How to Name a Plane? The purpose of this article is to provide insight on how mathematicians name planes.
There are several ways to name a plane. There are at least three traditional methods used in geometry textbooks, all using coordinate axes.
Which method one uses depends on the coordinates one chooses for the origin and the number of coordinates used in the system being employed.
The first coordinate system, called the “Cartesian coordinate system”, is named after René Descartes.
It uses two coordinates: a horizontal x-coordinate and a vertical y-coordinate. The point (x, y) is the location of a point on the plane relative to the origin.
The origin is the point (0, 0).
The second coordinate system is called “polar coordinate system”. It uses a single coordinate: an angle θ measured in radians from a fixed line called the “pole” to the point (x, y) on the plane.
The point (x, y) is the location of a point on the plane relative to the pole. The angle θ is the direction of moving from the pole around toward (x, y) on the plane.
Because it uses only one coordinate, there is no need for an axis; all points are equally valid polar coordinates of a point in 2-dimensional space.
The pole is the point (0, 0).
The third coordinate system is called “spherical coordinate system”. It uses two coordinates: an angle θ measured in radians from a fixed line through the origin to point (x, y) on the plane;
and a distance ρ that measures how far away (x, y) is from the origin on a plane called a “sphere”. The sphere has radius ρ from its center at (0, 0).
While spherical coordinates have been used for centuries by astronomers, they are less familiar to mathematicians who do not work with celestial objects so often.
In addition, the sphere is only an infinite surface which means that points on the sphere can be infinitely close to (0, 0) or infinitely far away.
This complication is resolved by using a coordinate “hat” symbol over the angle θ, ρ when it is written in spherical coordinates, to remind us that it is a coordinate on a sphere and not on the plane.
The point (x, y) is the location of a point on the plane relative to the origin.
The symbol ∈ is used to indicate that a point is in a particular coordinate system. For instance, (2, 3) would be written as (2, 3) ∈ Cartesian coordinate system.
It is important to remember that naming a point in one coordinate system does not affect its name if it is named again in another coordinate system.
For instance, since the origin is (0, 0) ∈ Cartesian coordinate system, then it is also true that (0, 0) ∈ polar coordinate system and (0, 0) ∈ spherical coordinate system.
The point (0, 0) is called the “origin” of a coordinate system.
It is never correct to say that a point in one coordinate system is given by coordinates in another coordinate system. For example, it would be incorrect to say that the Cartesian coordinates of the origin are also polar coordinates and spherical coordinates.
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The origin has a unique set of Cartesian coordinates, polar coordinates, and spherical coordinates. Every other point on the plane has a different set of coordinates in each coordinate system.
The three most common coordinate systems are:
• Cartesian coordinate system: uses two coordinates: x-coordinate and y-coordinate
• Polar coordinate system: uses a single coordinate: angle θ measured in radians from a fixed line called the “pole” to the point (x, y) on the plane.
• Spherical coordinate system: uses two coordinates: angle θ measured in radians from a fixed line through the origin to the point (x, y) on the plane; and a distance ρ that measures how far away (x, y) is from the origin.
It is generally easier to work with Cartesian coordinate system for mathematics because it uses two coordinates and can be used to graph functions or find line integrals.
However, the spherical coordinate system is used in physics and engineering the most frequently because the equations of physics use ρ and θ more than x and y.
An excellent visualization of how mathematicians and physicists view spherical coordinate systems differently was done by Bill Thurston on his blog.