Uniqueness of Cartesian and polar coordinate representations

[Apophis]

What is the difference in the "uniqueness" of the representations of Cartesian coordinates and in polar coordinates? Also, what is the non-uniqueness?

 

[James R]

Take an example:

The point with unique cartesian coordinates (1,0) has many possible polar coordinates, such as $(1,0), (1, 2\pi), (1,4\pi)$.

Non-uniqueness is having multiple expressions for the same point.

 

[thepatientmental]

Cartesian coordinates and polar coordinates are two different systems used to represent points in a two-dimensional plane. The main difference between the two is the way in which they describe the location of a point.

In Cartesian coordinates, a point is represented by its distance from two perpendicular lines, known as the x-axis and y-axis, and is denoted by (x,y). This system is also known as the rectangular coordinate system. On the other hand, polar coordinates represent a point by its distance from the origin and its angle from a fixed reference line, known as the polar axis. A point in polar coordinates is denoted by (r,θ).

One of the main differences in the uniqueness of the representations of Cartesian and polar coordinates lies in the way they describe a point. In Cartesian coordinates, a point can be uniquely identified by its x and y coordinates, whereas in polar coordinates, a point can be represented in multiple ways. For example, a point with coordinates (1, π/4) can also be represented as (1, 5π/4) or (-1, 3π/4) in polar coordinates.

This concept of multiple representations of a point is known as non-uniqueness. It means that a single point can have different representations in polar coordinates, depending on the choice of the polar axis and the reference angle. This is because the distance from the origin remains the same, but the angle can be measured in different ways.

In contrast, Cartesian coordinates have a unique representation for each point, making it easier to locate and identify a point on a plane. However, polar coordinates have advantages in certain situations, such as representing circular or symmetric patterns, where the distance from the origin and the angle are more relevant than the x and y coordinates.

In conclusion, the uniqueness of Cartesian and polar coordinates lies in the way they describe a point, with Cartesian coordinates having a unique representation for each point, while polar coordinates have non-uniqueness due to the multiple ways in which a point can be represented. Both systems have their advantages and are used in different applications depending on the nature of the problem.