# Locally approximating a circle by a line segment?

The book falls naturally into two parts. Part I is concerned with the general theory of fractals and their geometry. Firstly, various notions of dimension and methods for their calculation are introduced. Then geometrical properties of fractals are investigated in much the same way as one might study the geometry of classical ﬁgures such as circles or ellipses: locally a circle may be approximated by a line segment, the projection or ‘shadow’ of a circle is generally an ellipse, a circle typically intersects a straight line segment in two points (if at all), and so on. There are fractal analogues of such properties, usually with dimension playing a key role. Thus we consider, for example, the local form of fractals, and projections and intersections of fractals.

What's the meaning of the bold text? How does one approximate a circle with a line segment? I've searched a little for it but didn't have success.

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Locally you can approximate any curve by a line segment. Just stick lots of points on your circle and draw chords connecting consecutive points. You get a polygon (with lots of sides) that approximates your circle. (The degree to which you have a "good" approximation depends on continuity/smoothness of the curve and, of course, how close your points are.)

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Locally, any function with a Taylor series can be approximated by a line segment.

In case of the circle, it's easy to visualize. Imagine a regular $n$-gon circumscribed by a circle, and look at what happens as $n \to \infty$. Each local piece of the circle will be approximated by a line segment, and the precision will be better as $n \to \infty$.

The idea of a local approximation is that it only works in a small neighborhood.

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If you zoom in on a small arc of a circle, it will appear to be more and more as a straight line interval as your magnification becomes greater and greater. In fact, you can define differentiability this way; see for example Keisler. Fractal curves are not differentiable and they don't "straighten out" to look like line segments when you zoom in. Alternatively, this can be expressed in differential-geometric terms by saying that when you magnify the circle, its geodesic curvature tends to zero.

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