Estimating Distance

A couple of weeks ago, I was watching a football game with my family. It was near sunset and the sky was illuminated by the last rays of the setting sun. Far overhead, jetliners could be seen crossing the sky, their contrails bright and warm although I knew that they existed in deadly cold.

One of them was not at cruising altitude. Older Grandson asked “How high is that plane?”

“About 25,000 feet,” SiL responded. I thought it was closer to 10K AGL.

“I don’t think so; I think it’s closer to 10,000.”

A short debate ensued and it was interesting. I respect SiL’s opinion given he works with aircraft as a professional.

Later, on the road Monday, the interchange recurred to me. As I drove along, I wondered whether there was a simple way to estimate the altitude. After thinking about it a few minutes, I came up with the following.

For small angles, the angle and the sine of the angle are approximately equal. Therefore the angle subtended by the aircraft can be used to estimate its distance from the observer. If the approximate angle from the horizontal is also known, then the elevation can be estimated.

The width of a human thumb at arm’s length is about a half-degree, or thirty minutes of angle. The pinky fingertip is about half that.

As I thought about my observations, my estimate was about three of the aircraft would be the width of my pinky finger at arm’s length… or maybe half that. That would be between two and a half and five minutes of angle, or one-twelfth to one-twenty-fourth of a degree. The sight distance to the aircraft would then be 12 to 24 times the apparent width of the vehicle.

My guess is that the apparent width of that aircraft was on the order of 120–200 feet. Therefore, for S&G’s, say it was 150 feet. At twenty times the apparent width, that would be 30,000  feet. My estimate for the angle-to-horizon was less than 45 degrees, so I’ll use that as an upper limit. The sine of 45 is about 0.7. So, the elevation is about 70 percent of the length of the hypotenuse — or sight distance.

My estimate of the altitude of the aircraft is about 20,000 feet. SiL was closer to correct than I was. Trust the professional.

Happy Pi Day, Belatedly

Light-Up PiI’m running a day late and a dollar (or two) short. But, I wanted to write about Pi Day before too much time gets away from me.

Every year I remember Pi Day (among a few other special days). It’s a fun reminder about the beauty and power of mathematics. One definition for pi is the ratio of the circumference of a circle to its diameter. Pi is one of the transcendental numbers, that is, it is an infinite sequence of non-repeating digits. Fun stuff…

There are a number of celebrations for Pi Day, but this year I elected to go to a geocaching meet, Pi Day with the Professor down in Gardnerville. I would meet a few geocachers, celebrate a moment of transcendence, and collect a geocaching souvenier. So the Girl and I packed out a bit early to scout some locations for landscape photographs on our way to Gardnerville.

It was a fun gathering and I was there before 9:26AM. That’s the other fun part of this particular Pi Day, yesterday, at 9:26:53AM, it was Pi Day to nine decimal places (3.141592653). This occurs only once each century, and the next one will be in 2115. I will not be there for that one.

Unfortunately, the noise and energy in the small coffee house was too much for me. I left not long after the special time, saying thanks and goodbye to my fellow lovers of the outdoors and the scavenger hunts known as geocaches. It was fun, but it was also time to get home and get after other tasks.

LaTeX?

Apparently, WordPress has a \LaTeX interface for producing mathematics and math-like symbols. If it does, then this is something I’ve wanted in my weblog for a long time.

Alright, it works. The background is white and the foreground is black (duh), so it doesn’t quite work with my current color scheme, but it will do. At least I can now produce a decent-looking equation.