Rose Curves

(Video) Graphing Roses

by ossmteach

Watch this video by ossmteach to learn more about graphing rose curves!

Rose curves follow the forms of the equations:

$$r = a\cos(n\theta)$$ $$r = a\sin(n\theta)$$

When $n$ is odd, there are $n$ leaves.

When $n$ is even, there are $2n$ leaves.

$a$ is the distance from the center to the end of the leaf.

For example, with the equation $r = a\cos{3\theta}$, we will see three leaves on the rose curve.

With the equation $r = a\cos{4\theta}$, we expect to see 8 leaves.

To graph a rose curve, the main strategy is to plug in easy $\theta$ values, then solve for r and plot the points.

Let's work through the example in the video together!

$$r = 2\sin(2\theta)$$

From the equation, we know there are 4 leaves and the distance from the end of the leaf to the center is 2.

When $\theta = 0$, $r=0$. When $\theta = \frac{\pi}{4}$, $r=1$. Keep trying different $\theta$ values and then you can plug these numbers in on the graph and connect them to develop the rose. The unit circle trigonometric values will be very useful.

We've learned that we can use the polar equation to graph circles, limaçons, and cardioids. We can also use polar equations to graph roses!

Rose Curves

Rose curves sound exciting! You may be wondering if they actually look like roses. Well, you be the judge for yourself!

This looks like a flower! That's cool. We can make this flower more complex.

Rose curves, also known as rhodonea curves, is a type of sinusoid that is graphed with polar coordinates. The typical equation used to graph rose curves is: $$r=a \cos (k\theta)$$ , where $r$ is the radial coordinate, $k$ is the angular frequency, $a$ is the amplitude, and $\theta$ is the polar angle. $k$ will tell you the number of "petals" in the rose curve. If $k$ is even, the rose curve will have $k$ petals. If $k$ is odd, the rose curve will have $2k$ petals.

If you ever encounter a situation where the coordinates aren't in the form you need, don't sweat! Remember that you know how to [convert from polar to rectangular and from rectangular to polar!

A rose curve will always be symmetrical about the origin and has a period of $2\pi$.

We can also express a rose curve using sine, which is the cosine curve but rotated counterclockwise by $\frac{\pi}{2k}$ radians.

Here is the graph of the following equation: $r=3\cos(1.5\theta)$

Let's say we want to increase the angular frequency to $4$, yielding the equation: $r=3\cos(4\theta)$. Our graph looks like:

As you can see, the radius and amplitude stayed the same, but the number of petals have increased!

In the next lesson, you'll learn about lemniscates!

Rose Curves

A rose curve is a graph that looks like a flower with petals. You can graph rose curves using a polar equation, like so:

$$r=a\cos(k\theta)$$ $$r=a\sin(k\theta)$$

Cartesian coordinates will not work when graphing rose curves.

In the equations, $a$ will tell you the length of each petal and $k$ will tell you the number of pteals.

If $k$ is odd, the rose will have $k$ petals. If $k$ is even, the rose will have $2k$ petals.

For example, $r=3\cos(2\theta)$ makes a rose with $4$ petals, each $3$ units long.

The petals are spaced evenly around the origin and always pass through it. Rose curves are symmetric and look like real flowers when drown on polar graph paper. These curves are called "rose curves" because of how they look, but they follow simple math rules.