Wednesday, September 7, 2016

Solar Geometry On The Flat Earth


No matter when you look, the sun is directly above one and only place on earth, and in that place an object standing upright will cast no shadow.

Suppose we put a marker at that place, and we call that place M. Since the sun is always moving, M is always moving, too. But at any given time, M is the place directly below the sun.

In the diagram at right, S represents the sun, M marks the place directly below the sun, and the line through M and P represents the surface of the earth.

A vertical object in any place other than M will indeed cast a shadow, and this diagram (which is clearly not to scale) shows the geometry of that shadow. T represents the top of the object, B represents the bottom of the object, and P marks the end of the object's shadow.

And my questions for today are: Can we determine the distance from M to P? And if so, how can we do it?

The answer to the first question is, "Yes!" But the answer to the second question depends on the shape of the earth. With the Globe Earth model, it's fairly difficult. But with the Flat Earth model, it's easy.

It's easy because we can use the geometry of similar triangles. As the diagram shows, angle SPM is the same as angle TPB, and SMP and TBP are both right angles, 90 degrees each. Therefore, angle MSP and angle BTP are equivalent, and triangle SMP is similar to triangle TBP, which means the corresponding sides of these triangles are proportional.

We can simplify things a bit here by introducing a few abbreviations:

As shown in the diagram at right, we can let:

H = the height of the stake above the ground, or the distance BT.

L = the length of the stake's shadow, or the distance BP.

K = the height of the sun above the earth, or the distance SM.

Q = the distance from the mark under the sun to the end of the shadow, or MP.

Because the two triangles are similar, Q/K = L/H.

If we we multiply both sides of this equation by K, we can see that Q = K*L/H.

We can measure L. We can measure H. According to the Flat Earth model, K is constant. And we have a simple equation for Q in terms of L, H, and K.

Therefore we can easily find Q. And this is what I decided to do.

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