Longitudewith

Longitudewith is a geometric measure used to quantify how long an object appears in a specified direction. It captures the extent of a shape when projected onto a line defined by a direction, rather than measuring its intrinsic length alone.

Definition and interpretation

For a compact set S in Euclidean space and a unit vector u, longitudewith(S, u) is defined as the difference between the maximum and minimum projections of S onto u:

Longitudewith(S, u) = max{ x·u : x ∈ S } − min{ x·u : x ∈ S }.

Equivalently, it is the length of the orthogonal projection of S onto the line spanned by u. In practice, for many objects the extrema occur at boundary or hull points, so computation often reduces to projecting the vertices of the convex hull of S.

Example

Consider a line segment from (0,0) to (3,4) and a direction u along the x-axis, u = (1,0). The projections are 0 and 3, so longitudewith(S, u) = 3. The actual segment length is 5, illustrating that longitudewith depends on direction.

Relations and computation

Longitudewith is a directional variant of width and diameter. The maximum longitudewith over all directions equals the object’s diameter, while the minimum over directions relates to the minimal projection length. For convex objects, efficient computation uses convex hulls and support points.

Applications

Used in computer graphics, CAD, robotics, and packaging to assess how an object fits or aligns with a given orientation, and to inform orientation-aware bounding boxes and collision checks.

Origin and terminology

The term combines “length” with “direction,” emphasizing its directional nature. It is sometimes referred to as directional length or projection length. See also projection, width, diameter, and oriented bounding boxes.