The Boon Companion has been experimenting with long exposure times and intentional camera movement, of late. She was just about to discard the motion-blurred cyclist above as a failed experiment when something about the image caught my eye.

In the thirtieth-of-a-second exposure, the bicycle wheel has rolled a short distance. But why do the spokes look curved? Why don’t the curves point towards the centre of the wheel? And why is the effect only visible in the lower half of the wheel?

So I sat down to figure out the trajectory of a bicycle spoke as the wheel rolls along the ground. As you do.

Any point on wheel rolling across a flat surface without slipping follows a curve called a trochoid (from Greek trochos, “wheel”). I won’t pester you with the relevant equations (they’re at the other end of the link above). Here’s what the trochoid curves look like for the two ends of a radial spoke, spanning the distance between a wheel hub and a thick wheel rim:

The shape of the wheel is plotted in grey dashes, with a single vertical spoke marked. As the wheel rolls left or right, the ends of the spoke follow the curved trochoid trajectories, with successive positions marked at 20º intervals.

But bicycle wheels don’t (usually) have radial spokes, and I felt obliged, going into the problem, to look at the position of real bicycle spokes. Here’s a very common pattern:

Thirty-six spokes, laid out in what’s called a “three cross” pattern. Fundamentally, there are only two different spoke alignments in this pattern—leading and trailing.

For a wheel rolling from right to left, the spoke I’ve highlighted in red is leading, and the blue spoke is trailing. They’re simply mirror images of each other. One side of the wheel has nine leading spokes, spaced 40º apart, and nine trailing spokes in the same pattern. The other side of the wheel is laid out exactly the same way, but with the pattern rotated by 20º. The final result is called a “three cross” pattern because each trailing spoke crosses three leading spokes on its side of the wheel (and vice versa).

In this pattern, the anchor point for the spoke at the hub is offset 60º relative to its attachment at the rim. So to see the trajectory of a representative bicycle spoke, I need to slide the trochoid curve for the hub 60º out of alignment with the rim curve. Here its, with the spoke drawn in at 20º intervals:

This is the trajectory of a trailing spoke for a wheel rolling right to left, and a leading spoke for a wheel rolling left to right.

We can already get a hint of why some sort of spoke pattern shows up in the lower half of the wheel, but not the upper. In the upper half, the spoke is moving rapidly sideways, as it pivots across the top of the wheel; in the lower half it performs a sort of dipping motion, arcing downwards towards the point at which the wheel contacts the road, and then arcing back up again.

Now, I figure a cyclist moving at a reasonable speed for a shared-use path will rotate the wheels through about 30º during a thirtieth-of-a-second exposure. Here’s a more detailed trajectory for a trailing spoke (for a bicycle moving right-to-left) during a 30º rotation:

You can see that the spoke slides along itself, to some extent—different parts of the spoke occupy the same spatial position at different times. These are the only parts of the spoke that will show up as a dense “shadow” during a prolonged exposure that blurs the other parts. In the final photograph we’ll therefore see something like the arc I’ve sketched in red, while the rest of the spoke is smeared into a blur:

Something similar happens for a leading spoke as it passes through the same position:

So, actually, while the leading/trailing distinction slightly changes the details, both kinds of spoke produce a curved shadow during a prolonged exposure.

If we catch a spoke that’s higher in its trajectory, we get another arc:

Now we can put these images together, showing the curved shadows of several spokes at once. Each of them will lie on a different set of trochoid arcs, shifted laterally according to how far the spoke lies from the lowest point of its trajectory. Like this:

I’ve marked the visual centre of the wheel, for reference. Notice how the partial shadow arcs formed by the lower spokes seem to point below the centre, while the arcs of the higher spokes point above the centre. It happens because none of the spokes are radial, and because the centre of the wheel is never stationary, but shifts horizontally as the spokes sketch out their curved arcs.

So. Although I was baffled by the photograph when I first saw it, I began to get an inkling of what was going on as I made the first trochoid sketches, and was then pleasantly surprised by how things began to fall neatly into place as I added more detail. I’m hoping you’re as pleasantly surprised as I am.