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Rectifiable Curve

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A rectifiable curve has a finite arc length.

Nonrectifiable curves, like fractals, have infinite length. They could also be called curves that are rectifiable by straight lines; their arc lengths can be expressed as elementary functions of their endpoints [1]. Many basic curves are rectifiable, including cycloids. Non-rectifiable curves include the Koch snowflake, circle and ellipse.

rectifiable curve

A cycloid, generated by a rolling curve, is a rectifiable curve. Image: Zorgit| Wikimedia Commons.


koch snowflake

Fractals, like this Koch snowflake, are nonrectifiable.


Defining a Rectifiable Curve

Formally, we can define a rectifiable curve as follows:

If a first derivative for a function is a continuous function on the closed interval [a, b], the graph is rectifiable between (a, f(a)) and (b, f(b)). Functions that meet this criteria are continuously differentiable on the specified closed interval and have a smooth curve between the two points [2].

Dealing With Nonrectifiable Curves

In calculus and real analysis, we’re mostly interested in rectifiable curves of finite length as they are straightforward to analyze. In a way, calculus can “force” a curve to be rectifiable: it’s length can be measured by very small line segments, leading to extremely accurate “approximations.”

The humble parabola is nonrectifiable, but we normally only deal with intervals of a parabola in calculus; In the rare case when you want to analyze a parabola over its entire (infinite) length, it can be made rectifiable by adding or subtracting its arc length to the length of an auxiliary parabola [1]. Nonrectifiable curves can also have their arc lengths expressed in terms of transcendental functions. Nonrectifiable curves also show as solutions to half-linear differential systems [3].


References

[1] Moll, V. et al. (2002). Bernoulli on arc length.
[2] Larson, R. & Edwards, B. (2009). Calculus, 9th Edition. Cengage Learning.
[3] Naito, Y. et al. (2018). Rectifiable and Nonrectifiable Solution Curves of Half-Linear Differential Systems. Math. Slovaca 68. No 3. 575-590. Retrieved April 11, 2021 from: https://search.proquest.com/openview/60993ad5055fb6bd783b70fcb571d77d/1?pq-origsite=gscholar&cbl=2038886
Cycloid image: Zorgit, CC BY-SA 3.0 , via Wikimedia Commons.

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Stephanie Glen. "Rectifiable Curve" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/rectifiable-curve/
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