Saturday, January 23, 2021

How Tall Is The Arctangent?

This is a follow on to yesterday's humiliating face-plant wherein I tried to calculate the height of a plant with my face.

Or something like that.

I claimed that similar triangles would allow me figure out the height of a distant tree, using my phone's camera and knowing my distance from the tree. Tim let me know I didn't know my right foot from my left when pacing off the distance and Ohioan proved once again that he is the superior mathematician.

I deeply resent them both and vow to get even with acts of vengeance that would make a Klingon nauseous with horror.

Wait, I was only supposed to think that, not type it. Oh well.

So here's the new calculation, as yet unmanifested in a Google Sheets calculation.

  • h is the height of the tree in feet
  • d is my distance to the tree. I'm going to stick with 3' per step and accept any inaccuracies.
  • 5.5 is the height of the camera off the ground, in feet. And yes, I'm taking into account the fact that I'm not looking at it with the top of my head.
  • alpha is the viewing angle of the camera, which for my Pixel 3 is 76 degrees
  • theta is the angle of the camera lens to a spot on the tree 5.5' up from the ground. This gives me a right angle for the resulting triangle
  • beta is the external angle of the right triangle I make with the tree

I can calculate beta as soon as I know my distance to the tree.

Theta is just 76 minus beta.

The height is then my distance to the tree times the tangent of beta plus the 5.5' accounting for the height of my eyes, assuming my eyestalks aren't fully extended.

Going back to yesterday's tree, if I thought it was 52', that means I was 13' from the tree, so d is 13.

  • beta is then 22 degrees
  • theta is 54 degrees
  • The tree is then 23' tall
Using the car in the picture, I could go along with the tree being about 4 SUVs high which would make it about 23'.

Good enough?


Mostly Nothing said...

Yeah, that looks about right. It is higher up from the street from the SUV, so if its base is about 5 feet above the street, then I'd put it in the 24ft region.

Seeing this made me realize I missed yesterday's bop around the internet. I'm going to go back to the 15 other pages I check everyday. Otherwise, I could have told you how dumb you were too, yesterday. :-)

So, when I read this, I immediately thought of teaching Boy Scouts how to estimate heights. Page 330 of the current Boy Scout handbook provides 2 methods.

1) Stick method.
Have a friend who's height you know stand next to the tree. Take stick, hold it out and close one eye. Line up the top of the stick with the top of the head of your friend. Place your thumb nail on the stick where the base of the tree/friends feet are.
Now move the stick one length at a time until you get to the top.

2) Felling method.
Back away from the tree. Hold a stick upright at arms length. Adjust so the top of the stick is at the top of the tree and your thumb is at the bottom.
Turn the stick 90 degrees, keeping the thumb at the base of the tree.
Get someone to stand at the other end of the stick at the same distance away.
Step off the distance between the two.

In that method, the stride is important. My stride used to be pretty close to 3', heel to toe. Now it is more like 30-32". And consistancy is important, and difficult on uneven terrain.

There is also a couple great methods for figuring out the width of a river (for example), one using a compass. Unfortunately, compass work seems to be a lost art in the world of the GPS.

Mostly Nothing said...

My son came down and saw me typing. He thinks more like 30'. He thinking that the SUV is closer to 6'. That appears to be a Highlander. Looking that up, it is 68". And the tree appears 5 heights of the SUV.

Tim eisele said...

That looks plausible, now you just need an object of known height to double-check against. I've made enough of these kinds of calculation mistakes myself that I'm not really comfortable with a result until I've tested it out.

My students regularly gleefully correct my math mistakes in class. The weird thing is this seems to make them trust the other things I tell them more, not less.

Ohioan@Heart said...

This seems like a sensible number and the method seems bullet proof (given the approximations).

Never being one to just leave well enough alone, I decide to try to compute the height of a Sequoia that I'd taken a picture of last August. Problem: I have only a vague idea of the distance I was standing back. Partial Solution: I have one picture that by sheer serendipity has my son (6'2") standing almost next to a tree, and that I'm fairly confident that the bottom of the tree and his and my feet are almost on a plane surface, but it's a Douglas Fir, not a Sequoia.

Step 1) hold camera (phone) at shoulder height parallel to the ground pointing down at my kitchen floor. Line up image to make the 1' square tiles look square and line one grout line up with the edge of the image. Result: 5' high = 5.25' wide. That's an aperture of ~58 degrees.

Now use your method using my son's eyes at 5.5' and the top of his head at 0.5' above the level base plane. Turn crank. I was about 78' away and the tree is 101' tall. The diameter of the tree is about 2' to 3'. Look up on google... Douglas Firs get to 250 feet tall with a diameter of 5' to 6'. Close enough...

I was also able to back out the angle the camera was being held at, which worked out to about 27.5 degrees - that makes sense, since the bottom of the picture was just below the level plane.

K T Cat said...

Mock me at your own peril, MN. I have a pair of chihuahuas and I know how to use them.

Thanks again for the feedback, all.

MN, I figured the felling method involved an axe.

One Brow said...

Tim eisele
My students regularly gleefully correct my math mistakes in class. The weird thing is this seems to make them trust the other things I tell them more, not less.

Same here. I think it helps take some ownership of the class, in a little way.

Mostly Nothing said...

Threaten all you want KT. I am protected by Gypsy, and she has the strongest claim to the throne of the Feline Theocracy.