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Concept

Arc length

The total distance travelled along a curve, defined as the supremum of the lengths of polygons inscribed in it. That supremum is finite only for some curves, and it belongs to a parametrisation rather than to the set of points, since a route traced twice is twice as long.

Named by 3 essays across one field — each of them below, with the objects they name alongside it.

Four staircases against a quarter circle, all of length 2. A quarter circle with staircases of 1, 2, 4, 16 steps drawn over it; each hugs the curve more closely than the last and every one of them is exactly 2 long.

The staircase that is not the diagonal

A staircase can be made to follow a quarter circle as closely as anyone likes. Its length is 2 at every stage and the arc's length is 1.5708, and no amount of refinement closes the gap — which is a fact about length rather than about staircases.

analysis · Arc length Inscribed polygons in a quarter circle, and the length they climb towards. Four polygons inscribed in a quarter circle with increasing numbers of corners, each drawn over the curve, with its length beneath it — the lengths increase towards the curve's own.

Which curves have a length at all

A length is defined as a supremum over inscribed polygons, which behaves because every refinement is longer than the last. It is also sometimes infinite — and the condition separating the two cases is a sum of absolute differences that either settles or does not.

analysis · Arc length One curve, 3 parametrisations, 3 lengths. Several maps from an interval with the same image drawn side by side, each with marks at equal parameter steps and its length beneath it — the same point set reported at different lengths.

The length belongs to the journey

Three maps from an interval with exactly the same image, and three different lengths. The picture of a curve is the set of points it passes through, and that set does not determine how far anything travelled along it.

analysis · Arc length

Named alongside it

The objects these essays reach for when they reach for this one.

ApproximationContinuityCounterexampleLimitConvergenceDerivativeUniformityGibbs' phenomenonMonotonicityParametrisationRiemann sumSupremum

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