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The Equiangular Spiral - Plotting a Spiral Curve

Lang Moore, David Smith, and Bill Mueller

Now that we have a formula R(t) for the growth of the radial shell measure, we will convert this formula to a polar-coordinate formula in the plane. Corresponding to the point counter


we have the polar angles


The values of n are also values of the continuous variable t, so the relationship between theta and t is


  1. Use this relationship and the function R(t) already defined to construct a function r = r(theta) that describes the shell radius as a function of polar angle.

  2. Test your function by making a polar plot of r = r(theta) for theta between - 2 pi and 4.5 pi, with equal scales in the horizontal and vertical directions. The result should look something like the green curve in Part 2.

  3. Find parametric equations, x = x(theta) and y = y(theta) to describe the same curve. Plot the parametric equations to confirm that you really have the same curve.

  4. Change the left end of the theta interval from - 2 pi to - 10 pi, and plot again. How does this change the graph?

  5. Zoom in on the graph several times until you are looking at parts of it much closer to the origin. What do you notice? (To zoom in, you need to make the upper bound for theta negative as well. If you need to, use values of theta smaller than - 10 pi.)

  6. Usually when you zoom in on a continuous curve, you see a very different behavior than you have just seen with the equiangular spiral. What is that different behavior?

Lang Moore, David Smith, and Bill Mueller, "The Equiangular Spiral - Plotting a Spiral Curve," Convergence (December 2004)