4.4 From Cartesian to polar coordinates

In Cartesian coordinates, we need two numbers to specify a point. If we restrict ourselves to the unit circle, each point is uniquely determined by one number, the angle φ from the positive x-axis given in radians such that 0 ≤ φ < 2π. We lost the need for a second number by insisting that the point has distance 1 from the origin.

More generally, let P = (a, b) be a nonzero point (that is, a point that is not the origin) in R². Let r = √(a² + b²) be the distance from P to the origin. The point

is on the unit circle. There is a unique angle φ 0 ≤ φ < 2π that corresponds to Q. With r, we can uniquely identify

Figure 4.22: Polar and Cartesian coordinates

(r, φ) are called the polar coordinates of P. You may sometimes see the Greek letter ρ (rho) used instead of r. ρ`italic polar coordinates...