# Exploring Polar Curves: Petals, Limacons and More

• MHB
• Eissa12
In summary, the graphs show that the maximum magnitude of r occurs when cos(2theta) is equal to 1 or -1, resulting in theta values of 0, pi/2, pi, and 3pi/2. For a general rose curve, the number, length, and position of petals can be determined by the values of a and b in the equation r = a sin b (theta). A polar curve of the form r = d + a sin (theta) can have variations such as a cardioid or a limacon with an inner loop, and these variations occur when certain conditions are met, such as when d is equal to a or when d is greater than a, respectively.
Eissa12
View attachment 8979A) Find all values on [0,2pie) such that (thita0) produces the tip of a petal (maximum magnitude of r) all values for which r=0, and sketch a graph?

a) r = 5 sin 2 (thita0)

a) r = 5 sin 3 (thita0)

a) r = 5 sin 4 (thita0)

B) considering what you can observe in the previous graphs, what are general guidelines for number, length and position of petals for a general rose curve: r = a sin b (thita0).

c) A polar curve in the frome r = d + a sin (thita0) is called a limacon and has several distinct variations, including a cardioid and a limacon with an inner loop. Create general guidelines for when these variations occur and explain what causes them to occur?

#### Attachments

• thumbnail_image1.jpg
69.1 KB · Views: 83
Last edited:
tip of the petal $\implies$ $\cos(2\theta) = \pm 1$ ...

$0 \le \theta < 2\pi \implies 0 \le 2\pi < 4\pi$$r = 5\cos(2\theta) \implies \cos(2\theta) = \pm 1 \implies 2\theta \in \left\{ 0, \pi, 2\pi, 3\pi \right\} \implies \theta \in \left\{0, \dfrac{\pi}{2}, \pi, \dfrac{3\pi}{2} \right\}$

Last edited by a moderator:

## 1. What are polar curves?

Polar curves are mathematical representations of points on a graph using polar coordinates, where the distance from the origin and the angle from a reference line are used to plot the points.

## 2. What are some common types of polar curves?

Some common types of polar curves include cardioids, limacons, roses, and lemniscates. These curves can have different numbers of petals or lobes depending on the equation used.

## 3. How are polar curves different from Cartesian curves?

Polar curves use a different coordinate system than Cartesian curves, which use x and y coordinates. Polar curves are plotted using the distance from the origin and the angle from a reference line, while Cartesian curves are plotted using x and y coordinates.

## 4. How are polar curves used in real life?

Polar curves have many applications in real life, such as in engineering, physics, and astronomy. They can be used to model circular motion, calculate orbits, and design structures with rotational symmetry.

## 5. What are some interesting properties of polar curves?

Polar curves have several interesting properties, such as symmetry about the origin, the ability to create intricate designs with varying numbers of petals, and the ability to represent complex shapes using simple equations.

Replies
5
Views
1K
Replies
2
Views
2K
Replies
3
Views
1K
Replies
3
Views
447
Replies
4
Views
10K
Replies
3
Views
2K
Replies
2
Views
1K
Replies
9
Views
3K
Replies
3
Views
6K
Replies
3
Views
2K