Polar Equation of Equiangular Spiral

Consider a curve AB(Figure 1) which has the polar equation

a and b being constants. Let the angle between a radius OB and a tangent to the curve at the end B of the radius be . Then

From the equation to the curve,


Accordingly, is a constant. Since r increases with , we obtain aspiral curve:

This is the polar equation of an equiangular spiral. The independent variable may have any value from to, so that the curve is unlimited in lenght.

Using the polar equation,it is a simple matter to make a rough sketch of a portion of the spiral with ruler and compasses only, if we accept circular arcs as approximations to the actual curve. Alternatively, polar graph paper will give amore accurate result. Consider three radiii seperated by rightangles (Figure 2),


Figure 2. Logarithmic and Rectangular Spirals

whence . Thus, OB is mean proportional between OA and OC, whence the angle ABC is a right angle. It follows that the rectangular spiral may serve as the basis of the logarithmic spiral of Figure 2.

The value of the angle is at our disposal. An extreme case occurs when . Then degenerates into a circle: r = a