In parametric equations, is "$r$" conventionally defined as the length of curve? If not what exactly is $r$?

Question

What is $r$ exactly? I know it can be tabled to get $x$ and $y$ points. Is $r$ a function to get the length of the curve for any angle put into it?

Answer 1

A. No, in your example, r has nothing to do with arc length. Rather, r is the (radial) distance to the point, measured from the origin. Typically, r will be a function of θ, which is the angle that the radius makes to the x-axis. Here, the function is of the simplest nature: f(θ)=θ−1. Also, when the function’s value is negative, you have to walk in the opposite direction to the direction pointed at by θ.

B. r and θ are traditionally used as the variables in polar coordinates. r measures the distance from the origin and θ measures the angle counterclockwise from the positive x axis.

C. To be clear, arc length is distance along a curve, by following the curve. In your exercise, and most polar coordinate, r is typically direct distance from the pole, otherwise known as the origin to the point. Does that make the distinction from radius clear?

Credit to @Lubin and Ross Millikan for first "answers".