Circular Coordinates

Why

We identified points in $\R ^2$ with elements of the plane in a natural way.¹

Definition

Let $(x, y) \in \R ^2$. Then $(r, \theta ) \in \R ^2$ is the polar form or circular form of $(x, y)$ if

\[ x = r \cos \theta \quad \text{ and } \quad y = r \sin \theta . \]

In this case we call $r$ and $\theta $ the circular coordinates or polar coordinates.

Since $\sin$ and $\cos$ polar coordinates are not unique.

Non-uniqueness

A difficulty with polar coordinates is that there are many elements of $\R ^2$ that correspond to the same point in the plane. For example, consider the points

\[ (5, \pi /3), (5, -5\pi /3), (-5, 4\pi /3), (-5, -2\pi /3). \]

Each of these specifies the same point in $\R ^2$.

Future editions will expand on this in the genetic approach, and likely reference celestial motion. ↩︎