Chapter 3

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                                              CHAPTER  3
                                

Curve Tracing

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A picture is worth a thousand words. A curve which is the visual image of a functional relation gives us a whole lot of infonnation about the relation. of course, we can also obtain this infonnation by analysing the equation which defines the functional relation. But studying the associated curve is often easier and quicker. In addition to this, a curve which represents a relation between two quantities also helps us to easily find the value of one quantity corresponding to a specific value of the other. Most of the function considered here for tracing are taken from +2 level textbooks of state boards of education in India.

Let us plot the following curve y =x³  

Generally, the characteristics of a curve that are of interest  are

1          1) Domain and extent of the function (value of x for which y exists and range of y values)

2          2) Symmetry about x-axis and y-axis

3          3) Asymtotes  ( as x tends to some finite number x₀ , y tends to infinity and vice versa)

4          4) Number of branches

5          5) Loops  in the curve

If function involves square root or any other higher order root we must be watchful about the range of x for which root function has real values. This is an important factor which decides the domain of the function.  

Symmetry can be easily checked analytically. If y replaced by –y in the given equation results in same expression , the curve is symmetric around x-axis and vice versa.

If expression for y is having numerator and denominator terms , we must check for asymtotes. 

Let us trace y =x³ 

Since this function rises  steeply with x, let us plot x in the range -2 to +2

Step 1: In cell A1 enter x and in B1 enter y.

Step 2: In Cell A2 enter 2, in cell A3 enter -1.5 . Select these two cells and using fill handle drag till A10. You will see a series with a common difference of 0.5.

Step 3:  In Cell B2 enter =(A2)^3  or simply =A2^3 and drag the fill handle till B10 . “^” stands for power(exponent) operation. Select Column A and B and choose scatter plot. We will obtain a curve like the one given below

 

Experiment No:2   y =x³  + 1 

Do the same step as in experiment No:1. Only change is in the formula. Note that the figure obtained is a ‘1 unit raised version’ of the previous one. 

 

Polar Co-ordinates 

In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a fixed point (origin) and an angle from a fixed direction. One of the applications of polar graphs is to describe the characteristics of microphones.

In the polar coordinate axis system, the point (r, θ) is defined as follows. Along the (horizontal) polar axis, go out r units, then rotate in a positive direction (anti-clockwise) about the pole by an amount θ. This is shown in the following diagram.

Example: The point (7, 120°) in polar coordinates is plotted below. We go 7 units out the polar axis, then rotate around the pole by 120°.

 

We normally use polar graph paper to plot such points. Here is the same point (7, 120°) plotted on a polar graph grid:

 Example 6 : Let’s draw the curve r = sin θ using polar coordinates.

If θ = 0°, then r = sin 0° = 0. So the starting point is (0, 0°). [Note: In polar coordinates, the point (0, 0°) means the distance from the origin is 0 and the angle θ is 0°.]

Let’s now choose angle θ = 30°. For this point, we need to move out r = sin 30° = 0.5 from the origin, then rotate 30°.

For a 3rd point on our graph, we choose θ = 45°. For this point, we need to move out r = sin 45° = 0.7071 from the origin, then rotate 45°.

Choosing another point θ = 80°, we need to move out r = sin 80° = 0.9848 from the origin and rotate 80°.

Here are the 4 points we have considered so far:

Θ                        0°                        30°                         45°                         80°

 r = sin θ            0                        0.5                          0.7071                  0.9848

Example: 7 polar plot .  Find polar plot of  r=1+sin( θ)

Since in Excel we can draw only x-y plot , we convert the curve into x-y coordinates as x= r *cos(theta)  and  y = r * sin(theta) . This curve is cardiod 

 

Example 8   Draw  polar plot of  r=1+sin( 4θ) 

Polar plot r with θ is equivalent to x-y plot with x=r*cos(theta) and y=r*sin(theta). Here we have x=sin(4*theta) cos(theta), y=sin(4*theta)sin(theta) 

Example 9    Spiral of Archimedes  r=2θ

 

Example 10  Equiangular Spiral    r= ae^θcotb

The equation for the equiangular spiral was developed by Rene Descartes (1596-1650) in 1638. This spiral occurs naturally in many places like sea-shells where the growth of an organism is proportional to the size of the organism.

The general polar equation for the equiangular spiral curve is

                                                a = 0.5 and b = 1.4 (radians)

Example 10   Polar plot of a face,  

Echidna

The shape of this one reminds me of the echidna (the Australian spiny anteater that rolls itself up into a ball when threatened). The function involved is: r = sin 44θ − 2 cos θ

Polar Graphs and Microphones

Different microphones have different recording patterns depending on their purpose.

a. Omani-directional Microphone: This microphone is used when we want to record sound from all directions (for example, for a choir).

The recording pattern is almost circular and would correspond to the polar curve r = sin θ that we met above.

                                             Omani-directional microphone 

The following diagram shows a real recording pattern for an Omani-directional microphone, graphed on polar graph paper. The different curves are for different frequencies, and the placement of the microphone is at the center of the circle. At low frequencies the pattern is almost circular, but at higher frequencies it becomes less so and more erratic. 

b. Cardioid Microphone:  

This is a uni-directional microphone, which means we only want to pick up sounds from in front (one direction). The recording pattern is a cardioid, which we met above. In the following image, I have used the graph of r = 1 + sin θ. (There is some “spill”, where sounds immediately behind the microphone are also detected.)

                                            Cardioid microphone

 

 

c. Shotgun Microphone: 

 This is a "super-directional" mike, where we only want to pick up sounds from directly in front of the mike. The graph used for this example is r = θ².

                                              Shotgun microphone

d. Bi-directional Microphone: 

 This is used in an interview situation, where we want to pick up the voices of the interviewer and the person being interviewd.

  

                                          Bi-directional microphone image:

So next time you see a microphone (in your mobile phone, notebook computer, in a recording studio or wherever), remember that the shape of its recording pattern is an interesting application of graphs using polar coordinates!