Comment by crdrost

I think this particular function was selected because it happened to be convenient to program and the visual effect was pleasant enough.

The actual "correct" thing to do would probably be to have the point maintain constant speed in 3D space like a real boat sailing on a globe, right? But that's a rather bigger lift:

    const degrees = Math.PI / 180;
    const bearing = 5 * degrees; // or it might be 85 degrees? Not sure off the top of my head
    const k = Math.tan(bearing);
    const v = 0.001 // some velocity, adjust as needed
    const phi = (t) => v*t/Math.sqrt(1 + k*k) // the sqrt is not strictly needed
    const theta = (t) => k*Math.ln(Math.tan(phi(t)/2)) // this is the annoying one haha

with outputs,

    const x = (t) => Math.sin(phi(t)) * Math.cos(theta(t))
    const y = (t) => Math.sin(phi(t)) * Math.sin(theta(t))
    const z = (t) => Math.cos(phi(t))

I doubt that they did the ln(tan(phi/2)) thing though, but it's what you get when you integrate the k d{phi} = sin{phi} d{theta} equation that you have here.