FLAT MAP FROM SPHERICAL EARTH.

We know the importance of google map today. How to make a map? Earth is spherical, how to flatten it? When we look at a piece of land, it looks flat. Because the area is small. Hence making a map of a small town is easy. When we stand at the sea-shore and look at the ocean, we see horizon where the sky seems to meet the ocean. Horizon appears because of earth's curvature.
A map is should be fairly accurate. It should help long distance traveller (ship, airplanes). It should also meet eduction needs. We are going to use projection method for making a map.
Imagine a 'earth globe' made out of glass. And a transparent real map is drawn on it. We also put a powerful LED lamp at the centre of the glass globe. Put a cylindrical lampshade made out of white p[aper around it. The lampshade should be just touching the sides of the globe. Now switch on the LED lamp. A map is projected on the cylinder. But north and south pole cannot be projected. Since the cylinder is open at both the ends. Now cut the cylinder and unroll it. You get a fairly accurate flat map which is real good enough for all practical purposes. This techniques is called Mecator's projection. It is still used and popular one. If we want the map of north and south pole one can use conical shades.

There are many projection methods. Nothing is perfect. There is some distortion in Mercator's method also. The regions near the pole's appear larger and the regions near the equator appears smaller. There is always some error when spherical surface is flattened. (can you flaten the skin of an orange fruit).
Even measuring distance on a spherical surface pose some difficulty. The units of distance measurement like kilometer, mile, yards -all fail on curved surface. Hence we use'nautical mile'.

Globe or sphere is a rotated circle. A circle has angle of 360 degrees. A degree is split into 60 minutes. One minute angle suspend an arc or distance on the surface of the earth. The length of the arc is 1852 meters. That is one nautical mile.
Our world is mostly curved one.
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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

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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

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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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