I haven’t forgot my little problem about working out where the apparent edge of the world was, from my visit to the Sleeping Bear Dunes in northern (lower) Michigan. What I have been is stuck on a way to do all the calculations in a way that’s clear and that avoids confusion. I realized the calculations were reasonably clear to me but were hard to describe because I could put into similar-looking symbols a bunch of things I wanted to describe.
So I’ve resolved that the best thing I can do is take some time to describe the things I mean, and why they’ll get the symbols that they do. The first part of this is drawing a slightly more mathematical representation of the situation of standing on top of the dune and looking out at the water, and seeing the apparent edge of the dune as something very much closer than the water is. This is what’s behind my new picture, a cross-section of the dune and a person looking out at its edge.
The crossed black lines are meant to represent the axes of a Cartesian coordinate system. The intersection of the two gets called the origin, and points on the plane get described in terms of how far they are to the right or the left, and above or below, that origin. If we want to talk about the coordinates of some arbitrary point, without pinning down which point we mean exactly, we usually use the label x to describe how far to the right of the origin the point is, and the label y to describe how far above the origin is, and since this is the 21st century and we’re comfortable with negative numbers, a negative x means we’re talking about a point to the left of the origin, and a negative y means we’re talking about a point below the origin.
There’s a sandy-orange shape meant to represent the dune, although all that I’m really interested in is the upper edge of the dune. This traces out some curve in the plane. Here I’ve drawn it as a parabola, but I don’t have any special reason to suppose that it is; a parabola just looks dune-shaped to me. I could have drawn a semicircle, or an ellipse, or a more wiggly figure. The more wiggly figure is probably truer to the actually existing dune, but it’s harder to work with. A parabola might be so much easier to work with that it’s worth the mistake made by using it. I’m also supposing that the sea is level with the origin. It has to be somewhere, after all, and zero is a nice round number, generally not difficult to deal with.
I am going to need to describe the edge of the dune, the boundary between the sandy-orange spot and the great emptiness above (and to the left of) it. This boundary between dune and non-dune is a collection of points, a set of paired values of x and y. And I’m assuming there’s some relationship between x and y, that is, that there’s some equation, containing x’s and y’s, which is satisfied whenever the point described as “a distance x to the right of the origin, and a distance y above the origin” is on the boundary, and which isn’t satisfied whenever that point is not.
This relationship could be simple, such as for a straight line, which might be something like . It could be a bit complicated, such as something like . It could even be a real mess, something like . I’m going to suppose that it can be written out in a form like , where it’s easy to separate the terms in the equation with a y in it from the ones with an x or anything else.
The blue dot on the right represents the viewer’s head: it’s somewhere to the right of the origin, and not just above the origin but also above the dune’s boundary. Let me say the person’s eyes are a distance h off the ground. That means the x- and y-coordinates of the blue dot have to satisfy the equation .
The line of sight going to the edge of the dune is the green line. If the person’s looking at the edge of the dune, this line has to touch some point — the red dot — that’s also on the boundary between dune and non-dune. The x- and y-coordinates of the red dot satisfy, again, .
That line, though, is also a collection of points. We only know what two of them are right now — those of the blue dot and those of the red dot — but that’s enough. If you have the coordinates of some point on a line — call those coordinates x0 and y0 — and you know the slope of the line — and for slightly obscure historic reasons that’s often called m — then one way to write an equation corresponding to that line is . (There are many ways to write this; don’t worry about that.)
The coordinates for the blue dot we should know. After all, it’s where we happen to be standing, and if we don’t know that, we’re lost. But we can say say that its x-coordinate is xb and its y-coordinate is yb, so we can work out everything we’d like to know without having to commit ourselves to some particular numbers. The red dot is a challenge. We don’t know just where it is offhand, but we can say say that its x-coordinate is xr and its y-coordinate is yr.
But we’ll be able to find where it is, because the green line is a tangent line to the dune’s boundary. A tangent line is, well, a line, and — at least near the point of tangency, here, the red dot — it only touches the curve it’s a tangent to the one time. (It might cross back again, if the curve wiggles around enough. The dune I drew doesn’t wiggle around enough, but it’s imaginable.) And here we draw calculus into things.
One of the first things one learns in calculus is how to calculate the derivative of a function at a point. This has many interpretations, but the one that’s useful here is as a measure of how a function changes. Imagine a bead confined to move along the curve described by , and starting from the coordinates of the red dot. If you slide the bead a little bit left or right, it also has to move a little bit up or down so as not to fall off the curve. And the ratio of how that y-coordinate changes with a little x-coordinate change turns out to be exactly the slope of the tangent line at the point of tangency.
There are a couple of notations to describe the derivative of a function at an arbitrary point with coordinate x, among them, , or (which makes the assumption that the curve is described by the equation , or even more simply, . The derivative of the dune’s boundary curve at the point with x-coordinate xr is then .
So the green line — which has to go through the blue dot, with coordinates of xb and yb, and which has the slope f'(x_r) — has to be described by an equation like .
And this has got all the symbols in play that we need to work out the question of how far the apparent edge of the dune is, and how far away the water past that — the yellow dot, which I hope you’ll let me describe as being at the x-coordinate xy and y-coordinate yy — is.