intersectcross

Intersectcross is a term used in computational geometry to describe a structured way of capturing both the points where two families of geometric objects intersect and the relational information formed at those intersections. It serves as a hybrid representation that combines point data with relational context, useful in analysis, rendering, and pattern recognition.

Definition and formalism

Let A and B be finite sets of geometric primitives in the plane, such as line segments or curves. The intersectcross IC(A,B) is a pair (I, R), where:

- I is the set of intersection points p in the plane such that p lies on some a ∈ A and b ∈ B and a ∩ b ≠ ∅.

- R is a relation subscripted by pairs (a,b) ∈ A×B with a ∩ b ≠ ∅, encoding characteristics of the intersection, such as whether it occurs in the interiors of a and b (a true cross) or at endpoints (touching).

Computational approach

To compute IC(A,B), one enumerates pairs (a,b) from A×B and tests for intersection. Each detected intersection contributes a point to I and an entry to R describing its nature. The framework can be extended to include weights, timestamps, or directional attributes, producing weighted or oriented intersectcross constructs.

Variants and extensions

Intersectcross can be extended to three-dimensional objects, where I comprises intersection curves or surfaces and R records 3D crossing types. It also supports probabilistic or approximate intersections, useful in noisy data or real-time applications.

Applications

Applications include collision detection, mesh merging, geometric pattern recognition, and analysis of spatial relationships in computer graphics and CAD systems. See also: intersection, crossing number, arrangement of curves, contact graphs.