semiplanes

A semiplane, also known as a half-plane, is a geometric figure that is formed by dividing a plane with a straight line. This line is called the boundary or edge of the semiplane. The semiplane includes all the points on one side of the line, but not the points on the line itself. It is essentially half of a plane, hence the name.

Semiplanes are fundamental concepts in geometry and have various applications in mathematics and other fields. They are often used to define and analyze other geometric shapes and concepts, such as angles, polygons, and transformations. For example, the interior of an angle can be considered a semiplane, and the concept of a semiplane is used in the definition of a polygon as a plane figure that is bounded by a finite chain of straight line segments.

In three-dimensional space, the analogous concept is a semi-space, which is formed by dividing space with a plane. Like semiplanes, semi-spaces include all the points on one side of the plane, but not the points on the plane itself.

Semiplanes are also used in the study of linear inequalities. A linear inequality in two variables can be represented as a semiplane, where the boundary line represents the equality, and the shaded region represents the solution set of the inequality. This visual representation is a powerful tool for understanding and solving linear inequalities.