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Polar Curves

- Calculus
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Have you ever looked closely at a sunflower? The curves you see crisscrossing each other in a sunflower are actually arranged in logarithmic spirals, a type of polar curve.^{1}

Other natural phenomena with shapes that can be described using logarithmic spirals include nautilus shells and even some galaxies.

This article will introduce polar curves, including important examples of polar curves, some of their symmetries, and how to graph them.

You might be used to write functions in the form

\[ y=f(x),\]

where the relation between \( x \) and \( y \) leads to a curve in the plane. When a function is written in terms of \( x \) and \( y \), we say that it is written in** cartesian coordinates, **also known as **rectangular coordinates.**

On the other hand, polar curves are curves written in terms of **polar coordinates** \( r \) and \( \theta.\) To better describe polar coordinates, consider a point on the plane.

Next, draw a segment that goes from the origin to that point.

Polar coordinates describe the position of a point on the plane in terms of \( r \) and \( \theta,\) where

\[ r = \sqrt{x^2+y^2},\]

is the distance from the origin to a point on the plane (the length of the segment shown in the diagram), and

\[ \theta = \arctan{\frac{y}{x}},\]

is the angle between the positive \(x-\)axis and the line that connects the origin to the point.

Another way of describing polar coordinates is through the equations,

\[ x = r\,\cos{\theta}\]

and

\[ y = r\,\sin{\theta}.\]

A **polar curve** is a function described in terms of polar coordinates, which can be expressed generally as

\[ f(r,\theta).\]

Please note that the functions described by polar coordinates will usually **not pass** the vertical line test. The vertical line test only applies to functions that are written as \( y=f(x)\)!

The equation

\[ r = 3\sin{\left(\frac{2}{3}\theta\right)}\]

defines a polar curve. Its graph looks like this,

You can see the above curve as the set of all pairs \( (r,\theta)\) that satisfy the equation

\[ r = 3\sin{\left(\frac{2}{3}\theta\right)}.\]

For more information on polar coordinates, what they are, and how to convert between polar and rectangular coordinates, see our articles about Polar Coordinates and Deriving Functions Written in Polar Coordinates.

Some types of curves can be expressed more naturally in polar coordinates than in rectangular coordinates. Six important classes of such curves are,

- Limaçons
- Cardioids
- Rose curves
- Archimedean spirals
- Logarithmic spirals
- Lemniscates

A limaçon is a polar curve defined by the equation

\[ r = a \pm b \sin{\theta}\]

or

\[ r = a \pm b \cos{\theta},\]

where \( a \) and \( b \) are constants. There are three types of limaçons,

- Looped (limaçons where \( a < b \)),
- Cardioids (limaçons where \( a = b \)),
- Dimpled (limaçons where \( a > b\)).

For example, the limaçon defined by

\[ r = 2 + 3 \sin{\theta},\]

is a looped limaçon because \( 2 < 3. \) Here is its graph.

As another example, the curve

\[ r = 4 - \cos{\theta}\]

is a dimpled limaçon because \( 4 > 1. \) It looks like this,

Figure 5. A dimpled limaçon

Note that limaçons defined using cosine are symmetric about the horizontal axis, while limaçons defined using sine are symmetric about the vertical axis.

Limaçon is for sure a weird name, isn't it? There is a reason for why this name is used!

The name limaçon, translated from French, literally means 'snail.' This name comes from certain types of limaçons that look like snail shells.

Cardioids are special instances of limaçons named for their heart-like shape. The equation for a cardioid is

\[ r = a(1-\cos{\theta}),\]

where \( a \) is some constant. For example, the cardioid

\[ 1- \cos{\theta}\] looks like this,

Rose curves are polar curves named by Guido Grandi, an Italian mathematician who studied them in the early 1700s. They are defined by equations of the form

\[ r = a\sin{\left(n\theta\right)},\]

or

\[ r = a\cos{\left(n\theta\right)},\]

where \( a \) is a constant that determines the size of the flower and \( n \) is a constant that determines the number and placement of the petals. For example, the rose curve

\[ r = \cos{\left( 9\theta \right)},\]

looks like this,^{3}

Meanwhile, the rose curve

\[ r = 3 \sin{\left( 9 \theta \right) } ,\]

looks like this,

Both roses have the same number of petals. This is because \( n \) is set to 9 in both equations. However, one rose is scaled by a factor of 3, corresponding to the fact that it has \( a=3.\) You can also note that one rose is rotated with respect to the other, which corresponds to the fact that one is defined in terms of sine function, while the other is written using the cosine function.

