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Curvature

Overview

FrontIntrinsicOps.jl computes three types of discrete curvature:

  1. Signed curvature $\kappa$ — for polygonal curves in $\mathbb{R}^2$.
  2. Mean curvature $H$ — for triangulated surfaces.
  3. Gaussian curvature $K$ — via the angle-defect formula.

Signed curvature of a polygonal curve

For a vertex $v_i$ on a closed polygonal curve with adjacent edges $e_{i-1} = (v_{i-1}, v_i)$ and $e_i = (v_i, v_{i+1})$:

Turning angle:

\[\theta_i = \text{signed angle from } \hat{t}_{i-1} \text{ to } \hat{t}_i\]

where $\hat{t}_e$ is the unit tangent of edge $e$.

Discrete signed curvature:

\[\kappa_i = \frac{\theta_i}{\ell_i^*}\]

where $\ell_i^* = (\ell_{i-1} + \ell_i)/2$ is the dual length at $v_i$.

Verification (circle of radius $R$, $N$ vertices):

  • Total length $\to 2\pi R$ as $N \to \infty$.

  • \[\kappa_i \to 1/R\]

    for all vertices.
mesh = sample_circle(1.0, 128)
geom = compute_geometry(mesh)
κ    = curvature(mesh, geom)    # ≈ 1.0 for unit circle

Mean curvature of a triangulated surface

Mean-curvature vector

The mean-curvature normal at vertex $v_i$ is defined via the Laplace–Beltrami applied to the position vector:

\[\vec{H}_i = \frac{1}{2} \Delta_\Gamma \vec{p}\big|_i = \frac{1}{2} (L\vec{p})_i\]

where $\vec{p} = (p_x, p_y, p_z)^\top$ is the vector of coordinate values and $L$ is applied column-wise.

Discrete formula:

\[\vec{H}_i = \frac{1}{2A_i^*} \sum_{j \in N(i)} w_{ij} (p_i - p_j)\]

This is the cotangent-weight mean-curvature vector of Pinkall and Polthier (1993).

Scalar mean curvature

\[H_i = \|\vec{H}_i\| \cdot \mathrm{sign}(\vec{H}_i \cdot \hat{n}_i)\]

The sign correction aligns the scalar mean curvature with the outward normal convention: $H > 0$ for a convex surface (sphere), $H < 0$ for a saddle.

Verification (sphere of radius $R$):

On a sphere $H = 1/R$ at every point (principal curvatures $\kappa_1 = \kappa_2 = 1/R$, $H = (\kappa_1 + \kappa_2)/2 = 1/R$).

R    = 2.0
mesh = generate_icosphere(R, 4)
geom = compute_geometry(mesh)
dec  = build_dec(mesh, geom)
H    = mean_curvature(mesh, geom, dec)
println(maximum(abs, H .- 1/R))   # → 0 as mesh is refined

Mean-curvature normal vector field

Hn = mean_curvature_normal(mesh, geom, dec)  # Vector{SVector{3,T}}

Gaussian curvature — angle-defect formula

The discrete Gaussian curvature at an interior vertex $v_i$ is the angle defect normalised by the dual area:

\[K_i = \frac{2\pi - \displaystyle\sum_{f \ni i} \theta_{f,i}}{A_i^*}\]

where $\theta_{f,i}$ is the interior angle of face $f$ at vertex $i$.

Derivation: On a smooth surface the Gauss–Bonnet theorem for a small disk of radius $\varepsilon$ around $v_i$ gives

\[\int_{\text{disk}} K \, dA \approx K_i A_i^* = 2\pi - \text{(sum of angles)}\]

The formula uses this relation as a definition.

Verification (sphere of radius $R$):

\[K = 1/R^2\]

at every point; $\int_\Gamma K \, dA = 4\pi$ for any radius.

K     = gaussian_curvature(mesh, geom)
intK  = integrated_gaussian_curvature(mesh, geom)   # Σ K_i A_i^*
println(intK)   # ≈ 4π for sphere

Gauss–Bonnet theorem

For a closed orientable surface:

\[\int_\Gamma K \, dA = 2\pi \chi\]

where $\chi = V - E + F$ is the Euler characteristic.

Surface$\chi$$\int K \, dA$
Sphere$2$$4\pi \approx 12.566$
Torus$0$$0$
Genus-$g$ surface$2 - 2g$$2\pi(2-2g)$

Discrete identity: The angle-defect formula satisfies Gauss–Bonnet exactly (before floating-point rounding):

\[\sum_{i=1}^{N_V} K_i A_i^* = 2\pi\chi\]

Residual diagnostic:

gb_res = gauss_bonnet_residual(mesh, geom)   # |∫K dA - 2π χ|
# gb_res < 1e-12 on well-constructed meshes

Accessing curvature fields

geom_with_curv = compute_curvature(mesh, geom, dec)
# geom_with_curv.mean_curvature    Vector{T}
# geom_with_curv.gaussian_curvature Vector{T}

Or directly:

H  = mean_curvature(mesh, geom, dec)         # scalar mean curvature
Hn = mean_curvature_normal(mesh, geom, dec)  # mean-curvature vector
K  = gaussian_curvature(mesh, geom)          # angle-defect Gaussian curvature

References

  • Pinkall, U. & Polthier, K. (1993). Computing discrete minimal surfaces and their conjugates. Experimental Mathematics 2(1):15–36.
  • Meyer, M., Desbrun, M., Schröder, P., & Barr, A.H. (2003). Discrete differential-geometry operators for triangulated 2-manifolds. In Visualization and Mathematics III, pp. 35–57.

See also