# Does focal length affect coordinate estimation?

I am using a webcam mounted on the ceiling to track some markers. I measured the area on the ground that the webcam is able to see, it is roughly 2.61m x 1.46m.

The output of the coordinates right in the corners of that boundary is however not 2.61 and 1.46, but rather 1.90 m and 1.10 meters. (Measured in the same coordinate space, so that is not the error).

I was curious how these values are obtained, so I noticed:

• 1.90/2.61 is roughly 0.73
• 1.10/1.46 is roughly 0.75.

The webcam I'm using is a Logitech Tessar / Carl Zeiss 2.0f/3.7mm, so the focal length should be 7.4mm, if I understand right.

If I attribute the error to measurement uncertainty (which is definitely there, I was not very accurate measuring the boundary frame), then this number looks suspiciously like the factor (in cm) by which the coordinate estimates are wrong.

Is this true, or is it just a coincidence in my case? I know close to nothing about cameras.

• "2.0f/3.7mm", so the focal length should be 7.4mm, if I understand right." Why do you think that? Did you multiply 3.7 with 2.0 to arrive at that result?
– null
Jun 27, 2016 at 11:37
• Yes, I read that the f value is the focal langth divided by lens diameter, which I assume this value means. Is that not correct? Jun 27, 2016 at 11:38
• The focal length of your lens is 3.7mm. The f-number is 2.0. That means the diameter of the entrance pupil is 1.85mm. Jun 28, 2016 at 8:34
• Is the camera pointed straight down so that the optical axis of the lens is perpendicular to the floor? Jun 28, 2016 at 8:36
• Yes it is pointed straight down. Jun 28, 2016 at 9:12

Your coverage area is 2,46 by 1.46 meters is almost an exact match of the aspect ratio 16:9 – the current standard for HD TV. I attempted to find the published angle of view for your unit which is likely model C905 720P. I was able to find that manufacture stated the diagonal angle of view is 75⁰.

Now the lens labeling tells me that the focal length is 3.7mm and that the speed of the lens is f/2.

Knowing the diagonal angle of view, and the aspect ratio, I am able to calculate:

The imaging chip format is approximately: Height 2.8mm thus with a 3.7mm lens mounted, the vertical angle of view is 41.5⁰

Length 5mm – horizontal angle of view 68.2⁰

Diagonal 5.74mm – diagonal angle of view 75⁰

If a longer lens is mounted, the angle of view will be reduced. If a shorter lens is mounted, the angle of view increases.

I figured it out. The mistake was that the camera was poorly calibrated. That the scaling factor was about the focal length was thus just a coincidence.