THE MATH OF STYLE WRITING

We have so many style of fonts in any language. They are easily scalable. Their file size is very small in kilo bytes. How is it possible? Let us explore the interesting idea that create fonts.
We know how to plot a point in a graph and how to draw straight line. Two points are enough to make a st.line. Every line has a slope m. Slope is found by dividing the ,rise, of the line by 'run' of the line. The 'rise' is change in Y-coordinate between two points. The 'run'is change in X-coordinate.
Let the point be X1, Y1
Than the equation of a line is Y-Y1 =m(X-X1)

our case:
Slope of the line given in the figure is m = rise/run = 8 - 3/6-1 = 5/5=1
using slope and one point we can write the equation of a line

slope = m=1
point = (X1,Y1) = (6,8)
so Y-8 = 1(X-6)
Y-8 = X-6
Y=X-6+8
Y=X+2. This is the equation of the line. The short segment of line can be written as
Y=X+2, from X=1, to 6
Hence, to draw line, we do not need hundreds of points, just this one line is enough.

Now let us take the font of English letter 'V' and draw it on a graph sheet.
We get two inclined lines. Find their slopes and write their equations in point-slope form.
Line I:
slope m = 5-1/1-3
m = 4/-2= -2 (negative slope)
The equation of the line considering the point (1,5)
y-5 = -2(x-1)
y-5 = -2x+2
y = -2x+2+5
y=-2x+7
The equation of the line segment is y=-2x+7 from x=1 to 3

Similarly equation of the line II is
y=2x-5 from x=3 to 5

Just these 2 equations compactly represent the font 'V'. Suppose you double the coordinates of a point. The point (1,5) becomes (2,10). The font size increases Hence you can easily scale the fonts.
But the fonts are not made of straight lines but also by curves. But curves can be represented by polynomial equations.

Example: The equation 3x^2+2x+1 represents a parabola. 4x^3+3x^2 +2x+2 represents another curve. Using these kind of equations, we can construct fonts made of curves.
We can say, this is the simplest use of (analytical geometry).
Math is everywhere.
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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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