Cartesian and Polar Coordinates

[Poweranimals]

What are the differences in the "uniqueness" of the representations in Cartesian coordinates and in polar coordinates?

 

[robphy]

Do you mean this?

Distinct rectangular coordinates map to distinct points.
However, distinct polar coordinates don't always map to distinct points.
For example, $$(r = 0, \theta=0)$$ and $$(r = 0, \theta=\pi/3)$$ map to the same point.

 

[wais]

Cartesian and polar coordinates are two different systems used to locate points on a plane. In Cartesian coordinates, points are represented by their distance from two perpendicular axes, the x-axis and the y-axis. This results in a unique representation of a point as an ordered pair (x,y). On the other hand, polar coordinates represent a point by its distance from the origin and the angle it forms with a fixed reference line. This results in a unique representation of a point as an ordered pair (r,θ).

The main difference between the uniqueness of representations in Cartesian and polar coordinates lies in the way they measure distance and direction. In Cartesian coordinates, distance is measured along two perpendicular axes, resulting in a rectangular grid. This allows for precise measurements and calculations, making it a preferred system in mathematics and physics. However, it does not take into account the direction of a point, as both the x and y coordinates can be positive or negative.

On the other hand, polar coordinates take into account both distance and direction, making it a more intuitive system in some cases. The distance from the origin is measured as the radius (r), while the direction is measured as the angle (θ). This allows for a unique representation of a point, as the radius and angle values cannot be negative. However, polar coordinates can be more complicated to work with in certain situations, especially in calculations involving trigonometric functions.

In conclusion, the uniqueness of representations in Cartesian and polar coordinates depends on the context and the purpose of the representation. Cartesian coordinates provide a precise and efficient way of representing points on a plane, while polar coordinates offer a more intuitive and unique representation of a point's distance and direction. Both systems have their own advantages and are useful in different scenarios.