Planes in Euclidean Geometry     Plane Definition

A plane is a flat, two-dimensional surface that can extend infinitely far. This means that there are no constraints in a plane. The examples of these planes can be seen in coordinate geometry and are very common in our world. The plane math definition or plane definition geometry is the same. Planes have no thickness or width, which makes it completely two dimensional. A mathematical plane can consist of a point, a line, or/and a three-dimensional space. Planes could also be subspaces of higher dimensional spaces, like the walls of a room, being extended infinitely or they can also be independently existing. This can be seen in Euclidean Geometry. Here you can understand the plane geometry definition and example.

Important Terms Plane

Point:

Point is an element in any dimensional space. The whole of Euclidean Geometry is based on points. They are defined by axioms and are said to have no length, area, volume, or dimensional attributes.

Line:

Lines can be considered as a set of points that have no curvature and are straight objects. They have the only dimension as length. They can be infinite or can be bounded between 1 point (ray) or two points (line segment). Lines also do not have any predefined definition and are described using Euclidean Axioms.

2-Dimensional Space:

This is a geometric aspect where two values or parameters are required to find the position of a point, line, or shape. The two-dimensional space can be represented as R2. Normally, the two parameters in coordinate axes are taken as x and y parameters.

3-Dimensional Space:

This is a geometric aspect where three values or parameters are required to find the position of a point, line, plane, or object. The three-dimensional space can be represented as R3. Normally, the three parameters in coordinate axes are taken as x, y, and z parameters.

Axiom:

Axiom, postulates, or assumptions are statements that are taken as true without proof to use in other proofs. They are accepted without controversy or questioning since they are well known.

Euclidean Geometry:

This is a mathematical system constructed on the basis of dimensions and axioms. Euclid described these ideas in his textbook: the Elements.

Example of a Plane:

In our three-dimensional world, finding examples of planes is very hard. You can consider a sheet of paper with very negligible thickness as a plane. The surface of a table or a flat surface can be considered as a plane if they can be considered to have negligible thickness (which is almost impossible in the real world!)

Properties of Planes According to Euclidean Geometry

In any dimension planes are determined by the following:

• Presence of three non-collinear points (points that do not lie on the same line)

• A-line and a point that does not lie on that line.

• Two lines that are distinct but intersect each other at a point.

• Two lines that are distinct but parallel to each other.

Here are some properties of planes for specifically three-dimensional planes:

• Two planes that are distinct can either be parallel or intersect each other in a line.

• A line can either be parallel to a plane, can intersect it at a single point, or can be contained in the plane.

• Two lines that are distinct and perpendicular to the same plane should be parallel to each other.

• Two planes that are distinct and perpendicular to the same line should be parallel to each other.

Planes can be found if there are different parameters given. For planes in three-dimensional spaces, they can be represented in many ways.

Point Normal Form:

If  r0 is the position vector of a point P0 = (x0, y0, z0)  and n=(a,b,c) is a non-zero vector, the plane can be determined.

The dot product of n and (r - r0) will be 0.

After expanding, this becomes:

a (x-x0) + b (y-y0) + c (z-z0) = 0

This form is the point-normal form of the plane. The same can be represented as a linear equation.

ax + by + cz + d = 0

where,

d = -(ax0 + by0 + cz0 )

Point with a plane and two vectors lying on it:

We can represent this as

r = r0 + SV + tw,

where ‘s’ and ‘t’ belong to real numbers and v and w are linearly independent vectors lying on the plane. r0 is the position vector of a fixed point on the plane.

Note: Plane and Plain are different things.

Geometrically, a plane is a flat surface with no width or thickness, whereas a plain is a flat expanse of land that is used in geographical terms to describe the terrain of a place. A plane can also be considered as an airplane.

1. How to Find Out the Distance Between a Given Point and the Plane?

We can easily find the distance from the point to a plane if they have provided the coordinates of the given point and the equation of the plane.

If the equation of the plane given is ax + by + cz + d=0 and a point P₁ = (x₁, y₁, z₁ ) is provided, then the shortest distance from point P1 to the given pane is

D = | ax + by + cz + d |/   ( a² + b² + c² )

P will lie on the plane only if D=0.

2. Where are Planes Used in Mathematics and Physics?

Planes are used in various areas of mathematics and physics since it forms the basis of Euclidean Geometry along with a few other parameters.

• Geometry and trigonometry are performed in two-dimensional planes and are of prime importance in the mathematical and physical world to understand objects and their working.

• Graph theory (the study of graphs) are plotted on two-dimensional coordinate planes.

• The concept of a topological plane can be used to construct surfaces. Differential geometry views planes as a two-dimensional object that has a differential structure. Although there is no distance, the smoothness or roughness of a plane can be known by its differentiability.

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