In this post the central point of the theme is the convex region, which are fundamentals for a plane in geometry.

Notions of convex region

Definition (Convex Region): a set of points $\Gamma$ is convex if, and only if, any two distinct points $A$ and $B$ of these sets are one of the possibilities:

  • endpoints of a line segment ($\overline{AB}$) contained in this set $\Gamma$
  • $\Gamma$ is unitary
  • $\Gamma$ is empty

Convexa

Example 1: A line is a convex set of points.

Reta Convex

In fact, the line segment $\overline{AB}$ is contained in the line $r.$

$$\overline{AB} \subseteq r. $$

Example 2: A plane $\alpha$ is a convex region.

Convex Plane

In fact, the line endowed with points $A$ and $B$ ($\overleftrightarrow{AB}$) is contained in the plane $\alpha$. From this, it implies that the line segment $\overline{AB}$ is contained in the plane $\alpha$.

$$\overleftrightarrow{AB} \subseteq \alpha \quad \Rightarrow \quad \overline{AB} \subseteq \alpha$$

Notions of concave region

Definition (Concave Region): if a region is not convex, then it is a concave region.

Concave Plane

Pac-man

Imã

Separation of the points of a plane

Next, we present a postulate for separating points of a plane.

Let be a line $r$ of a plane $\alpha$ and separate this plane into two sets of points $\gamma$ and $\beta$ and it satisfies the requirements:

  • the intersection of these sets of points are empty
    • $\gamma \cap \beta = \emptyset$
  • the two sets of points $\gamma$ and $\beta$ are convex
  • For $A \in\gamma$ and $B\in \beta$ it implies that the intersection of the line segment $\overline{AB}$ and the line $r$ is void.
    • $A \in\gamma$ and $B\in \beta \quad \Rightarrow \quad \overline{AB}\cap r \neq \emptyset$

Separating the points of a plane

semiplane

Definition (Half-plane): each of the two sets $\gamma$ and $\beta$ is an open half-plane.

  • the sets $r \cup \gamma$ and $r \cup \beta$ are semiplanes.
  • the line $r$ is the origin of each one of the planes.
  • $\gamma$ and $\beta$ are opposite half planes.

This is a small introduction to the concepts of plans, wait for the next posts.