semiplanesregions

A semiplane is a geometric concept referring to one of the two regions into which a plane is divided by a straight line. Imagine a line drawn on an infinite flat surface. This line acts as a boundary, and the line itself and the two distinct, unbounded areas on either side of it constitute the semiplane.

These regions are often referred to as open semiplanes, meaning they do not include the dividing line. If the dividing line is considered part of the region, it is called a closed semiplane. The boundary line is known as the edge or boundary of the semiplane.

Semiplanes are fundamental in various areas of mathematics, particularly in coordinate geometry and linear algebra. For example, the inequality $ax + by > c$ represents an open semiplane in the Cartesian coordinate system, where the line $ax + by = c$ serves as the boundary. All points $(x, y)$ that satisfy the inequality lie in one of the two semiplanes defined by this line.

The concept of semiplanes is also crucial in the study of convex sets. A closed semiplane is a convex set, meaning that any line segment connecting two points within the semiplane lies entirely within that same semiplane. This property is important in optimization problems and the theory of inequalities. The union of two closed semiplanes can form a larger region, and their intersection always results in a convex set.