Polar equation with spread sheets by Jiyoon Chun

Why do we use spread sheets?

Spread sheets are very useful when we want to estimate the shape of curve since it gives coordinate of the points. By using this, we can predict the shape of the graph, and the order which the curves are to be drawn.

r=cosθ

I already did this part using spread sheet, but I will do it one more time to give you the basic idea of the usage of the spread sheet.

Since I did not want to deal with the long numbers, I rounded the results. The third and fifth column have rounded values.

As we see the table, five numbers are repeated: 1, 0.86, 0.7, 0.5, 0 (rounded values). I do not think it is proper to use "quadrant" since it is not cartesian system. Well, to make sense, I will just use it, but I will put the word quadrant in " ".

From 0 to 90 degree, we see the values are decreasing.

From 90 to 180, the values are negative. Therefore, the points will be plotted on " the 4th quadrant". Since the absolute values are the same, the points will be reflection of the points of the first 4.

From 180 to 270 degree, the points will be located on "the 1st quadrant" since the values are negative. Since the absolute values are the same, the points will be on the points of the first four.

From 170 to 360 degree, none of the values are negative. Therefore, they will be on "the 4th quadrant". Since the absolute values of the points from 90 to 180 are the same with the points from 270 to 360 degree, the points will be exactly on the previous four points.

Let's see the graph.

r=-cosθ

Since we know that r=ncosθ will look like the graph below with different size, I want to investigate what the drawing order or the position of the curve when n is negative. Here is the spread sheet.

As we see on the table above, it will starts at (-1,0). Therefore, the curve will be located on the left side. See the graph below.

In addition, the curve will be drawn counterclockwise because the r-coordinate is on the extended ray θ, and θ moves counterclockwise.

r=cos2θ

I love this result. Look at the spread sheet below.

Since the number turned out pretty simple, it is very easy to anticipate the shape of the graph. For convenience, I will divide the polar plane into four quadrants just like Cartitian plane

When θ is in between 0 to π/2, and r is positive, the point is on the "1st quadrant". However, if r is negative, it is on the "3rd quadrant". The table below shows where the points are according to the signs of r the range of θ.

(r,θ) |
Quadrant |
(r,θ) |
Quadrant |

(+,0<θ<π/2) |
1 |
(+,π<θ<3π/2) |
3 |

(-,0<θ<π/2) |
3 |
(-,π<θ<3π/2) |
1 |

(+,π/2<θ<π) |
2 |
(+,3π/2<θ<2π) |
4 |

(-,π/2<θ<π) |
4 |
(-,3π/2<θ<2π) |
2 |

The table below, I marked the each quadrant of the each point.

As we can observe at the last column, the curve is to drawn starting from 1 to 3 to 4 to 2 to 3 to 1to 2 to 4. The flash below will help us to understand the order.

r=-cos2θ

The shape of the graph is the same, but the order will be changed. See the table below.

The curve will be drawn from 3 to 1 to 2 to 4 to 1 to 3 to 4 to 2 in this order. See the flash below.

r=1+2cos3θ

I want to investigate all the points since I know this curve is little bit complicated.

Let's start with the angles between 0 to 200°.

With all these data, we can not only find the orders but also the sizes of the leaves and period.

1) From 0° to 40°

Since the value of r is positive, the points are on the '1st quadrant".

2) From 40° to 80°

Since the value of r is negative and 0<θ<π/2, the points are on the "3rd quadrant".

Here, we can observe the different patterns of the data. From 40° to 60°, the data decreases from 0 to -1. From 60° to 80°, the data increases from -1 to 0. In addition, the numbers appeared symmetric over 20° in the table. Thus, we can say that we will have a closed symmetric closed curve, and its period is about 40°.

Moreover, the same pattern of data appear again from 160° to 200°. Therefore, we can conclude that we will have the same curve from 160° to 200°. We know sometimes curve are overlapped and cannot figure it out whether it is two different curve or a single curve. However, the curve form 40° to 60° is not overlapped by the curve from 160° to 200° because difference of the angles are neither 180° nor 0°.

Let's see if we have the same shape of the curve. The table below is from 200° to 360°.

As we can see in the table above, we have the same shaped curve from 280° to 320°.

Let's take a look at the data from 200° to 280°. It starts from 0 and increase to 3. When the angle hits 240°, The data decrease from 3 to 0. Therefore, we can conclude that we have bigger closed curve of the period 80°. We do have two other bigger closed curve; form 80° to 160°, and 320° to 40°.

In the table below, I will summarize the shape of the curve and the location of it.

I marked the order in the graph below.