If evaluated over points, the calculated perimeter for each point measures the perimeter of the region of its share of the surface as discussed above. If the accumulation is done per piece or throughout then parts of the perimeters of point surface shares that are common between two points in the same piece are excluded.

If evaluated over points, the calculated area for each point measures the perimeter of the region of its share of the surface as discussed above.

The calculated volume is only meaningful for closed surfaces. This is because it is estimated as the sum of signed volumes of tetrahedra with a common (automatically chosen) apex based at individual surface triangles. Open surfaces may return random looking results which may even be negative.

The complete description of the curvature of a surface at a point takes more than a single number. Briefly put, at an “ordinary” point of a smooth surface, there is a direction of movement along which the surface locally curves most sharply outwards (equiv. least sharply inwards). It turns out that if one rotates that direction around the point normal by 90 degrees one finds the direction along which the surface curves least sharply outwards (equiv. most sharply inwards). These two orthogonal directions are called the “principal” directions and their respective amounts of curving, measured as (signed) numbers are called the “principal” curvatures, where the first one represented by a larger number than the second one and with positive and negative values respectively indicating outward and inward curving. The two principal directions and their respective principal curvatures, combined, carry all the information there is about the curving of the surface at a given spot. The various types of curvatures generated by this operator are all derived from these.

Planes through surface normal cut the surface at curves of various curvatures. The largest (red) and the smallest (blue) are the two principal curvatures and their tangents at surface point are principal directions.

On a smooth surface where value (function) is given everywhere, the gradient of the value at a point on the surface is a vector tangent to the surface which points in the direction along which the value increases most rapidly locally. The magnitude of the gradient vector is proportional to how fast the value increases in that direction. For example you can use the magnitude of the gradient of the height on a polygonal terrain to distinguish locally flat regions (near zero length gradients) from sloped ones (large length gradients).

The Laplacian is a measure of conformity of a function to its surrounding region. Intuitively the Laplacian at a point of the surface estimates its “deficit” for evening out its value difference with its neighborhood. For example, repeatedly add a small fraction of an attribute’s Laplacian has the effect of smoothing the attribute. Conversely, repeated subtracting small fractions of the Laplacian tends to sharpen the features. Note however that the Laplacian of the function changes as the values of the function are modified and unless the modifications are done in small increments the process may be unstable and can generate noise.