[Little Demos] 04 – What Curve Do Shadows Trace (On a Spherical Earth)?

This was a cancelled “Work-At-Home” experiment for physics undergraduates, due to local weather conditions, and a second thought that this project is better suited for high school students.

Conic Sections

You may skip to the next section if you are comfortable with analytic geometry or related mathematical fields.

If you use a plane to cut the surface of an infinitely extending cone, the resulting curve of intersection is called a conic section. Depending on the relative positions of the plane and the cone, you might get an ellipse, a parabola, or a hyperbola.

On the Cartesian plane, any quadratic equation of x and y (as long as it’s not degenerate) corresponds to a conic section. That means that all conic sections follow the form:

Ax² +Bxy+Cy² +Dx+Ey+F =0

Merits of Geocentrism

The earth orbits the sun with a persistent axial tilt of roughly 23.5 degrees. As a result, the latitude at which the sun directly shines overhead at solar noon (the subsolar point), oscillates between -23.5 and 23.5 degrees during the course of a year. You might have seen these photos from tropical regions, such as Hawaii, where the shadows of things disappear on certain days at noon, giving you a surreal feeling.

On this note, the long-superseded Geocentric view of the universe did leave one practical legacy: for the slow-moving creatures on the surface of the planet such as ourselves, thinking like Ptolemy of Alexandria gives us a useful framework to describe the paths that celestial bodies take during the day.

A sphere is locally flat, and so we picture the ground on which we stand as the x − y plane, with positive x- direction being east and positive y-direction being north. Readers in the Southern Hemisphere can simply flip the signs.

We then regard the rest of the universe a large sphere that surrounds us. It has two poles that are directly above the earth’s north and south poles, and the ground plane cuts into it, forming a hemispherical “sky dome”. All celestial objects, the sun included, trace almost circular paths on the dome as a result of the earth’s rotation. 

This coordinate system is illustrated below.

The northern celestial pole conveniently has a bright star system nearby – Polaris – and deducing one’s latitude would be as simple as measuring the elevation angle of that star. That’s not relevant to today’s little demo, of course.

The Shadow

In the coordinate system defined above, we can consider a rod of unit length standing upright relative to the ground.

Assuming the earth is perfectly spherical, it is not hard to show that the tip of the rod’s shadow will move on the ground according to the following equation,

(􏰀x² +y² +1) 􏰁sin²a=(ycosb−sinb)²

This is a family of conics (one or a pair) parametrised by two angles:

• a, the latitude of the sub-solar point,
• b, the latitude of the rod.

Henceforth, if we can get the curve traced by the shadow on the East-North coordinate system, we may employ nonlinear curve fit to find a and b. In other words, we can locate ourselves in space and in time (restricted to somewhere on contemporary earth …) by staring at shadows.

Of course, to see shadows change with your own eyes, you must be very bored and wait for hours. So we stare at the model instead, like happy little theorists that we are.

For most of us, notice that the a and b combinations usually describe a hyperbola, going readily to infinity to both sides, which corresponds to sunrises and sunsets. At the same time. it never hurts to remind ourselves that, when the length of the shadow is comparable to the curvature scale of the earth, our model breaks down.

For high latitudes, it’s actually possible for the tip of the shadow to trace an ellipse, which corresponds to the Midnight sun, where the sun never sets during summer. Of course, the transition case of a parabola is also possible. Question for you: Where and when does that occur?

I’ve prepared an interactive notebook (with one extra simplification model where we work out a values according to day of the year):

Demo on Desmos Graphing Calculator.

I skipped a lot of historical contexts in the interest of time, but as you may have guessed, what I described here is exactly how sundials work…

Lastly…

Take this, flatearthers!
(I was one of them until I realised some F. E. believers were actually serious… Is this a hint of the next Aperiodical blog post?)