To make the virtual “painting” above, I plotted simple and moderately-complex trigonometric functions on a single coordinate plane, as shown below, using Geometer’s Sketchpad. I then erased all the text, etc., copied-and-pasted a screenshot into MS-Paint, and used that program to make the finished image above.
[Note: If you haven’t yet read part one, it is recommended to read it first, before reading this part. Part one is the most recent post here, before this one.]
In part one, a special circumsinusoidal region was described, bounded by parts of semicircles, and a sine or cosine wave. As previously described, this requires that the wave’s amplitude be exactly one-fourth the wavelength. Of course, most sinusoidal waves do not have amplitudes and wavelengths that fall nicely into a 1:4 ratio. What happens, then, with the majority of waves — the ones with with other amplitude:wavelength ratios? Can they still be used for forming circumsinusoidal regions? The answer is yes — but the cost required is that semicircles may no longer by used in their construction.
A semicircle can be thought of as a 180 degree arc, if the diameter, often considered part of its perimeter, is ignored. When the sine or cosine wave’s amplitude:wavelength ratio is not equal to 1:4, the only necessary adjustments for the circular arcs needed to define particular circumsinusoidal regions are identical changes in the angular size of each arc used, plus translations of the centers of the circles which contain each of these circular arcs. These translations move those circle-centers away from the line representing the wave’s rest position, in a direction perpendicular to that line. There are two cases to consider: shorter-amplitude waves (those with an amplitude:wavelength ratio smaller than 1:4), and taller-amplitude waves (where the same ratio is greater than 1:4). The first picture below shows the shorter-amplitude case.
For every half-wavelength which starts at the rest position, an arc through three points is needed for the outer bound of the circumsinusoidal regions, which are shown in yellow. Those three points are two consecutive points (such as A and C above) where the sinusoidal wave crosses the rest position, plus the point at the top of the wave crest (such as B), or the point at the bottom of the trough, exactly half-way along the sinusoidal curve, in-between the two consecutive rest-position points under examination. In this example using points A, B, and C, above, the circle containing those points was constructed by drawing segments AB and BC, and then constructing the perpendicular bisectors of those two segments. Thone perpendicular disectors intersect at some point D, which is the center of a circle containing A, B, and C. The part of this circle which is not used for the arc through A, B, and C is shown as a dashed arc, while the arc used is shown as a solid curve. For other half-waves, to the left or right of this circle and arc, the construction would proceed in the same fashion, but is not shown, for the sake of clarity.
The taller-wavelength case also can be constructed, using the same procedure, as shown below.
These regions, of course, have area. To determine the exact area of any circumsinusoidal region requires integral calculus, and this area is equal to the difference in the areas under two different types of curve. Without calculus, the best that can be found for these areas are mere approximations, not exact answers. I am leaving this find-the-area problem for mathematicians who have a better understanding of calculus than I possess.
The inner boundary of the yellow regions above is a sine curve (technically, a cosine curve, but that’s the same thing, just with a phase shift). The outer boundaries are semicircles. In order for this to work, to form these yellow regions, the semicircle centers (centers of the circles they are each half of) must be directly below peaks, and above troughs, of the sine (or cosine) curve, and vertically positioned at what would be called the rest position in physics. (I’m resorting to use of some physics terminology here, simply because I don’t know the corresponding mathematical terms).
In addition, each semicircle involved must have a radius equal to one-fourth the wavelength of the sine or cosine wave. The two sets of curves cross each other at the rest position, and are tangent to each other at each peak and trough, producing four of these yellow regions per wavelength.
In this case, semicircles could used because I adjusted the wavelength, making it exactly four times the amplitude of the wave. My goal was to compare the two curves, simply to see how well one simulates the other (answer: not very well at all). Then, however, I became more interested in the discrepancy between the two, represented by the yellow regions which are outside the true wave, and inside the semicircles which contain that wave. Until and unless I find that such regions already have a different name, I am naming these two-dimensional curved shapes “circumsinusoidal regions.” There are four of them per wavelength of the wave, and two per semicircle. Each circumsinusoidal region has two vertices, but the two paths connecting them are distinct curves. No part of either path contains any length which is a straight segment.
It would be possible to generate interesting solids by rotating circumsinusoidal regions around vertical or horizontal lines, such as the x- or y-axes, or around diagonal lines. Many such solids would be variations of a torus, including the central hole of a torus, but with circumsinusoidal cross-sections replacing a torus’s circular cross-sections. Unfortunately, I do not have the software I would need to generate pictures of such solid figures.
If the wavelength used for a given sinusoidal wave is not exactly four times the wave’s amplitude, semicircles won’t work to enclose the wave with the same points of tangency, but it is still possible to generate circumsinusoidal regions — using something, in their place, other than semicircles. This will be described in part two, which will be the next post on this blog.
A few minutes ago, I wondered how to write a function whose graph would be a sine curve, but one that undulated above and below the diagonal line y=x, rather than the x-axis, as is usually the case. How to accomplish such a 45 degree counterclockwise rotation?
Well, first, I abandoned degrees, set Geometer’s Sketchpad to radians, and then simply constructed plots for both y = x and y = sin(x). Next, I added them together. The result is the green curve (and equation) you see above.
This only half-solves the problem. Does it undulate above and below y=x? Yes, it does. However, if you rotate this whole thing, clockwise, one-eighth of a complete turn, so that you are looking at the green curve going along the x-axis, you’ll notice that it is not a true sine curve, but a distorted one. Why? Because it was generated by adding y-values along the original x-axis, not by a true rotation.
I’m not certain how to correct for this distortion, or otherwise solve the problem. If anyone has a suggestion, please leave it in a comment. [Note: an astute follower of this blog has now done exactly that, so I refer the reader to the comments for the rest of the story here.]
The last post made me curious about other trigonometric functions’ graphs, in a polar coordinate system. They were not what I expected. Here they are.