Euclidean plane

In mathematics, a Euclidean plane is a special kind of space where everything is flat and two-dimensional. It is part of a larger idea called Euclidean space and has a dimension of two. This means that to find the exact spot of any point on this plane, you need two numbers, like on a map with up-down and left-right directions.

The Euclidean plane works like an affine space, which means it follows rules about parallel lines — lines that stay the same distance apart forever and never meet. It also has special metrical properties that let us measure things. For example, we can use a distance to draw perfect circles and to find the size of angles between lines.

When we add a Cartesian coordinate system to the Euclidean plane, it becomes called a Cartesian plane. This system uses pairs of real numbers, written as R², to describe every point clearly. This way of looking at the plane is very common and is often called the Euclidean plane or the standard Euclidean plane, because all Euclidean planes are the same as this one in shape and size.

History

See also: Euclidean geometry § History

Books I through IV and VI of Euclid's Elements explored two-dimensional shapes and ideas like the Pythagorean theorem, equal angles, and areas.

Later, the plane was described using a Cartesian coordinate system. This system uses two numbers to show where a point is on the plane. These numbers are distances from two lines that cross at a central origin. This idea was developed by Descartes and Pierre de Fermat.

In geometry

See also: Euclidean geometry

Coordinate systems

Main articles: Rectangular coordinate system and Polar coordinate system

In math, we can describe any point on a flat surface using two numbers. Imagine two lines crossing at a center point, called the origin. These lines are called axes, usually labeled x and y. By measuring how far a point is from each axis, we can find its exact spot.

Another way to describe a point is by how far it is from the origin and the angle it makes with a horizontal line.

• Cartesian coordinate system

• Polar coordinate system

Embedding in three-dimensional space

Polytopes

Main article: Polygon

In two dimensions, we can draw many flat shapes called polygons. Some of the simplest are regular polygons, which have all sides and angles equal. Examples include triangles, squares, and pentagons.

Circle

Main article: Circle

A circle is a special shape in two dimensions. It is all the points that are the same distance from a center point. This distance is called the radius. The space inside a circle is called its area.

Other shapes

Main article: List of two-dimensional geometric shapes

There are also many curved shapes, such as ovals, parabolas, and hyperbolas.

In linear algebra

Another way to think about a two-dimensional space is through linear algebra. In this view, the plane has two dimensions because the length of a rectangle does not depend on its width. Every point in the plane can be described using two independent directions, called vectors.

The dot product helps us understand angles and distances between vectors. It connects the idea of multiplying numbers with the geometry of arrows in space. This tool is important for measuring how vectors relate to each other and for finding the length of a vector.

In calculus

Main article: Fundamental theorem of line integrals

Main article: Green's theorem

In calculus, we study how to measure and understand changes in two-dimensional space. One important idea is the gradient, which tells us how a quantity changes in different directions on a flat surface.

We also learn about line integrals and double integrals. A line integral helps us add up values along a path, while a double integral lets us add up values over an area. These tools are useful for solving many real-world problems, like figuring out the total flow of a liquid or the amount of material in a shape.

In topology

In topology, the Euclidean plane is special because it is the only 2-manifold that is "contractible." This means it can be continuously shrunk to a point without tearing.

If you remove one point from the plane, the space that remains is still connected — you can still draw paths between any two points — but it is not "simply connected." This means there are loops you can make that cannot be shrunk to a point without crossing the missing spot.

In graph theory

In graph theory, a planar graph is a type of graph that can be embedded on a flat surface, like paper, without any lines crossing except at their ends. This special drawing is called a plane graph. Each point in the graph is placed on the plane, and the lines between points are drawn as smooth curves that only meet at their starting and ending points.