joonelõik

Joonelõik (Estonian for line segment) is a basic concept in geometry. It represents the finite part of a straight line that lies between two endpoints. The joonelõik AB consists of all points on the line AB that are between A and B, including the endpoints A and B.

It is denoted by AB or segment AB, with A and B serving as its endpoints. Unlike an infinite line, a joonelõik has a definite length, defined as the distance between its endpoints. The length is nonnegative and is written as |AB|.

In coordinate geometry, if A = (x1, y1) and B = (x2, y2) in the plane, the length AB = sqrt((x2 − x1)² + (y2 − y1)²). In 3D space, with coordinates (x, y, z), AB = sqrt((x2 − x1)² + (y2 − y1)² + (z2 − z1)²). The segment AB can also be described parametrically as the set of points tA + (1 − t)B for t in [0, 1].

The midpoint M of AB is the point that divides the segment into two equal parts, with coordinates M = ((x1 + x2)/2, (y1 + y2)/2) in the plane. The joonelõik is a convex set: any point on the line segment between two points on AB also lies on AB. It lies on the unique line AB but does not extend beyond its endpoints.

Applications of joonelõik include measuring distances between points, constructing shapes (such as polygons and polyhedra), and serving as fundamental building blocks in proofs, geometry software, and computer graphics. The concept is fundamental in Euclidean geometry and coordinates-based approaches.