Is this true? Sine curves are continuously differentiable, and if you if you take one with a very small period you’ll get a long length in a small area. It’ll “look like” a straight line if the amplitude is small enough.
That’s right, Snarky Kong. You can make a bunch of smooth curves (in the sense of smooth functions from [0, 1] to space) which get, pointwise, closer and closer to tracing out a small circle, or whatever else you want, yet their length branches off towards whatever arbitrarily large value you like. It’s not a question of smoothness; it’s just that, as Hari Seldon said, length just isn’t continuous in this sense.
In other words: The length of the pointwise limit of a bunch of curves in this sense needn’t equal the limit of their lengths. Knowing the values of a curve to precision epsilon doesn’t really put any upper constraint on its length at all; it could be oscillating infinitely wildly within that epsilon, and build up whatever length you like. Length just isn’t continuous in this sense. It just isn’t. It’s perhaps continuous in other senses. But not this sense. In this sense, length is not continuous, and all these examples illustrate exactly why.
Yes, you can use Fourier series to approximate any reasonable function. The infinite series you generate will converge to the function, but again, there’s no sense that taking the limit of a series of functions preserves arc lengths.
The bit about tangent lines is important; when you derive the integral for calculating lengths of general arcs, you do it basically by using the Pythagorean theorem on a small section of the arc - the sides at a right angle being the changes in x and y and the hypotenuse being a short secant line that comes closer to approximating the tangent line as you get smaller and smaller. The argument used in the OP’s drawings could be extended to “prove” that any line’s length could just be found by taking the distance between its end points.