## CSI: Toronto

11.4.11 | Posted by
OM |
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So a guy came into the subway with absurdly large shoes. So large in fact that I had to take a picture. Unfortunately the iPhone’s extremely wide angle lens skews the perspective of the picture considerably – although still large in the picture, the shoes were much larger in real life. Without the picture, there is no way for me to convey how hilariously massive this individual’s shoes were… or is there?

Well actually there is but it involves a bit of math. If you don’t like math here is the answer: 15 inches. The guy’s shoes were 15 inches long. Isn’t that wild? If you want to know how I got that read on.

So basically, an image passes through a lens at the same angle of view in which it comes in at. If that made no sense look at this diagram:

This creates a pair of similar triangles. That means that the ratio between the size of the virtual shoe projected onto the sensor and the distance between the sensor and focus point is proportional to the size of the actual shoe and the distance between the shoe and the focus point. Expressed mathematically:

Solving for the real shoe size:

Of course the shoe was not shot head on but instead has an X and Y component, so:

where:

and the same for

*s*. That leaves two variables to be solved: the size of the shoe on the image sensor, and the distance between the camera and the shoe.

_{ry}Finding the size of the “virtual shoe” is pretty straightforward.

Meaning that the projected shoe measured 0.969mm wide. The same was done to solve for

*s*.

_{vy}Finding the focus distance (f2) was a bit trickier. I was sitting on the left-most side of a three-seater, and shot with my phone in my right hand. The shoes seem to be in line with the right side of the three-seater, meaning that the X distance would the width of the three-seater minus the distance between the seat and my right hand. The shoes seem to be around the midpoint of the car, so the Y distance would be half of the car width minus the distance between the side of the train and my phone. If that made no sense look at this diagram:

Also my phone was around 2.5’ from the ground. Solving for a vector in three-space:

which gave a focus distance of around 1475mm. I had everything I needed to solve for size of the real shoe:

which gave a total length of 379.65mm which is around 15 inches. So yes, his feet were huge.

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