The value \(n\) does not need to be an integer. For example, the rose curve

\[ r = 3 \cos{\left(\frac{4}{7}\theta \right)},\]

looks like this,

Rose curves with irrational values for \(n\) (like \(\pi\), for example) can be particularly interesting because they never actually complete. Given an irrational rose curve \(f\) and an arbitrary point \(P\) in the disc containing \(f\), no matter how close you get to \(P\), there is a point of \(f\) closer to \(P\) than you are. \(f\) is an example of a **dense set**; another example is the set of rational numbers on the real line.

An Archimedean spiral is a polar curve defined by the equation

\[r = b\theta.\]

There is a generalized version of the Archimedean spiral, called a neoid, defined by the equation

\[ r = a + b\theta, \]

where \( a \) and \( b \) are constants. The constant \( a \) determines the value of the curve at \( \theta = 0 \) (the positive \( x-\)axis), and \( b \) corresponds to the size of the spiral.

Archimidean spirals have another particularity, as there is a constant separation distance between consecutive segments of the spiral which is equal to \(2\pi b\). For example, the neoid

\[ r = 1 + 3\theta,\]

looks like this,^{4}

Archimedean spirals are polar curves named after the ancient Greek mathematician Archimedes. Archimedes wrote an entire book, *On Spirals*, about these curves and their applications.

Another important type of spiral is the logarithmic spiral. Logarithmic spirals are polar curves defined by the equation

\[ r = a e^{b\theta},\]

where \( a \) and \(b \) are constants. Logarithmic spirals receive their name from the fact that you can isolate \( \theta \) and write it in terms of the natural logarithm of \(r.\)

For example, the logarithmic spiral

\[ r = 2e^{\frac{1}{3}\theta},\]

looks like this,

The logarithmic spiral is also called spira mirabilis, which means “miraculous spiral” in Latin.

Cassini ovals are polar curves discovered in 1680 by Giovanni Domenico Cassini. They are defined by the equation

\[ r^4 = 2a^2r^2\cos{\left(2\theta \right)} + b^4 - a^4,\]

where \( a \) and \( b \) are constants. The ratio of \(a\) to \(b\) determines the shape of the Cassini oval as follows,

- If \(\frac{b}{a} > 1\), the oval has 1 loop,
- If \(\frac{b}{a} < 1\), the oval has 2 separate loops,
- If \(\frac{b}{a} = 1\), results in a lemniscate.

For example, the Cassini oval

\[ r^4 = 2(2)^2r^2\cos{\left(2\theta\right)}+(3)^4-(2)^4,\]

looks like this,^{4}

Lemniscates are special types of Cassini ovals that look like figure eights. Lemniscates are defined by the equation

\[ r^2 = a^2\sin{\left( 2\theta\right)},\]

or

\[ r^2 = a^2\cos{\left( 2\theta\right)},\]where \( a \) is a constant. These curves are obtained by setting \( a= b\) in the equation for a Cassini oval. For example, the lemniscate

\[ r^2 = 4\cos{\left( 2 \theta \right)},\]

looks like this,

Lemniscates defined in terms of sine are identical to those defined in terms of cosine, just rotated by 45 degrees.

Polar curves turn up in some interesting and perhaps unexpected places. For example, you may have heard of the Mandelbrot set. The Mandelbrot set is an example of a fractal, a shape that contains copies of itself on different scales.

The Mandelbrot set contains several of the curves we have talked about. For example, the main bulb of the set is in the shape of a cardioid.^{5}

Zooming in to the edge of the set, you can see logarithmic spirals.^{5}

The boundary of the Mandelbrot set, while it is incredibly complex, can actually be constructed using a sequence of lemniscates.

