Schattenüberdeckung

Schattenüberdeckung is a term used in German-language discussions of geometry and related fields to describe the idea of covering a set by shadows produced through projection along a family of directions. In this context, a shadow (Schatten) is the projection of a set onto a fixed hyperplane or surface when viewed from a direction in a given set of directions.

Formal idea: Let A be a subset of a Euclidean space and D a set of directions (often represented by unit vectors). For each d in D, the shadow S_d(A) is the projection of A onto a chosen ground hyperplane along the direction d (commonly taken as an orthogonal projection onto the hyperplane perpendicular to d). The Schattenüberdeckung of A with respect to D is the union of these shadows, ∪_{d∈D} S_d(A). If this union covers a target region or the original set A, one speaks of a Schattenüberdeckung with respect to D.

Variants and questions: Different projection types (orthogonal versus perspective), varying the ground plane, and using continuous versus discrete sets of directions lead to different behaviors. Key questions include the minimum number or arrangement of directions needed to cover a given shape, and the computational complexity of finding such coverings.

Applications: The concept appears in computer graphics to approximate silhouettes from multiple directions, in visibility analysis and sensor coverage planning, and in illumination design where shadows are used to study how well an area is covered by light.

See also: Projection, Shadow, Covering problem, Shadow volume. Note that the term is not completely standardized and may be described under related notions in different sources.