Map-Estimation and Arc-Length

Whenever I spend a lot of time driving, navigating, and map-reading, I find myself making a lot of rough estimates of distances.  The process reminds me of how one estimates and using calculus, ultimately evaluates, the arclength of a curve.

To find the length of a curve, the basic idea is to approximate the curve with a bunch of line segments.  It’s relatively easy to find the length of a line segment, and so by sacrificing exactness, you turn a hard problem into an easy one.

I made my line segments equal in length to 40 miles on the map.  Now I just approximate the curve and add up the lengths of the little line segments.

Each of the seven line segments is (roughly) equal to 40 miles, so the approximate length of the path from Brooklyn to Burlington is 280 miles.  (Not a terrible estimate).  There are plenty of ways to improve the approximation (such as?) and the straightforward (but complicated) calculus-y approach eventually produces the arclength integral, also known as a line integral.

On the actual drive my approximations weren’t as good, as I was using an inferior distance estimator.

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