Polar curves often have symmetries that you can take advantage of while graphing them and studying their properties. Commonly, there are three symmetries involved when dealing with polar curves,

**Symmetry about the polar axis,****Symmetry about the pole,****S****ymmetry about the vertical axis.**

The table below summarizes the different symmetries and how to look for them.^{6}

Name | Axis of Symmetry | Test |

Symmetry about the pole | $$\theta = \frac{3\pi}{4}$$ | Replace \(r\) with \(-r\) or (equivalently) replace \(\theta\) with \(\pi + \theta\). |

Symmetry about the polar axis | $$\theta = \pi$$ | Replace \(r\) with \(-r\) and \(\theta\) with \(\pi - \theta\), or (equivalently) replace \(\theta\) with \(-\theta\). |

Symmetry about the vertical axis | $$\theta = \frac{\pi}{2}$$ | Replace \(r\) with \(-r\) and \(\theta\) with \(-\theta\), or (equivalently) replace \(\theta\) with \(\pi - \theta\). |

Symmetry about the polar axis is the same as symmetry about the line \( \theta = \pi,\) the horizontal axis. A polar curve is symmetric about the polar axis if you can flip the graph about the horizontal axis and get back the same graph you started with. Equivalently, this means that if the point \( (r,\theta)\) is on the polar curve, then the point \( (r,-\theta) \) obtained by flipping \( (r,\theta) \) around the horizontal axis is also on the curve.

There are a couple of tests you can use to check if a curve is symmetric about the polar axis. If you are given an equation for a polar curve and can replace every instance of \( \theta \) with \( -\theta, \) and get back the same equation, then the curve is symmetric about the polar axis. This is the same as checking whether, given a point \( (r,\theta) \) on the curve, the point \( (r,-\theta) \) is also on the polar curve.

Another test you can try is replacing \( r \) with \( -r \) and \( \theta \) with \( \pi - \theta.\) This test works because the point \( (-r,\pi-\theta) \) is the same as the point \( (r,-\theta),\) just represented slightly differently.

Check the rose curve

\[ r = 3 \cos{\left(2\theta\right)},\]

for symmetry about the polar axis.

**Solution**

Begin by \( \theta \) by \( -\theta.\) This will give you

\[ \begin{align} r &= 3\cos{\left(2(-\theta)\right)} \\ &= 3\cos{(-2\theta)}. \end{align}\]

The cosine function has the property

\[\cos{(-\theta)}=\cos{\theta}\]

for any value of \( \theta,\) so

\[ \begin{align} r &= 3 \cos{(-2\theta)} \\ &= 3\cos{(2\theta)}. \end{align}\]

Since this is the same equation as the one you started with, this test succeeds. So, this rose curve is symmetric about the polar axis.

If a polar curve passes a symmetry test, then it definitely has that symmetry. However, even if a polar curves fails all the symmetry tests you try, it may still have that symmetry. This comes from the fact that polar curves have many equivalent algebraic representations, and you may not always be able to tell just by looking at two equations whether they describe the same curve.

If a polar curve is symmetric about the pole, then flipping it about the line

\[ \theta = \frac{3\pi}{4}\]

does not change its graph. Equivalently, a polar curve is symmetric about the pole if, for any point \( (r,\theta)\) on the curve, the point \( (-r,\theta) \) is also on the curve. To test for symmetry about the pole, you can replace \( r \) with \( -r \) and see if you get back the same equation. You can also try replacing \( \theta \) by \( \theta + \pi.\)

Test the lemniscate

\[ r^2 = 4\sin{\left( 2\theta \right)},\]

for symmetry about the pole.**Solution**

First, try replacing \( r \) with \( -r, \) that is

\[ \begin{align} (-r)^2=4\sin{(2\theta)} \\ r^2 = 4\sin{(2\theta)}. \end{align}\]

Since the equation you obtained is the same as the original, this lemniscate is symmetric about the pole. You can also try the other symmetry test and replace \( \theta \) by \( \theta+\pi, \) that is

\[ \begin{align} r^2 &= 4\sin{\left(2(\theta+\pi) \right)} \\ &= 4\sin{(2\theta+2\pi)}. \end{align}\]

The sine function has a period of \(2\pi,\) which means that the identity

\[ \sin{(\theta+2\pi)}\]

holds true for any value of \(\theta,\) so

\[ r^2 = 4 \sin{(2\theta)}.\]

Once again the equation is the same as the original, so the curve has symmetry about the pole.

Finally, symmetry about the vertical axis is the same as symmetry about the line

\[ \theta = \frac{\pi}{2}.\]

To test for symmetry about the vertical axis, try replacing \( r \) with \( -r \) and \( \theta \) with \( -\theta.\) You can also try replacing \( \theta \) with \( \pi-\theta.\)

Test the limaçon

\[ r = 2+3\sin{\theta},\]

for symmetry about the vertical axis.

