Belts stretch, and circles come out as ovals

How compliance in a drive system becomes geometric error in the part.

A belt is not perfectly rigid, so a small amount of the motor's motion is spent merely stretching the belt rather than moving the carriage, and because that stretch changes with the load and the direction of travel, a circle commanded in one smooth sweep comes out very slightly egg-shaped rather than round, the error not constant but depending on how hard and in which direction the belt happens to be pulling at each point around the curve.

What is really going on

A drive belt is stiff compared to a rubber band but nowhere near perfectly rigid, and under the pull needed to accelerate a carriage it stretches by a small, real amount before the carriage actually starts to move. That stretch is not fixed, it grows and shrinks with how hard the motor is pulling at any instant, which itself depends on how fast the carriage is accelerating and which direction it is being asked to go. Tracing a circle means constantly changing direction, which means the pull on the belt is constantly changing too, so the small lag between where the motor thinks the carriage is and where the carriage actually is keeps changing shape as the circle is drawn. A lag that changes shape around a curve does not produce a uniformly smaller circle, it produces one flattened in whichever directions the belt happened to be under the most rapidly changing load, which is usually along the axis where the carriage has to reverse or turn hardest. Two separate belts drive most such machines, one for each axis, and because the two rarely stretch by identical amounts under identical loads, the flattening is not even perfectly symmetrical either, which is why a genuinely round test circle is one of the more demanding shapes a belt-driven machine is ever asked to trace, harder in practice than many shapes that look far more intricate on paper.

The elastic-lead comparison

Walking a dog on a lead with some genuine stretch in it shows the same lag on a scale anyone can feel through their own hand. On a straight path at a steady pace, the stretch settles to roughly the same amount the whole way and the dog trails along at a near-constant distance behind your hand. Turn a sharp corner, though, and the lead has to first take up the new direction of pull before the dog actually follows it around, so the dog cuts the corner rather than tracing the same sharp turn your hand made, because the elastic in the lead absorbed part of the sudden change in direction before transmitting the rest of it. A belt-driven carriage tracing a curve does the same cutting, absorbing part of each change in direction into stretch before the carriage itself catches up, and a shape made of continuously changing direction, a circle, ends up with that cutting happening a little differently at every point around it. A stiffer lead, one with almost no stretch in it at all, would let the dog track your hand's path far more faithfully around the same corner, which is exactly the reasoning behind choosing a stiffer belt, or a shorter run of it, wherever a machine's geometry allows the choice.

The number that matters here

A belt might stretch by only a few hundredths of a millimetre under normal running loads, a distance far too small to see on a straight line, but a circle commanded to a diameter of several centimetres can still come out measurably short of round by roughly that same small amount, because the error does not average away around the curve, it repeats at the same points on every single lap.

Where this stops being true

Slowing the machine down removes most of the effect, since a belt pulled by a smaller, steadier force stretches by a smaller and steadier amount, and a circle drawn slowly enough that the belt's stretch barely changes as direction changes comes out close enough to round that the error can no longer be measured by eye. This is exactly why the same machine can produce a passable circle at a cautious speed and a visibly oval one when pushed to move quickly, without anything about the belt or the motors themselves having changed at all between the two attempts. A straight line, needing no change of direction at all once under way, barely shows the effect regardless of speed, which is why the error is so often discovered only when a design finally calls for a genuine curve rather than the straight edges most early tests happen to use.