects

Complete Riemann Curvature Framework: R(∂_u, ∂v) = [G_u, G_v] - G[∂_u,∂_v] providing direct computation of Riemann tensor from connection operators, eliminating Christoffel symbols entirely

Three-Layer Computation Structure: Layer 1 (base connection) → Layer 2 (geometric normalization) → Layer 3 (curvature tensor), each with clear geometric meaning

Elimination of Coordinate Singularities: Normalized frame method achieves < 2% error everywhere, while traditional methods reach 1390% error at poles

Lie Group Algebraization: Coordinate system objects form Lie group SE(3)×ℝ³₊ with intuitive division/multiplication operations

Theoretical Rigor: Equivalence with classical Riemannian geometry proven via Cartan's structure equations; error bound O(h²) established; optimal step size h=10⁻³ theoretically derived.

Numerical Validation: Verified on spheres, tori, cylinders, cones. Sphere curvature error < 2%, torus error < 0.5%, step-size dependence matches O(h²) theory. Complete Riemann tensor computation achieves machine precision (0.000% error) on analytical test cases.

Software Implementation: Open-source C++/Python implementation, providing one-line curvature computation API, applied in computational geometry, general relativity, and mesh processing.