Kreisdimension

Kreisdimension, literally “circle dimension” in German, is not a single standardized concept in mathematics. The term is used in different contexts with varying definitions, and its meaning is often dependent on the field or author.

In differential geometry and basic geometry, a circle is a one-dimensional smooth manifold, so its intrinsic dimension is 1. In this sense, Kreisdimension may be used informally to emphasize the one-dimensional nature of a circle.

In the study of families of circles in the plane, the set of all circles can be regarded as a three-parameter family determined by the center coordinates (x, y) and the radius r. Some authors refer to this three-parameter space as the Kreisdimension of the family of planar circles.

In geometric measure theory and fractal geometry, Kreisdimension is sometimes used as a circle-based analogue of covering or fractal dimensions. A common approach is to define, for a subset X of the plane, N_K(r) as the minimum number of discs of radius r needed to cover X. The Kreisdimension d_K(X) is then the infimum of d ≥ 0 for which N_K(r) = O(r^(-d)) as r → 0, mirroring how Hausdorff or Minkowski dimensions are defined but using discs as the covering sets.

Applications mentioned in literature include circle packing problems, pattern analysis, and computer graphics. Because Kreisdimension is not a universally standardized term, its precise definition and properties are context-dependent and should be taken from the specific source in use. See also Hausdorff dimension, Minkowski dimension, circle packing.