Diametrien

Diametrien is a geometric descriptor used to quantify how much a planar shape's boundary deviates from a circle. It is defined for any closed plane curve or solid boundary by analyzing the radial distances from a chosen center to points on the boundary. The diametrien is the root-mean-square deviation of those radii from their mean radius. If all boundary points lie at the same distance from the center—as in a circle—the diametrien is zero.

Computation: Let c be the center, and r_i the distance from c to boundary point i along the boundary, with r_bar the average of the r_i. The diametrien D is the square root of the average of (r_i − r_bar)^2 across the boundary. For discrete boundaries, sum over sampled points; for smooth boundaries, integrate along the perimeter. The diametrien has the same units as length and can be normalized by r_bar to yield a dimensionless circularity index.

Properties: D scales linearly with uniform scaling of the shape. It is invariant under translations and rotations that preserve the center. It complements other shape descriptors such as the isoperimetric ratio and low-order Fourier descriptors of the boundary.

Applications: In image analysis and biomechanics, diametrien serves as a compact measure of roundness for cells, organelles, leaves, and manufactured disks. It is useful in quality control, pattern recognition, and shape classification.

See also: Circularity, Isoperimetric ratio, Radius variance, Radial function.