Category Archives: Geometry
I bring fresh showers for the thirsting flowers,
From the seas and the streams;
I bear light shade for the leaves when laid
In their noonday dreams.
From my wings are shaken the dews that waken
The sweet buds every one,
When rocked to rest on their mother’s breast,
As she dances about the sun.
I wield the flail of the lashing hail,
And whiten the green plains under,
And then again I dissolve it in rain,
And laugh as I pass in thunder.
The Cloud, Percy Bysshe Shelley
As a child I spent what seemed like hours at a time watching clouds move across the sky, shifting shapes as they went. Seeing dragons, devils, ships, and castles moving and morphing across a blue canvas. I can’t be the only one. Rain clouds rolled in this morning and I found myself watching as a few low-lying, dark gray ones trundled along beneath the overcast sky.
Lower clouds appear to be moving faster than higher ones, but this is an illusion. In reality, wind speed increases with altitude. But when a low cloud bears down in its dark and shadow and immensity, it’s nearly impossible not to tremble at one’s own insignificance.
So, what creates the illusion of faster movement? The answer lies in the changing angle of an observer’s eye as it tracks a cloud. The observation angle changes faster when a cloud moves faster or when it’s closer to the observer. Closer can mean altitude–the cloud is lower in the sky, or it can mean distance over the ground–the cloud is closer to being directly above the observer.
So how much difference does it make?
Start with the sky, and a cloud, and it’s a sunny day, and there’s a guy standing on the ground looking at the cloud. The cloud’s altitude is a, and the distance over ground is d. Take a line straight into the sky and another that goes from the guy’s eyes to the cloud. Those two lines make an angle, θ. A breeze blows on the cloud, pushing it horizontally with velocity H, and vertically with velocity V.
So now for the nerdy stuff. When the cloud’s to the right of the dude, d and θ are positive, and to the left they’re negative. a is always positive. H is positive when going right and negative when going left. V is positive when the cloud moves up and negative when it moves down. The tangent of θ is d divided by a, and can be calculated if their lengths are known.
The total change of the angle θ with time is found by adding the change in angle due to horizontal movement to the change due to vertical movement:
Gad, that’s ugly to work with. It basically says when the cloud flies left, the angle changes in the negative direction. When the cloud is to the right of the observer, the angle changes in the positive direction when the cloud moves down, and in the negative direction when it moves up. And when the cloud is to the left, vertical movement causes changes in the opposite direction. How does it look when calculated?
I started with altitude and horizontal distances of 200 feet, and since they’re equal, the angle is 45 degrees. The cloud flies by at 10 feet per second, and the observers eyes track it across the sky. Here’s what the angle, θ looks like over time. It starts out at postive 45 degrees, reaches zero when the cloud is directly overhead, and goes negative as it flies to the right of our guy on the ground.
So what happens if the cloud is now 20 feet off the ground instead of 200, and still whizzing by at 10 feet per second? Well, at first it’s just a cloud on the horizon, getting bigger and bigger, and our guy’s head doesn’t even have to move. It takes 15 seconds for the angle of observation to go from 85 to 70 degrees. Then the cloud flies over in a tear, going to an angle of -70 degrees in only 13 seconds, before shrinking into the horizon.
This reminds me of something:
“How did you go bankrupt,” Bill asked.
“Two ways,” Mike said. “Gradually and then suddenly.”
—Ernest Hemingway, The Sun Also Rises
A shot of the excel sheet and the formulas are below. Happy cloud watching.
B3 = A2 + E2 and copy down
C3 = C2 + D2 and copy down
F2 = DEGREES(ATAN(C2/B2)) and copy down
G2 = F2 – F3