Using trig to measure heritage trees (tricky)

[Peter Andrew]

Hi there,

I like applying math to other problems like the arts. I'm doing a project about a heritage tree and want to add a math component but this time I'm stumped.

Problem: To calculate the width of these heritage tree in 1910 and 1960 from old photos. (Once I have this I can use a log function to plot their growth and estimate its age).

Data: I have photos of the tree in front of a heritage building. The photos were taken in 1910 and 1960 and I also took my own photos. I know how big the tree is now, how big the buildings are and the distance between the tree and the building.

I have three known attributes:

• The 2014 tree width {W₂₀₁₄}
• The building width (constant) {B}
• The distance between the tree and the building (constant) {D_TB}

I can measure the old photos to give me two relative attributes from 1910/1960:

• Pixel width of tree in 1910 or 1960 {W_P-1910 or W_P-1960}
• Pixel width of building {B_P-1910 or B_P-1960}

I have one unknown attribute to calculate:

• The 1910 (or 1960) tree width {W₁₉₁₀ or W₁₉₆₀}

It should be simple to apply trig to solve this problem but I've spent two weeks and can't figure it. Does anyone have any suggestions?

The main problem I struck is that the distance between the tree and photographer isn't constant. But the photos are all taken from the same angle and the distance between the tree and building obviously hasn't changed.

Any help would be greatly appreciated!

 

[stapel]

...the distance between the tree and building obviously hasn't changed.

But this distance has changed (though perhaps not by much). When the tree was a sapling, the side of its trunk was not as close to the side of the building as it was a hundred years later, after having widened (along with having gotten taller).

 

[Ishuda]

Just mumbling out loud: Appears to me that the real problem is that the photos were taken from different distances. If you could, get an estimate of the distances from which the photos in 1910 and 1960 were made and take two more pictures; one from each distance. You could then try to project pixel widths 'back' to a common distance and see if you can get some kind of agreement for the 1910-present pair of photos and the 1960-present pair of photos. All sounds a little iffy but might work out. I think you are going to have to do something akin to that to get anywhere.

Another thought to try to accomplish the same type thing. Take photos at various distances to see if you can get a match on width of building.