# Is there a mathematical theorem that states the equivalence of curved and straight lines

Any curve can be considered to be made of sufficiently small straight lines. What is the name of the theorem which states this fact?

Thank you.

I asked the same question at Hacker News.

[Edit: in the Hacker News question, the OP included the following link, which I think clarifies his/her intent:

http://musr.physics.ubc.ca/~jess/hr/skept/Math/node10.html

--PLC.]

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Can it, really? – Robin Chapman Oct 5 '10 at 18:06
Let's be a little generous and perhaps offer assumptions needed for this to have a meaningful interpretation and to be true, recognizing that it comes from a non-mathematician seeking help. E.g., "all piecewise differentiable curves are rectifiable." – whuber Oct 5 '10 at 18:08
Is any curve "piecewise differentiable"? Is any rectifiable curve "made of sufficiently small straight lines"? – Robin Chapman Oct 5 '10 at 18:18
I started a discussion about this on meta because I would like it re-opened if possible. – anon Oct 5 '10 at 20:21
On the assumption that it is useful to have a permanent link to the relevant discussion on meta posted here: meta.math.stackexchange.com/questions/901/… – Rahul Oct 5 '10 at 21:05

For many calculations that one might want to perform on a "reasonable" curve (e.g., one coordinatized by functions that are differentiable, or piecewise differentiable/smooth/analytic/nice), taking a finely spaced mesh of points on the curve and computing the result on the polygon joining those points, instead of the curve, will produce an answer close to the one for the curve. The smaller the spacing, the closer the result will be to the result for the curve.

Quantities approximable in this way include area enclosed by a curve, arc length, integrals of given functions along the curve, winding number around a point, splines, parametrizations, and others. Quantities not approximable in this way include curvature, which will be zero on the sides of any polygon used as a substitute for the curve, and integer-valued "global" quantities such as number of tangent lines.

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"Calculus is the acknowledgment that anything but a linear function is far too complicated for us to handle. Differentiation provides the techniques to tame functions by making them locally linear, and integration comprises the rules for sticking the local pieces together again." -- Alf van der Poorten, Notes on Fermat's Last Theorem (Lecture IX).

From this point of view, one might nominate the Fundamental Theorem of Calculus (to which van der Poorten is alluding) as an answer to this question.

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As a variant to this answer, one could also say that, although there is no one particular theorem expressing precisely this idea (that a curve can be considered to be made up of straight line segments), calculus can be regarded as a set of techniques and theorems that are to a large extent collectively based on this principle. (See also T..'s answer for more elaboration on exactly what aspects of the curve can be studied in this way.) – Matt E Oct 6 '10 at 3:18
@Matt E: Interesting. I had a slightly different interpretation of this quotation: it does not seem to refer to approximations by piecewise linear functions (which, after all, works well only in one dimension anyway--it fails for surfaces), but rather to the idea that an integral is "adding up" infinitesimals. Alternatively, we can take a more sophisticated view of "locally linear" as not being linear, but only guaranteeing an arbitrarily good approximation to linearity within sufficiently small neighborhoods (looking at the germs of functions, really). – whuber Oct 6 '10 at 3:31

Maybe what you need is the Stone-Weierstrass Theorem, which you can use to show (roughly) that a continuous function on a compact set can be approximated arbitrarily well by continuous piecewise-linear functions. The boolean ring version of the theorem is directly applicable here.

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Piecewise-linear?! – Robin Chapman Oct 5 '10 at 18:42
Anyone who can understand the statement of that theorem knows that not all curves can be "considered" to be made of sufficiently small straight lines! – Mariano Suárez-Alvarez Oct 5 '10 at 18:43
@Robin Piecewise-linear!! [Hewitt & Stormberg, Real and Abstract Analysis, example (d) on pg. 97] – Jyotirmoy Bhattacharya Oct 5 '10 at 18:47
Polynomial, not PL. PL is an easy intermediate case used to prove the polynomial result. – T.. Oct 5 '10 at 19:20
@T. The general S-W theorem is stated in terms of algebras or lattices of functions, so polynomial and PL are both applications. AFAIK Weirstrass's original result was just for polynomials. But you are right, it is easy to prove PL from first principles. – Jyotirmoy Bhattacharya Oct 5 '10 at 19:29