Physiogram shapes and figures

Simple principles

When a light bulb swings above a camera, after each elliptical orbit the light returns to approximately the same position as last time, where it has about the same velocity. In a conventional physiogram, with the camera stationary, this gives a trace for each orbit near to the previous one, and it is this sequence of near-superimposition that builds an attractive picture out of simple ellipses.

With the camera moving as well, then without special attention, when the light bulb has completed an orbit, the camera is somewhere else. No longer is the result built from repetitions of a simple shape. The result is unlikely to be very attractive. For example, in the following photographs, the camera was placed on a record turntable while the light swung in a straight line above it, with no attempt to match the period of swing to the rotation of the turntable. The second had a longer exposure than the first. A very long exposure would have resulted in the entire circle being filled up. These photographs are "interesting" rather than "attractive". (The first one has been published twice, but probably for curiosity reasons).

Camera on record turntable

The lesson is that the period of the camera should be related to the period of the light bulb. This doesn't mean they must be exactly the same. On the contrary, that would make it pointless to move the camera at all! But simple ratios, such as 1-to-2, 1-to-3, 2-to-3, and 3-to-4, normally give the best results. (A ratio of 2-to-4 is the same as 1-to-2, of course. On this page, period-ratios use integers without a common factor except 1).

Another good principle is to avoid combining too many motions at once. For example, except for the "record turntable" photographs above, the camera was always swung without any rotation or twisting about the lens axis. Throughout its orbit, it would be parallel to its position elsewhere, so that its base always faced the same wall.


Lissajous Figures

Swing the light in a straight line. Swing the camera in a straight line at a right-angle to the swing of the light. The result is a Lissajous Figure. It is the shape seen on an oscilloscope when a sine-wave is fed in. The camera's motion corresponds to the vertical scan of the oscilloscope, while the swing of the light bulb corresponds to the horizontal motion of the spot representing the signal.

Two parameters are particularly important. These are the ratio of the periods, and the "relative phase". The latter identifies whether there is ever a time when both of them are at the end of their swing at the same time.

First below are three 1-to-1 Lissajous Figures with different phases. In the first, both are at the ends of their swing at the same time. In the third, when one is at the end of its swing, the other is at the centre of its swing. Then there are some more interesting ratios! Where I have them, I show photographs that (approximately) match.

Lissajous Figures, 1-to-1
   
Lissajous Figures, 1-to-2
 
Lissajous Figures, 1-to-3
Lissajous Figures, 2-to-3
   
Lissajous Figures, 3-to-4
   

Ratios can become ever more complicated, but by this time the result is getting too cluttered. If the result showed exactly one complete cycle, it would probably look OK, but with either less or more it looks confusing.

Other parameters that affect the result include the relative sizes of the swings, and whether the swings really are at right angles:

Lissajous Figures, 2-to-3,
not at right angles

Twin-elliptical Figures

The real power of swinging the camera arises when both light and camera are swung in ellipses, not in straight lines! (Straight lines and circles are actually both extreme types of ellipse). In all of these cases, the camera was swung without any twisting about the lens axis. Once again, this works best when the ratios are simple. But not too simple, as these traces show!

Twin-elliptical, period-ratio 1-to-1
Same direction Opposite directions

Not only are the ratio of periods, and to a lesser extent the relative phase, important, but whether they are swung in the same direction or opposite direction matters. There are simple rules to use as the starting point when trying to predict the result:

Rule 1: If they swing in the same direction, the main "cusps" will be on the inside of the shape, and the number of them will be equal to the difference of the ratio-integers.

Swinging in the same direction
1-to-2 1-to-3 1-to-4 2-to-3 3-to-4
   

(Note that with the more complicated ratios there are smaller cusps added to the main cusps. These tend to result in cluttered figures, and unless particular care is taken, the simpler ratios work best).

Rule 2: If they swing in opposite directions, the main "cusps" will be on the outside of the shape, and the number of them will be equal to the sum of the ratio-integers.

Swinging in opposite directions
1-to-2 1-to-3 2-to-3 3-to-4 3-to-5

The above sequence raises the question: "why is the 7-cusp photograph treated as 3-to-4 rather than 2-to-5 or 1-to-6?" (After all, each of those figures also has 7 cusps). The answer is in the sequence of the cusps traced by the light. Number the cusps from 1 to 7. Then trace the line, noting the sequence of the cusps visited:

7-cusp results
1-to-6 2-to-5 3-to-4

Cusp sequence:
1 7 6 5 4 3 2 1

Cusp sequence:
1 6 4 2 7 5 3 1

Cusp sequence:
1 5 2 6 3 7 4 1

   

The photograph has the sequence: 1 5 2 6 3 7 4 1. It is not a coincidence that, in the photograph, the 5th cusp is 4 away from the 1st cusp measured clockwise, and 3 away from the 1st cusp measured anti-clockwise! (And so on for the entire sequence). Mathematics can often help elucidate the beauty of natural phenomena!

The relative sizes of the swings of the light and the camera matter a lot. If the slower-moving object has a bigger swing than the other, the cusps diminish. And vice-versa. (I won't repeat any photographs in this table).

Period 1-to-2, same direction
Size 3-to-1 Size 2-to-1 Size 1-to-1 Size 1-to-2 Size 1-to-8
Period 1-to-2, opposite directions
Size 4-to-1 Size 2-to-1 Size 1-to-1 Size 1-to-2 Size 1-to-8
Period 2-to-3, same direction
Size 3-to-1 Size 2-to-1 Size 1-to-1 Size 1-to-3 Size 1-to-8
Period 2-to-3, opposite directions
Size 4-to-1 Size 2-to-1 Size 1-to-1 Size 1-to-2 Size 1-to-6
Period 3-to-4, opposite directions
Size 1.3-to-1 Size 1.2-to-1 Size 1-to-1 Size 1-to-1.5 Size 1-to-3

It also matters how much the orbits are ellipses rather than circles. These show the effects of eccentricity in the slow-moving object. (Some of the traces have been rotated to match the photographs).

1-to-2, swinging in opposite directions
Eccentricity 3.5 Eccentricity 2 Eccentricity 1 Eccentricity 0.3 Eccentricity 0.1
   

Sometimes several different factors combine, and it can take considerable experimentation with the spreadsheet to work out later what (probably) happened! (Some of the differences arise from such things as friction in the pivot of the compound pendulum, etc).

2-to-3
opposite directions
relative size 1-to-5
eccentricity 10
+ phase shift
1-to-2
opposite directions
relatives size 5-to-2
eccentricity 0.3
+ phase shift

Some of the results remain a mystery!