# If a camera optical axis is perpendicular to a flat surface, does the area covered by a pixel is the same for all pixels?

My question is as follows: if a (calibrated) camera is facing a perfectly flat surface (i.e. the optical axis is perpendicular to the surface), is the area represented by a pixel independent of the position of the pixel on the sensor?

I simplified the problem (1D instead of 2D) and made a figure of what I think models the issue:

The first horizontal line represents the camera sensor and the bottom one represents the flat surface. They are parallel and the camera optical axis (vertical line) is perpendicular to the two because of the hypotheses. The variable $$c$$ is a constant representing the width of a pixel, and $$y$$ is the projected length. The variable $$x$$ represents the position of the pixel on the sensor.

The question is now formulated as: does $$y$$ depends on $$x$$?

I applied two Thales theorems and found that $$y = \frac{b c}{a}$$, so $$y$$ seem to be independent of $$x$$:

Since $$\frac{x}{d} = \frac{a}{b} = \frac{a_1}{b_1}$$ and $$\frac{c}{y} = \frac{a_1}{b_1}$$,

then $$\frac{c}{y} = \frac{x}{d}$$ and $$\frac{x}{d} = \frac{a}{b}$$

so $$y = \frac{c d}{x}$$ and $$d = \frac{b x}{a}$$

finally $$y = \frac{b c x}{a x}$$ = $$\frac{b c}{a}$$

Are my modeling and my reasoning valid?