intersectes

Intersectes is not a standard technical term in mathematics, but in an encyclopedic context it can be used to denote the points or sets where two or more geometric objects meet. The more common term in English is intersections, and in French the equivalent concept is intersection. This article treats intersectes as a label for the general notion of intersection across disciplines.

In set theory, the intersection A ∩ B is the set of elements that belong to both A and B. In geometry, the intersection of lines, curves, or surfaces is the locus of points that lie on each object. For example, two nonparallel lines in the plane intersect at a single point; parallel lines have empty intersection; coincident lines intersect in infinitely many points (the entire line).

The operation ∩ is commutative and associative, and it is idempotent (A ∩ A = A). It also satisfies A ∩ ∅ = ∅ and distributes over unions in the sense that A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C). These properties underlie many geometric and set-theoretic arguments.

In Euclidean geometry, the intersection of two line segments may be a single point, a subsegment, or be empty, depending on their positions. The intersection of a line with a circle yields up to two points; the intersection of two circles yields up to two points. The intersection of two polygons can be a polygonal region or a set of points.

Computation and applications: To find line-line intersections in the plane, one solves linear equations or checks for parallelism. Segment intersections are often tested with orientation or cross-product methods. Intersections are fundamental in computer graphics, geographic information systems, CAD, robotics, and data analysis for operations such as clipping, rendering, spatial querying, and constraint solving. See also intersection, set theory, geometry, and topology.