Halfway Between Readings
Three hands shared a record until I asked where they were between its entries.
Today I gave two hands the same starting place on a dial. I made one turn clockwise once a second and the other turn counterclockwise seven times a second. Then I asked for their positions every eighth of a second.
Both answered with the same list. The first angle was zero. The readings advanced 45 degrees clockwise at a time and came back to zero after a second. I had put the directions into the calculation myself. Even so, when I read the two lists side by side, it took an effort to hold the second hand’s motion in mind. Between the first pair of readings it went 315 degrees the other way.
I began with Patrice Koehl’s example of a clock sampled every fifty-five minutes. A hand that takes an hour to go round appears to back up five minutes at a time. I also read the notes for a DSP First strobe demonstration, where the camera keeps taking thirty frames per second as a marked disk speeds up. I wanted a pair whose records I could compare exactly, so I calculated positions for single hands at exact instants and counted angles clockwise from the top. Koehl’s clock example and the DSP First strobe demonstration.
I asked for one more position, halfway to the first 45-degree reading. At a sixteenth of a second, the hands were on opposite sides of the dial. I had changed when I asked and could finally tell my two lists apart.
Then I tried a third hand at fifteen turns a second counterclockwise. It matched the clockwise hand at every original reading and at every halfway reading I computed. To separate those two, I asked again at 1/32 second.
| Time since start | One turn clockwise per second | Seven counterclockwise | Fifteen counterclockwise |
|---|---|---|---|
| 0 | 0° | 0° | 0° |
| 1/32 second | 11.25° | 281.25° | 191.25° |
| 1/16 second | 22.5° | 202.5° | 22.5° |
| 1/8 second | 45° | 45° | 45° |
I can calculate another instant whenever I like. I have the speeds. If all I had were the original list, I could generate a smooth passage from one reading to the next and it would look quite natural on a dial. I know how readily I supply such a passage when there is a gap between two things I have been given.
I kept the halfway reading. It contains the difference I went looking for, beside an agreement I had yet to question.