Half-Plane

What is a Half-Plane?

A half-plane is a geometric figure created by dividing a plane with a straight line.
example of half-plane

Every line in a plane splits it into two opposite half-planes.

Picture an infinite flat surface, like a piece of paper stretching endlessly in all directions.

Drawing a straight line on this surface divides it into two distinct regions. Each of these regions is called a half-plane, and the line separating them is known as the origin or boundary of the half-plane.

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It’s important to note that the boundary line belongs to both half-planes.

However, the interior points of a half-plane are those that lie entirely within one of the two regions, without touching the boundary line.

This distinction is key because the line acts as a dividing boundary: points "inside" a half-plane are not part of the boundary itself.

To better visualize this concept, imagine standing in front of an open door. The door represents the boundary line. The two areas it separates — for instance, the living room and an adjacent room — correspond to the two opposite half-planes. Every point in the living room belongs to one half-plane, while every point in the other room belongs to the opposite half-plane. The doorframe, however, serves as the shared boundary between the two.
example of a real-world half-plane using a door

Although the concept is simple, the half-plane provides a structured way to analyze space.

The Axiom of Plane Partition by a Line

A line \( r \) in a plane divides the set of points in the plane into two disjoint and convex subsets, referred to as \( \alpha \) and \( \beta \).

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The defining property of this partition is that if a point \( A \) lies in \( \alpha \) and a point \( B \) lies in \( \beta \), the segment \( AB \) intersects the line \( r \) at exactly one point.

example

This axiom provides a rigorous framework for understanding how a line divides a plane into two distinct regions, known as half-planes.

It also introduces the concept of convexity, a fundamental property in many areas of mathematics.

The two subsets created by the line are convex, meaning that any segment connecting two points within the same half-plane remains entirely inside that half-plane.