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.
- 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.
- 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.
- 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.
- 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.
- 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?
- Experiment with logarithmic plotting
of the data to determine the type of growth.
- Find a formula for a continuous
function r = R(theta) that reasonably approximates the measured data
points.
- 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.