Chapter 25
Coordinate Systems and Metric Coherence

Flow Algebra

flow "Coordinate Systems and Metric Coherence"

# ---------- CONCEPT ----------
# A coordinate system assigns numeric structure to geometry — mapping spatial relations into algebraic form.
# The metric defines how distances, angles, and orientations are measured within that coordinate framework.
# Together, they form the coherent bridge between geometric intuition and analytic representation.

# ---------- STATES ----------
state CoordinateSystem(C):
  BasisVectors:
    description: "Set of n linearly independent vectors {e₁, e₂, …, eₙ} defining orientation and scale."
  Coordinates:
    description: "Representation x = Σ xᵢ eᵢ — components of position or direction relative to the chosen basis."
  MetricTensor:
    description: "g_{ij} = eᵢ · eⱼ — inner-product structure defining distances and angles."
  OrthogonalSystem:
    description: "Basis with g_{ij} = 0 for i≠j — directions independent and mutually perpendicular."
  OrthonormalSystem:
    description: "Basis with g_{ij} = δ_{ij} — unit-length orthogonal basis; Euclidean standard."
  TransformationMatrix:
    description: "Linear map between bases: x' = A x, g' = Aᵀ g A — preserves or distorts metric depending on A."

# ---------- PARAMETERS ----------
params:
  dimension: n
  base: "Basis {eᵢ}"
  coordinates: "x = (x₁, …, xₙ)"
  metric_tensor: "g_{ij}"
  transformation: "A ∈ GL(n, ℝ)"
  orientation: "Right-handed if det(A) > 0"
  curvature: "Flat (Euclidean) or curved (Riemannian)"
  closure: true

# ---------- INVARIANTS ----------
invariants:
  - "Distance squared: ds² = g_{ij} dxᵢ dxⱼ."
  - "Angle between vectors: cosθ = (g_{ij} uᵢ vⱼ) / (‖u‖‖v‖)."
  - "If A orthogonal (AᵀA = I), metric preserved: g' = g."
  - "Coordinate transformation preserves geometric truth if g transforms covariantly: g' = Aᵀ g A."
  - "Jacobian determinant J = det(A) measures scale change of volume element."
  - "Metric invariance under Euclidean motion: x' = R x + d ⇒ ds'² = ds²."
  - "Orthogonality and normalization define coherent coordinate geometry: eᵢ · eⱼ = δ_{ij}."

# ---------- TRANSFORMS ----------
transforms:
  ChangeOfBasis(x,A):
    rule: "x' = A x;  basis vectors transform contravariantly."
  TransformMetric(g,A):
    rule: "g' = Aᵀ g A;  preserves inner products if A orthogonal."
  ComputeDistance(P,Q,g):
    rule: "ds = √[(x_P − x_Q)ᵀ g (x_P − x_Q)]."
  ComputeAngle(u,v,g):
    rule: "cosθ = (uᵀ g v) / (√(uᵀ g u) √(vᵀ g v))."
  VolumeElement(g):
    rule: "dV = √(det g) dx₁ dx₂ … dxₙ — metric scaling of volume."
  MetricInversion(g):
    rule: "Find g^{ij} = (g^{-1})_{ij} for raising/lowering indices."
  OrthogonaliseBasis(vectors):
    rule: "Apply Gram–Schmidt process to enforce eᵢ·eⱼ=0 for i≠j."
  NormaliseBasis(eᵢ):
    rule: "Scale each vector so that ‖eᵢ‖=1."
  EvaluateJacobian(A):
    rule: "J = det(A);  volume scales by |J| under transformation."

# ---------- MONITORS ----------
monitors:
  OrthogonalityCheck:
    measure: "eᵢ·eⱼ"
    output: "Zero ⇒ orthogonal basis."
  NormalisationCheck:
    measure: "‖eᵢ‖"
    output: "One ⇒ unit-length basis vector."
  MetricSymmetry:
    measure: "g_{ij} − g_{ji}"
    meaning: "Zero ⇒ symmetric metric tensor (required for real geometry)."
  DeterminantMonitor:
    measure: "det(g)"
    meaning: "Positive ⇒ orientation preserved; zero ⇒ degenerate system."
  InvarianceCheck:
    meaning: "Verify ds'² = ds² for orthogonal transformations."
  VolumeConsistency:
    meaning: "Track √(det g) under transformations to monitor scale preservation."

# ---------- EQUATIONS ----------
equations:
  - "x = \\sum_{i=1}^n x_i e_i"
  - "g_{ij} = e_i · e_j"
  - "ds^2 = g_{ij} dx_i dx_j"
  - "g' = A^{\\mathsf T} g A"
  - "dV = \\sqrt{\\det(g)}\\, dx_1 dx_2 … dx_n"
  - "\\cos\\theta = \\frac{g_{ij} u_i v_j}{\\sqrt{g_{ij} u_i u_j}\\,\\sqrt{g_{ij} v_i v_j}}"
  - "J = \\det(A)"
  - "e_i' = A e_i,\\; g_{ij}' = e_i'·e_j' = A^{\\mathsf T} g A"

# ---------- NOTES ----------
notes:
  - "A coordinate system maps geometric relationships into algebraic form — transforming geometry into analyzable structure."
  - "The metric encodes how coherence is measured — it defines distance, angle, and curvature across the coordinate field."
  - "Changing coordinates does not change geometry, only its representation; true coherence is invariant under valid transformation."
  - "Orthogonality simplifies geometry — axes remain independent and angles easily measured; orthonormality further enforces unity of scale."
  - "The Jacobian determinant expresses how transformations scale area or volume, the bridge between local and global coherence."
  - "In Flow Algebra, the coordinate system is a coherence frame, and the metric is its regulator — together they preserve truth as geometry flows between representations."