Halbsegments

Halbsegments are a geometric concept used to describe the two subparts of a line segment created by dividing the original segment at an interior point. If a closed segment AB is chosen and an interior point P lies on AB, the Halbsegments are the portions AP and PB. Depending on convention, the endpoints may be included in one or both subsegments; commonly AP is the portion from A to P and PB from P to B, each sharing the point P.

Basic properties include that the Halbsegments partition the original segment without overlap, and their lengths add up to the length of AB. If the distance from A to P is d and the length of AB is L, then the Halbsegments have lengths d and L−d. If P coincides with an endpoint, one Halbsegment degenerates to a point while the other remains the full segment. In the relative topology of the segment, removing an interior point P yields two open subsegments (in the subspace sense).

Applications of Halbsegments appear in elementary geometry, piecewise linear modeling, and computer graphics, where a segment is often subdivided to support rendering, collision detection, or hierarchical partitioning. They also serve as a simple example in proofs that rely on partitioning a set into disjoint pieces and analyzing their properties.

Relation to broader concepts: while the idea generalizes informally to higher dimensions as a division by a point, line, or plane, the specific term Halbsegment is predominantly used in one-dimensional contexts.