Asperity Density, Summit Radius, and RMS Height Calculator

Estimate areal RMS height, summit density, and local principal summit radii from a rectangular surface-height grid.

Engineering screening tool—verify inputs and results independently. Use at your own risk.

Surface and analysis inputs

Surface file (CSV or TXT)Choose a local rectangular numeric matrix. Files stay in this browser.Surface height matrix z_ijAt least 5 × 5 finite values; commas, tabs, spaces, or semicolons are accepted. Missing values are rejected rather than interpolated.

X spacing ΔxColumn-to-column sample spacing; must be greater than zero.Y spacing Δysame unitRow-to-row spacing; uses the selected lateral unit.Height unitUnit of every matrix height and height threshold.Leveling methodSubtracts a least-squares reference before statistics.Smoothing radius r_ssamplesSquare moving-average half-width, 0–5; zero means no smoothing.Summit neighborhoodA summit must be strictly higher than each selected neighbor.Minimum prominence p_minheight unitMinimum rise above the highest neighboring sample; non-negative.Minimum summit height h_minheight unitSigned elevation relative to the leveled mean plane; positive is upward.Radius-fit half-window r_fsamplesFits a (2r_f + 1) square neighborhood. Candidates too near an edge are excluded.

Surface statistics

Areal RMS height Sq—selected height unit

Accepted summits N_s—

Summit density η—1 / lateral unit²

RMS summit height h_rms—height unit

Median R₁—lateral unit

Median R₂—lateral unit

Median R_eq—lateral unit

Grid —Area — lateral unit²Rejected fits —Edge exclusions —

Processed height map; red rings mark accepted fitted summits. Color is relative within the current grid.

How to use

Upload a CSV/TXT surface or paste a rectangular matrix with at least 5 rows and 5 columns. Blank, missing, and non-numeric cells are not interpolated and must be corrected first.
Enter Δx and Δy and select their common lateral unit. Select the unit used by every height value in the matrix.
Select mean removal, plane leveling, or quadratic detrending. Use stronger detrending only when it represents form rather than roughness.
Set an optional moving-average radius, the 4- or 8-neighbor summit rule, minimum prominence, signed minimum summit height, and local fit half-window.
Select Calculate to update Sq, summit count and density, summit-height RMS, radii, diagnostics, and the detected-summit map.
Use Reset example to restore the supplied 9 × 9 analytical surface and its units and controls.
Interpret density in inverse lateral-unit squared, heights in the chosen height unit, and fitted radii in the chosen lateral unit. Review rejected fits and edge exclusions before reuse.
Repeat with defensible threshold, smoothing, and fit-window choices. Materially changing results indicate sensitivity that must be reported.
Do not use this model for nonuniform spacing, unordered point clouds, unresolved tip convolution, surfaces dominated by form, or safety-critical contact predictions without independent metrology and model validation.

Worked example

Use the supplied 9 × 9 paraboloid with Δx = Δy = 1 µm, heights in µm, mean removal, no smoothing, 8 neighbors, 0.01 µm prominence, and a 2-sample fit half-window. Expected results are Sq = 0.6540 µm, one accepted summit, η = 0.015625 µm⁻², R₁ = 5.000 µm, R₂ = 10.000 µm, and R_eq = 7.071 µm. The recovered radii identify the imposed principal curvatures; this analytical example is a code check, not a claim about a manufactured surface.

Equations used

Let z_ij be the preprocessed height at row i and column j, N the number of grid points, N_s the number of accepted fitted summits, Δx and Δy the physical sample spacing, and A = (n_x − 1)Δx(n_y − 1)Δy the sampled rectangular area.

Sq = √[(1/N) Σ_i,j z_ij²]

η = Nₛ / A

h_rms = √[(1/Nₛ) Σ_k=1^Nₛ h_k²]

Each accepted neighborhood is fitted by least squares to z = ax² + bxy + cy² + dx + ey + f. The local Hessian is H = [[2a,b],[b,2c]]. Both eigenvalues κ₁ and κ₂ must be negative for a concave-down summit. Principal radii use R_i = −1/κ_i after converting height and lateral coordinates to one unit. This calculator reports R₁ ≤ R₂ and R_eq = √(R₁R₂).

Positive height is upward from the leveled reference plane. Mean removal subtracts one constant; plane leveling subtracts α + βx + γy; quadratic detrending subtracts α + βx + γy + δx² + εxy + ζy². Optional smoothing is a clipped-edge square moving average. A strict local maximum must exceed every selected neighbor and both thresholds.

Theory and method

Discrete summit statistics are not intrinsic constants of a surface: sampling interval, instrument bandwidth, leveling, filtering, neighborhood definition, thresholds, and fit window all alter the reported population. The implementation therefore exposes these choices and reports edge and curvature-fit exclusions. It does not compensate for probe-tip convolution, measurement uncertainty, anisotropic acquisition artifacts, or scale-dependent roughness.

The sampled area convention spans the center-to-center extent of the grid. Sq uses the final leveled and optionally smoothed grid. Summit heights are measured from the leveled reference plane. Radius statistics include only finite negative-definite Hessian fits; saddle points and concave-up fits are rejected. The geometric mean is the only equivalent-radius convention used here.

Applicability limits: use rectangular uniformly spaced areal grids with finite values and adequate lateral/vertical resolution. Verify instrument calibration, remove physically justified form, inspect outliers, and repeat the analysis across plausible settings. ISO 25178-2 supplies areal surface-texture terms but does not make this discrete summit detector interchangeable with every commercial implementation.

References

ISO 25178-2, Geometrical product specifications (GPS)—Surface texture: Areal—Part 2: Terms, definitions and surface texture parameters.
J. A. Greenwood and J. B. P. Williamson, “Contact of nominally flat surfaces,” Proceedings of the Royal Society A 295 (1966), 300–319. doi:10.1098/rspa.1966.0242.

Continue in TriboSolver

Use defensible Sq and summit-radius statistics as screening inputs for a rough-surface contact study in TriboSolver. Preserve the preprocessing settings and sensitivity range so the contact model can be independently checked.