halplane

A halplane is a geometric concept referring to one of the two parts into which a plane is divided by a straight line. Imagine a flat surface, like a table. If you draw a single straight line across that table, the line divides the table into two separate sections. Each of these sections, along with the line itself, is called a halplane. The line that creates the division is known as the boundary of the halplanes.

Halplanes can be open or closed. An open halplane does not include the boundary line, while a closed halplane includes it. In mathematical terms, a line in a two-dimensional Cartesian coordinate system can be represented by an equation of the form ax + by = c. This equation defines the boundary line. The two open halplanes are then defined by the inequalities ax + by < c and ax + by > c. Adding the boundary back, ax + by ≤ c and ax + by ≥ c, defines the two closed halplanes. This concept is fundamental in various areas of mathematics, including linear programming and computational geometry, where it helps to define regions and constraints.