Messages: 1771 Location: Northern New Hampshire
Registered: November 2003

Center hand & Subdial hand intersections (or "Playing with your Chrono")

Thu, 15 September 2011 23:20

If you own a chronograph, it likely has a constant second hand as well as a center chronograph seconds hand. Once both hands are going, those hands will cross each other. At certain moments, there is a point on the dial which is under both hands simultaneously.

Ever wonder about that intersection? If you have a chrono, start it and watch the center second hand cross the subdial (constant) seconds. See the path the intersection point takes? How would you characterize that curve?

Maybe to you it is intuitively obvious where those curves come from, but not to me. I'm more of a plug-and-chug kind of guy, I like to plug in the numbers and see what comes out.

So let's set up a mathematical model (all groan). Here you can see the center seconds hand in red--at (0,0)--and a subdial seconds hand in blue centered at (0,1). I've also drawn half the subdial just to see the range of these curves:

OK, now follow me through some quick elementary trig and a bit of grade school algebra.

We want to find intersections between the red and blue lines and that means we need the equations for those lines. The slope of the center hand (remember "rise over run") is the adjacent side to the angle (rise) over the opposite side to the angle (run). That's just the cotangent. So the slope is cot(α) and the intercept is 0. Therefore, the formula for its line is

y = cot(α)x

The slope of the subdial hand is cot(β) and the intercept is 1. Therefore, the formula for its line is

y = cot(β)x + 1

The intersection is the value of x where both lines have the same y value. Since the 2 lines have the same y value, we can set the 2 equations equal to each other:

cot(α)x = cot(β)x + 1

We want the x value where this is true, so solve for x (here's where the algebra comes in).

subtract cot(β)x from each side:
cot(α)x - cot(β)x = 1

gather terms:
(cot(α) - cot(β))x = 1

divide each side by (cot(α) - cot(β)):
x = 1 / (cot(α) - cot(β))

Given two angles, it is easy to solve for x and substitute to find y and thus get the coordinates of the intersection point. However, the angles are changing as time goes by. Since the red line is a second hand, the angle α is simply (6t)° where t is time in seconds.

When the hands are synchronized, that is, when α = β, the lines overlap completely and the intersection is the entire line at t = 0 and t = 30, but there are no other intersections.

Now consider the situation where the center hand is started when the subdial hand is already at 5 seconds. In that case α + 30° = β, and things get interesting.

We can graph the intersection points as a function of t and discover this curve which describes the path the intersection point takes across the dial as the hands tick along. Here I've drawn in the entire subdial and the data points are 1 second apart.

But that is only one possibility. The center hand can be started at any time, so this procedure gives us an entire family of curves. Here are the curves with the second hands offset by 2.5s, 5s, 10s, 15s, 20s, 25s, and 27.5s:

These are the curves you can see on your own chrono. And if your subdial isn't at 12, just rotate this drawing. The pointy part will always point directly at the center hand pivot.

Now all that is fine, but now that we have a mathematical model, we can ask questions about things we can never see in real life. For example, in the above analysis, we don’t consider answers outside the subdial since the subdial hand cannot be longer than its distance from the center (or the hands would collide). But what if such an impossible creature could be built?

We can extend our first curve (where the subdial hand is already 5 seconds on when the center is started) to include ALL the solutions and we get this. If you saw this coming, you are better at math than me, because I was surprised when I saw this familiar shape:

But here's the cool part. This is the full story with representatives of all the possible curves spaced as above and with the whole curve present. You can still see the subdial there near the middle:

Each different offset gives a circle of a different diameter. When the hands are 0 or 30 seconds apart, the circles have infinite diameter (hence the straight line intersections). When they differ only a little, the circles are big. When they differ a lot, the circles are small--in fact, down to diameter = 1.

These curves cover any 2 hands moving at the same rate such as a center hour hand crossing a subdial hour hand. It doesn't cover the minute over minute hands because subdial minutes are usually only 30 minutes and thus minute hands don't move synchronously.

Bet you didn't think all that was happening on the dial of your watch!