IS THE DIGITAL PHOTO LYING?

Today, we live in a world of digital images. Ads, Media, movie, legal documents, CCTV or just picnic photos. Everywhere, there are images. We know, this images can be easily doctored and engineered by any body with the computer. Further, when the photos are taken at different perspectives, the lengths, the angles and their relationships are distorted. Then, how to find the truthiness of the image? There is a mathematical way. Let us go into detail.
Take a torch. Switch it on. Allow the cone of light to fall on the wall. If you hold the torch straight, you see a circular patch of light. If you tilt the torch slightly, you see an ellipse of light.
The circle is there in the light cone always. The circle is distorted into an ellipse, when the light patch hits the wall. I want to prove, the ellipse is only the projection or an image of the circle. Here we go. Draw a diagram as given below.

Here, when the beam of light is sliced vertical to it, you get a circle. But the wall slices off the light beam which is falling obliquely on it. Hence you see the ellipse. Now draw straight lines from O A,B,C and D of the circle to the ellipse and get O' A' B' C' and D'
The ratios OA/OB= O'A'/O'B' = 1
OC/OD =O'C'/O'D' = 1
The consistency and equality of the ratios proves that the ellipse is the projected image of the circle. (many other ratios are also possible in this case)
We will go to real life example.

Say, a man is seen from point O or captured by the camera from O. Light rays are emanating from the man and falling on the eye aiding vision. Four such rays are drawn here.
Suppose, a flat screen is held (straight or oblique) in the path of rays, a distorted image of the man falls the screen. (This is what happens inside the camera and the eye with the help of lens. But for convenience, we have shown the image outside).
The projective geometry says that,
(AC/CB)/(AD/DB) = (A'C'/C'B')/(A'D'/D'B')
This "cross ratio " holds good for any real life object anf its image. Cross ratio is the ratio of ratios using which we can verify the authenticity of any image.
We live in 3D world. But mostly images are in the 2D screen. Hence the 'cross ratio' is highly helpful.
Mathematics is true forever.

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