Here, I've added the draggable complex point q through which there's
a unique equiangular spiral going through (1, 0) and with "collapsed
radius" of 1. By "collapsed radius", I mean the radius of the circle
we get when the ray-intersection angle is a right angle.

It can be seen how for any ray, there's a family of spirals and that
bringing the test point onto the ray selects the spiral that matches
the rectangular-coordinates one.

Next, to find out how we can do the reverse---find the rectangular-
coordinates spiral that matches a given real-power-of-complex-point
spiral.

Also on the back burner: illustrating construction of spirals via
buildup using increasing number of rays, as described in http://
mathworld.wolfram.com/LogarithmicSpiral.html :