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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LISSAJOUS FIGURES

Definition: "When a particle is subjected to two sine wave motion or two oscillatory motion at right angles, the particle describes lissajous figures". We know sine wave motion and circular motion is basically same. Hence we draw two circles A and B perpendicular to each other. The circle B rotates twice faster than circle A. That is, frequency of circle B is two times than that of A. A particle at the intersection of two circles is subjected to two sine wave motion A and B at 90 degree simultaneously. The particle will describe figures depending on the frequency and phase of A and B . In our case, the ratio of frequency is 1:2 and the two waves are in phase. To draw lissajous figures : A moving point in both the circles are chosen. Here we should remember; during the time taken by the circle A to complete one rotation, circle B completes two. Hence the points are marked on the circles according to their speed. Then straight lines

THE PARABOLA

A jet of water shooting from a hose pipe will follow a parabolic path. What is the so special about parabola. Y= x^2 Draw a graph for the above equation. It will result in a parabola. This parabola is also called unit parabola. Any equation involving square will yield a parabola. Example: Y = 2x^2 +3x+3 (also called quadratic equation) X= 2 and -2, both satisfies the equation 4 = X^2. Parabolic equations always have two solutions. Any motion taking place freely under gravity follows parabolic path. Examples: An object dropped from a moving train, A bomb dropped from flying plane, A ball kicked upwards. If a beam of light rays fall on the parabolic shaped mirror, they will be reflected and brought to focus on a point. This fact is made use of in Dish Antenna, Telescope mirrors, etc. Inverted parabola shape is used in the construction of buildings and bridges. Because the shape is able to bear more weight. A plane

CASINO'S GAME

Let us find out how the casino survives with mathematics. Say, your friend invite you for a game of dice. You must bet (wager) 2 dollars. If you roll 'six' you will get back 8 dollars. The game will go on for 30 rounds. All sounds good. The probability of rolling 'six' is 1/6. Since the game will be played for 30 times, the 'expected win' is 30*1/6 = 5. That is, you are expected to win 5 rounds out of 30. Hence your gain will be 5 * 8 =40 dollars. ok. This also implies that you will loose 25 rounds. Hence your loss will be 25*2 =50 dollars. Your net gain will be gain-less = 40-50 = -10 dollars. For 30 rounds, the loss is -10 dollars, Hence, for one round =-10/30 = -1/3 dollars. There will be a loss of -1/3 or 0.33 dollars per round. It is not a fair game. Let us make a simple formula to calculate 'Pay out per round\. The probability for a win = p The pay-out in case of win = V No. of rounds = n The expect

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