As The Crow Flies

by Richard Glass, PhD

Yes To Mathematics October 24^th, 2003

http://www.matcmp.ncc.edu/~glassr/y2m_2005

Polar Graph Paper

Figure 1

            There is an old expression, as the crow flies. Well, how does a crow fly? Certainly not by only going left, right, forward and backwards (moving in xy directions) but by pointing its beak in a particular direction (angle measured from some reference line) and flying a distance from where it is (the origin). We will call the direction θ, the angle as measured counterclockwise from the positive x - axis and the distance and r the distance or radius from a given location. Just as one can create rectangular coordinate graph paper, one can create polar coordinate graph paper by drawing circles to represent the distances from the center and radial lines to represent the angles (see Figure 1).

            Anyone who has seen any movie about flying or airplanes might recognize Figure 1 a radar screen (please no blipping allowed). The angle θ would be measured in compass directions (north, east, south and west) while the distance (radius) r would be measured in miles.

There are old puzzles such as:

Jane leaves her house and walks 3 miles south. She then walks 4 miles west. She now walks 5 miles north and arrives home. Where does Jane live?

or

Jane lives in a house that has a window on each wall. All windows face south. Where does Jane live?

The answer to both those questions is that Jane lives at the North Pole (she rents the basement apartment in Santas workshop).

            Another example of polar graphs can be seen from the map of the South Pole (see figure 2) Polus Antarcticus (Antarctica) taken from Novus Atlas by Jan Jansson circa 1650.

Figure 3

Polar Graphs

            Plotting on polar graph paper is just as easy as plotting in rectangular coordinates. Rather that counting off little squares (or using a ruler), one measures an angle by using a protractor (or counting the predetermined angles on the paper) and then measuring the distance from the center. For example, to plot the point A, with coordinates (r,θ) = (1,45 ^∘), one moves to the 45 degree line, then a distance 1 (assume each concentric circle is one unit from the previous one) from the center. Note, we list the r coordinate first then the θ coordinate. If a point B had coordinates (2,180 ^∘), one would move to the left horizontal, 2 units from the center. To plot a point C with coordinates (-3,45 ^∘) would move to the 45 degree line then move 3 units in the opposite direction of the angle. This is the same point as (3,225 ^∘) and (3,-135 ^∘).

Figure 4. Graph of r = θ

Examples of Polar Graphs

            The polar graph is a circle. Since it doesn’t matter what angle we are at, the points plotted would always be a distance 4 from the center.

            The graph of is a spiral, where θ is measured in radians. At an angle of 0, we are at the origin. At an angle of π/6 (30 degrees), we are 0.52 units from the center. At π/4 (45 degrees) degrees, at π/2 (90 degrees), we are 1.57 units from the origin. Figure 4 plots the graph from θ = 0 to θ=2π. This graph is known Archimedes Spiral. This spiral was studied by Conon, and later by Archimedes in On Spirals about 225 BC. If we continued to increase θ, the graph would in fact spiral outward (Figure 5).

            What would the graph of look like? Certainly not a straight line but certain similarities exist. At an angle of 0, the spiral does not start at the origin but a distance 3 units away. At the angle increases, the distance away from the origin is double the angle. The graphs

Figure 5 Graph of r = θ, 0 ≤θ≤6π

Figure 6 Graphs of r=θ and r=2θ+3

of and together can be seen in Figure 6.

Figure 7

Sine and Cosine

            The sine and cosine of an acute angle of a right triangle are defined as follows:

 and , namely, the ratios of the sides of a right triangle. By considering the unit circle centered at the origin, we can plot what the graphs of the angle versus the adjacent side (and the opposite side) to obtain the graphs of where x is the angle θ (see Figures 8 and 9).

Figure 8 y=sin(x)

Figure 9 y=cos(x)

            Rather than plot the graphs in rectangular coordinates, let us consider what the graphs of sine and cosine look like in polar coordinates.

            The graph of (remember, we are plotting on polar graph paper) can be seen in Figure 10 while the graph of
is known as the cardioid, is shown in Figure 11.

Figure 10 r = sin(θ)

Figure 11 r = 1+cos(θ)

            Although the cardioid is somewhat of a different graph, the beauty of polar graphs begin when we consider functions such as .

Figure 12 r = sin (2θ)     Figure 13 y = sin(3θ)

Figure 14 r = sin (4θ)         Figure 15 r = sin(5θ)

Can you predict what the graphs of look like? If you guessed figures 16 and 17, you would have been correct.

  Figure 16 r = sin(6θ)    Figure 17 r = sin(7θ)

            You can now conjecture that for , for n even, you will have 2n petals while if n is odd, you will have n petals.

Plotting Polar Graphs In The Safety and Privacy Of Your Own Home

            Given the right software such as Maple and a painting program, one could make some fancy custom graphics for web pages. On the other hand, your TI graphing calculator would allow you to inspect what some graphs may look like.

Let us graph the function .

            You should now see a graph (Figure 18) that looks like:

Figure 18 r=1-2sin(4θ)

To create some polar graphs, begin with a template function which combines sine and cosine such as and let the constants A, B, C, D, E and n and m vary.

In the above example, I chose A= 1, B=-2, C=4, D = 0 and n =

Figure 19 r=2sin(4θ)+3cos(3θ)

1.

In Figure 19, A=0, B=2, C=4, D=3, E=3 and n and m = 1.

See more samples at http://www.matcmp.ncc.edu/~glassr/y2m_2005/samples.htm

.