mris_curvature: Estimate per-node curvature of a surface mesh

In ravetools: Signal and Image Processing Toolbox for Analyzing Intracranial Electroencephalography Data

Estimate per-node curvature of a surface mesh

Description

Estimates, at every vertex of a closed triangular mesh, the local mean curvature, Gaussian curvature, and the two principal curvatures, by fitting an osculating quadratic surface to each vertex's neighborhood.

Usage

mris_curvature(mesh, verbose = FALSE)

Arguments

mesh	triangular mesh of class 'mesh3d' (or coercible via ensure_mesh3d). The mesh must be closed, manifold, and genus-0 (watertight, no boundary or non-manifold edges, single connected component); mris_curvature raises an error if such defects are detected
verbose	logical; print progress messages. Default FALSE

Details

For each vertex, the function:

1. builds an orthonormal tangent frame (e_1, e_2, n), where n is the vertex's (already-computed) unit normal;

2. expresses each neighbor's offset from the vertex in this frame as tangential coordinates (u, w) and a height h above the tangent plane along n;

3. fits, by least squares over the vertex's 2-ring neighborhood, the osculating ⁠paraboloid⁠ h = a u^2 + b u w + c w^2;

4. derives the mean curvature H = a + c, the Gaussian curvature K = 4ac - b^2, and the principal curvatures k_{1,2} = H \pm \sqrt{\max(H^2 - K,\ 0)}.

Because the tangent frame is orthonormal and the ⁠paraboloid⁠ is fitted with zero gradient at the vertex (by construction, since n is the fitted normal direction), these are exactly the eigenvalues of the second fundamental form, i.e. the principal curvatures.

The sign of the result follows the orientation of the per-vertex outward normal: a locally convex ('⁠gyrus⁠-like') patch, where neighbors lie toward the surface's interior relative to the outward normal, has negative mean curvature, while a locally concave ('⁠sulcus⁠-like') patch has positive mean curvature. For example, a sphere of radius r with outward-pointing normals has uniform curvature H = -1/r and K = 1/r^2 everywhere.

Vertices whose 2-ring neighborhood is degenerate (fewer than three neighbors, or neighbor offsets that do not span the tangent plane, e.g. nearly ⁠collinear⁠) are reported with all four curvature values set to zero.

Value

A named list of four numeric vectors, each of length ncol(mesh$vb) (one entry per vertex, in vertex order):

mean

Mean curvature H = (k_1 + k_2)/2.

gaussian

Gaussian curvature K = k_1 k_2.

k1

First principal curvature (k_1 \geq k_2).

k2

Second principal curvature.

References

Cortical surface-based analysis II: Inflation, flattening, and a surface-based coordinate system. NeuroImage, 9(2), 195-207 (1999).

Examples

if (is_not_cran()) {

data("left_hippocampus_mask")
mesh <- vcg_isosurface(left_hippocampus_mask)

plot(mesh)

# Fix defects
mesh <- vcg_fix_defects(mesh, verbose = TRUE)

# Smooth
smoothed <- mris_smooth(mesh, verbose = TRUE)

res <- mris_curvature(smoothed)

range(res$k1)

col <- color_ramp_continuous(
  res$k1, clim = c(-1, 1), alpha = TRUE,
  cmap = c("black", "gray", "red"))

plot(smoothed, col = list(col),
     eye = c(-100, 100, 0), up = c(0, 0, 1))

}

ravetools documentation built on Aug. 21, 2026, 5:06 p.m.