Line Segment - in Real or Complex Vector Spaces

In Real or Complex Vector Spaces

If V is a vector space over or, and L is a subset of V, then L is a line segment if L can be parameterized as

for some vectors, in which case the vectors u and u + v are called the end points of L.

Sometimes one needs to distinguish between "open" and "closed" line segments. Then one defines a closed line segment as above, and an open line segment as a subset L that can be parametrized as

for some vectors .

Equivalently, a line segment is the convex hull of two points. Thus, the line segment can be expressed as a convex combination of the segment's two end points.

In geometry, it is sometimes defined that a point B is between two other points A and C, if the distance AB added to the distance BC is equal to the distance AC. Thus in the line segment with endpoints A = (a_x, a_y) and C = (c_x, c_y) is the following collection of points:

.