Perfect Circle

A circle is the set of points sitting the same distance from a centre. That is the entire definition, and it is also the entire scoring method: measure the distance from the dot to every point of your stroke, and see how much those distances vary.

Dividing by the mean is what makes the score fair across sizes. A wobble of five pixels is disastrous on a tiny circle and invisible on a huge one, so the error is measured relative to size.

The exponential is what makes the score mean anything. Scoring 1 − wobble directly is far too kind at the top: a radius swinging by 10% is visibly lumpy but still scores 90, so nearly every attempt lands in the nineties and the number stops distinguishing anyone. Decay spreads the top end out.

There is an obvious exploit in the maths above: draw a short arc and every point on it sits at almost exactly the same distance from the centre. A quarter of a circle scores close to 100% while looking nothing like a circle.

So before scoring anything, the stroke has to prove it went round. The angle from the centre to your pointer is tracked as you draw and the total turn is added up — a circle is 360°, so anything short of a full turn is rejected instead of rewarded. The stroke also has to close rather than spiral, which is checked by comparing the radius where you started against the radius where you finished — not the distance between the two endpoints, since overshooting the start by a few degrees leaves a perfectly closed circle.

Tracking that angle has its own catch. It flips from +180° to −180° as you cross the far side, which naively reads as a giant jump backwards. Any step bigger than half a turn is treated as that seam and unwrapped, otherwise every circle would score as having gone nowhere.

Most people score in the seventies on a first attempt and are surprised — circles feel like they should be easy. The reason they are not is mechanical:

• Pivot from the elbow or shoulder, not the wrist. The wrist has a limited, uneven range, so wrist-drawn circles come out as eggs tilted in the direction of least comfort.
• Go slower than feels natural. Fast strokes are ballistic — once launched you cannot correct them mid-flight.
• Go bigger. A larger circle spreads the same absolute error over a longer path, which the relative scoring rewards.
• Look at the dot, not your line. Watching the pointer makes you chase it. Fixing on the centre lets peripheral vision keep the radius honest.

The dashed circle drawn over your result is the one you were measured against. Comparing them shows exactly where the stroke drifted — almost always at the far side, where your arm has run out of comfortable travel.

There is an old story, told by Vasari in the sixteenth century, about the painter Giotto. A messenger from the Pope arrived asking for a drawing to prove his skill. Giotto dipped his brush, planted his elbow against his side to steady it, and drew a single freehand circle. That was the whole submission.

The messenger thought he was being mocked. The Pope, according to the story, understood immediately — an unaided circle shows control in a way an elaborate drawing can hide. Round as Giotto’s O survives in Italian as a phrase for something perfect.

Note the detail that gets left out when the story is retold: he braced his arm first. Even the anecdote agrees it is about the mechanics.