Widthwithout

Widthwithout is a term used in geometry and related design disciplines to denote the width of a geometric object after excluding a specified portion or obstacle. It is not a universally standardized concept, and its exact definition can vary by domain or application. In practice, widthwithout provides a way to measure how wide a shape remains once a chosen subset is removed.

Formally, for a bounded set S in Euclidean space and a direction u, the width of S in direction u is w(S, u) = max {⟨x, u⟩ : x in S} − min {⟨x, u⟩ : x in S}. If E is a subset of S that is removed, the width of the remaining set S \ E in direction u is w(S \ E, u) = max {⟨x, u⟩ : x in S \ E} − min {⟨x, u⟩ : x in S \ E}. The overall widthwithout of S with respect to E can be taken as the supremum of w(S \ E, u) over all directions u. In some contexts, a more restricted definition may be used, for example considering a fixed set of directions.

Examples help illustrate the idea. Let S be the unit square [0,1] × [0,1] and let E be the left half [0,0.5] × [0,1]. Then S \ E is [0.5,1] × [0,1], and the width in the x-direction becomes 0.5, whereas the original width in that direction was 1. This demonstrates how removing a portion can reduce the measured width, depending on the chosen direction.

Applications of widthwithout appear in computational geometry, collision detection, layout algorithms, and any scenario where the impact of removing obstructions on a shape’s extent is relevant. See also width, directional width, and set difference.