### The Equiangular Spiral (Alternate Version)

#### Part 2: Measuring a Spiral Seashell

The background figure in the following applet shows a cross-section of a nautilus shell with superimposed axes and 45° lines (in white). Our objective in this section is to determine a formula for the outer spiral curve -- the black crosshairs are located at one of the points where this curve crosses the x-axis. You will use the applet to measure polar coordinates of points on the curve. Then you will paste the measured coordinates into your worksheet and fit a curve to the points described by them.

1. Click the "Mark point" button to mark the starting point, whose coordinates are given in the upper right corner. The units for these measurements are pixels, the smallest distinguishable dots on your screen, and the x and y measurements are taken from the lower left corner of the picture (not from the origin marked by the white axes). Those measurements are automatically converted to r and theta measurements, which are from the marked origin. Thus, r = 20 and theta = 0 indicate that the starting point on the curve is on the x-axis and is 20 pixels to the right of the origin. Moving around the curve in a counterclockwise direction, click and mark a few more points, and notice how the numbers change.

2. Now click the "List points" button. This will open a smaller window (which you may resize as necessary) in which your marked points will be listed in proper format for pasting into your worksheet. Close that window for now -- we will open the list again when we have enough data points.

3. If you are satisfied with your marked points thus far, continue moving around the curve and marking points until you have at least 20 points. You may mark as many more as you wish. At any time, you may click on "Clear points" to start over. When you think you have enough points to describe the curve, click "List points" again, then copy the results and past them into your worksheet at the indicated place.

4. Note that each time you make a complete circle around the origin, the theta values "start over" -- that is, all measured values of theta are between 0 and 2 pi. Thus, you need to add 2 pi to each theta value after the first drop, 4 pi after the second drop, and so on. If you copy and paste, this will go very quickly.

5. Follow the instructions in your worksheet to plot the sequence of polar measurements, with r as a function of the theta. What sort of growth does this look like?

6. Experiment with logarithmic plotting of the data to determine the type of growth.

7. Find a formula for a continuous function r = R(theta) that reasonably approximates the measured data points.

8. Test your formula by superimposing the graph of R(theta) on the cartesian plot of the data. If the fit is not good enough, adjust your formula until it is. When you are satisfied with the fit, move on to the next part.