Chapter 26
Curvature and Gaussian Surfaces

Flow Algebra

flow "Curvature and Gaussian Surfaces"

# ---------- CONCEPT ----------
# Curvature measures how geometry deviates from flatness — how a surface bends relative to its tangent plane.
# Gaussian curvature encodes intrinsic bending, independent of embedding or orientation.
# Surfaces are coherent fields of curvature; their geometry arises from how these local deviations interconnect.

# ---------- STATES ----------
state CurvatureSystem(K):
  Surface:
    description: "A smooth 2D manifold S ⊂ ℝ³ parameterised by r(u,v) = (x(u,v), y(u,v), z(u,v))."
  TangentVectors:
    description: "r_u = ∂r/∂u, r_v = ∂r/∂v — span the tangent plane at each point."
  NormalVector:
    description: "n̂ = (r_u × r_v)/‖r_u × r_v‖ — perpendicular to tangent plane."
  FirstFundamentalForm:
    description: "I = E du² + 2F du dv + G dv², with E=r_u·r_u, F=r_u·r_v, G=r_v·r_v — metric on surface."
  SecondFundamentalForm:
    description: "II = e du² + 2f du dv + g dv², with e=r_{uu}·n̂, f=r_{uv}·n̂, g=r_{vv}·n̂ — curvature relation."
  GaussianCurvature:
    description: "K = (eg − f²)/(EG − F²) — intrinsic curvature."
  MeanCurvature:
    description: "H = (Eg − 2Ff + Ge)/(2(EG − F²)) — average bending (extrinsic measure)."

# ---------- PARAMETERS ----------
params:
  coordinates: "(u,v)"
  embedding: "r(u,v)"
  normalisation: "‖n̂‖=1"
  orientation: "determines sign of normal"
  differentiability: "r ∈ C²"
  closure: true

# ---------- INVARIANTS ----------
invariants:
  - "First form defines metric on surface; measures distances and angles."
  - "Second form encodes normal curvature; measures deviation from tangent plane."
  - "Gaussian curvature K depends only on intrinsic geometry — invariant under bending without stretching."
  - "Mean curvature H governs local equilibrium: H=0 ⇒ minimal surface."
  - "Principal curvatures κ₁, κ₂ are eigenvalues of shape operator S = I⁻¹II; K = κ₁κ₂, H = (κ₁+κ₂)/2."
  - "Surface normal evolution given by shape operator: dn̂ = −S(dr)."
  - "Gauss’s Theorema Egregium: K computable from I alone — intrinsic property of surface metric."
  - "For closed surfaces, ∫_S K dA = 2πχ(S) (Gauss–Bonnet theorem) — global topological invariant."

# ---------- TRANSFORMS ----------
transforms:
  ComputeTangentVectors(r):
    rule: "r_u = ∂r/∂u, r_v = ∂r/∂v."
  ComputeNormals(r_u,r_v):
    rule: "n̂ = (r_u × r_v)/‖r_u × r_v‖."
  ComputeFundamentalForms(r):
    rule: "I = [E,F;F,G], II = [e,f;f,g]."
  ComputeCurvatures(I,II):
    rule: "K = (eg−f²)/(EG−F²), H = (Eg−2Ff+Ge)/(2(EG−F²))."
  PrincipalCurvatures(K,H):
    rule: "κ₁,₂ = H ± √(H² − K)."
  SurfaceNormalFlow(S):
    rule: "n̂ evolves via dn̂ = −S(dr)."
  ApplyIsometry(F):
    rule: "Transform surface by distance-preserving map; K invariant, H transforms extrinsically."
  IntegrateGaussian(S):
    rule: "∫_S K dA = 2πχ(S) — total curvature equals topological constant."

# ---------- MONITORS ----------
monitors:
  NormalisationCheck:
    measure: "‖n̂‖"
    output: "Should equal 1."
  TangentOrthogonality:
    measure: "r_u·n̂, r_v·n̂"
    meaning: "Zero ⇒ tangent vectors orthogonal to normal."
  MetricDeterminant:
    measure: "EG−F²"
    meaning: "Positive ⇒ regular surface patch."
  CurvatureCheck:
    measure: "K = (eg−f²)/(EG−F²)"
    meaning: "Detects convex (K>0), saddle (K<0), or flat (K=0) geometry."
  GaussianInvariance:
    meaning: "Verify K unchanged under isometric deformation."
  SurfaceClosure:
    meaning: "For closed S, ∫ K dA = 2πχ(S) (Gauss–Bonnet)."

# ---------- EQUATIONS ----------
equations:
  - "r(u,v) = (x(u,v), y(u,v), z(u,v))"
  - "r_u = ∂r/∂u, r_v = ∂r/∂v"
  - "n̂ = \\frac{r_u × r_v}{‖r_u × r_v‖}"
  - "E = r_u·r_u, F = r_u·r_v, G = r_v·r_v"
  - "e = r_{uu}·n̂, f = r_{uv}·n̂, g = r_{vv}·n̂"
  - "K = \\frac{eg − f^2}{EG − F^2}"
  - "H = \\frac{Eg − 2Ff + Ge}{2(EG − F^2)}"
  - "κ_{1,2} = H ± \\sqrt{H^2 − K}"
  - "dn̂ = −S(dr), S = I⁻¹II"
  - "∫_S K\,dA = 2πχ(S)"

# ---------- NOTES ----------
notes:
  - "Curvature measures the deviation of a surface from its tangent plane — the geometry of bending and shape."
  - "The first fundamental form defines how distances and angles behave within the surface; the second describes how it bends in surrounding space."
  - "Gaussian curvature (K) is intrinsic: it depends only on the surface metric, not on how the surface sits in space."
  - "Positive K indicates convex curvature (sphere-like), negative K indicates saddle curvature (hyperbolic), and K=0 represents flatness (plane or cylinder)."
  - "Mean curvature (H) measures the net bending; H=0 defines minimal surfaces such as soap films."
  - "Principal curvatures capture curvature along the most and least bent directions; they form the eigenstructure of the shape operator."
  - "Gauss’s Theorema Egregium reveals that curvature is a property of the surface itself — geometry encoded intrinsically within its metric."
  - "In Flow Algebra, curvature expresses the self-interaction of coherence: how geometry folds through itself; Gaussian surfaces are its stable manifestations — closed fields of curvature where geometry and topology converge."