The Power of Tangency

When thinking about maps, it doesn’t take students long to realize that although they have a flat, 2D map, the Earth’s surface is actually curved, so why is a 2D map a pretty good estimate of a small region of Earth’s surface? We’ve already talked about how to communicate topography on a map, but not about why the map is a good estimate in the first place for a small region. So, let’s talk about tangency, first on the level of algebra, then calculus. We will start in 2D, then move onto 3D. In 2D, the Earth will be seen as a circle and our map will be a line. Remember, a tangent line is a line that touches a curve at exactly one point; no more, no less. But how do we figure that out?

To begin, a secant line is a line that intersects a curve at two different points, like below:

Screen Shot 2019-06-04 at 2.37.03 PM

Now, to find the tangent line that touches the curve at exactly (4.505, 4.505), I start moving the purple point closer and closer to the blue point:

Screen Shot 2019-06-04 at 2.36.29 PMScreen Shot 2019-06-04 at 2.38.49 PMScreen Shot 2019-06-04 at 2.38.59 PMScreen Shot 2019-06-04 at 2.39.08 PMScreen Shot 2019-06-04 at 2.39.21 PM

Notice, the closer the two points get, the closer my secant line gets to becoming a tangent. However, it is technically impossible to make the difference between the points zero because we need two points to produce a line. But you get the idea, right? Isaac Newton and Leibniz invented a way to overcome this problem, but that discussion will happen at the end of this post. For now, here is the perfect tangent line below:

Screen Shot 2019-06-04 at 2.45.59 PMBut why did we go off on this tangent about tangent lines (see what I did there)? Well, obviously the tangent line is not nearly the same as the circle, but notice what happens as we zoom in further and further:

Screen Shot 2019-06-04 at 2.45.59 PMScreen Shot 2019-06-04 at 2.47.19 PMScreen Shot 2019-06-04 at 2.47.26 PMScreen Shot 2019-06-04 at 2.47.34 PMScreen Shot 2019-06-04 at 2.47.42 PMOnce we zoom in far enough, the tangent line becomes indistinguishable from the edge of the circle. Therefore, the region very close to the point (4.505, 4.505) on the circle can be reasonably estimated by the tangent line. Well, turns out this same idea can be applied to a sphere (the Earth) and a plane (the map). Check it out:

Screen Shot 2019-06-04 at 3.02.54 PMScreen Shot 2019-06-04 at 3.03.15 PMScreen Shot 2019-06-04 at 3.03.40 PMUnfortunately, I am limited by software capability for this demonstration, but you get the idea that the closer in we zoom, the closer the flat plane gets to a good approximation of the sphere around a given point. This is why maps of a very specific region work! A map of Wellington, New Zealand, where I am currently based, is a fair approximation because Wellington is such a tiny region on the scale of the Earth:

Wellington GE