A curve whose equation in polar coordinates is r=aekθ, where a (>0) and k are constants. Let O be the origin and P be any point on the curve. The curve derives its name from the property that the angle α between OP and the tangent at P is constant. In fact, k=cot α. The equation can be written r=kθ+b, and the curve is also called the logarithmic spiral.
http://jwilson.coe.uga.edu/EMT668/EMAT6680.F99/Erbas/KURSATgeometrypro/related%20curves/related%20curves.html Animations of related curves, and an interactive manipulative of the spiral in Geometer's Sketchpad.