Grab the brown and black knobs. You can also dial some of the numbers on the right.
Spherical coordinates are defined by ρ (rho, the distance from the origin), θ (theta, rotation parallel to the xy-plane), and φ (phi, inclination from the north pole to the south pole). This interactive drawing shows how they relate to the Cartesian xyz coordinates. The key is the horizontal slice of radius r.
Which makes sense: when φ=0, we’re looking at the north pole, z=ρ and r=0. Then we’re left with the familiar equations:
In many textbooks, the definition of r is inserted into the definition of x and y, making them difficult to memorize (and the image above harder to see). In particular, defining r allows one to mentally construct spherical coordinates on top of polar coordinates, rather than as a separate entity. While we’re improving notation, remember that τ = 2π.