**Solution**

Begin by replacing \( r \) with \( -r \) and \( \theta \) with \( -\theta,\) that is

\[ -r = 2 + 3 \sin{(-\theta)},\]

where you can use the fact that the sine function is an odd function, which means that

\[ \sin{(-\theta)}=-\sin{\theta}\]

for any value of \( \theta,\) so

\[ \begin{align} -r &= 2 -3\sin{\theta} \\r &= -2+3\sin{\theta}. \end{align}\]

Since the above equation is not the same as the original, this test does not tell you whether the curve is symmetric about the vertical axis. However, if you try replacing \( \theta \) with \( \pi-\theta,\) you will get

\[ r = 2 +3 \sin{(\pi-\theta)},\]

and now you can use the identity

\[ \sin{(\pi-\theta)} = \sin{\theta},\]

so

\[ r = 2 + 3 \sin{\theta}.\]

The above equation is equivalent to the original equation, proving that this limaçon is symmetric about the vertical axis.

You have seen a wide variety of polar curves and their graphs. Let's now suppose you are given a formula and wanted to graph the polar curve.

One strategy is to learn the formulas for important examples of polar curves and understand their corresponding graphs. Many of the curves you may be asked to work with are variants of the polar curves discussed above, so knowing these curves, their equations, and their properties can be quite helpful. However, odds are that not every polar curve you run into will have a formula you recognize. Not to mention the task of memorizing all shapes and formulas!

Here you can look at some alternatives for graphing polar curves.

There are several strategies you can use when graphing polar curves manually. In these cases, one of the first things you can do is check for periodicity. If the function you are graphing is periodic, then you need only graph the function over one period. Then, find points on the curve for various values of \( \theta.\) Once you have graphed these values, connect the dots to approximate the curve.

You can also check for symmetries. This can immensely reduce the amount of work you need to do. For example, if you know that a polar curve is symmetric about the vertical axis, you must only draw the curve in one half-plane then reflect it across the axis to get the other half.

Graph the polar curve described by

\[ r = 1 + 2\cos{\theta}.\]

**Solution**

Begin by checking for symmetries. First, check for symmetry about the polar axis by replacing \( \theta \) with \( -\theta,\) so

\[ r = 1 + 2\cos{(-\theta)},\]

since the cosine function is an even function, this means that

\[ \cos{(-\theta)} = \cos{\theta},\]

so

\[ r = 1 + 2\cos{\theta}.\]

Since this equation is the same as the original, the curve is symmetric about the polar axis. Before checking for more symmetries, try plugging in a few nice values of \( \theta \) into the equation.

\[ \theta \] | \[ 1 + 2\cos{\theta} \] | \[ r \] |

\[ 0 \] | \[ 1 + 2 \cos{0} \] | \[ 3 \] |

\[ \frac{\pi}{4}\] | \[ 1 + 2 \cos{\frac{\pi}{4}}\] | \[ 1 + \sqrt{2}\] |

\[ \frac{\pi}{2}\] | \[ 1 + 2 \cos{\frac{\pi}{2}}\] | \[ 1 \] |

\[ \frac{3\pi}{2}\] | \[ 1 + 2\cos{\frac{3\pi}{2}}\] | \[ 1 - \sqrt{2}\] |

\[ \pi \] | \[ 1 + 2 \cos{\pi}\] | \[-1\] |

To graph negative values of \(r,\) just go to the opposite side of the \( \theta\) angle you are using. Plotting these points will give you.

Next, connect the dots. You should keep in mind that you are graphing a polar curve, so try connecting the dots in a curvy fission!

You found previously that the graph has polar symmetry, so reflect the graph along with what would be the \(x-\)axis.

There are many freely available online tools like Geogebra and Desmos for graphing polar curves. To graph polar curves in either of these tools, simply type the equation for the curve into the input field.

For example, to graph a rose with three leaves in Geogebra or Desmos using polar coordinates, type

\[ r = \cos{(3\theta)}\]

and press enter.

To type the \( \theta \) symbol in Geogebra, use the math keyboard, which can be opened by clicking the keyboard symbol on the lower left-hand side of the input field. To type the \( \theta\) symbol in Desmos, type the word 'theta'.

There are, of course, many other ways to plot polar curves. If you have a graphing calculator, it likely has a function that lets you plot functions in polar coordinates. You can also use programming languages and software like Python, Matlab, and Octave.

If you are having trouble graphing a polar curve, check the range of theta values that you plotted. Occasionally, you need to expand the range of theta values you're using to make sure that the curve is plotted correctly.

Suppose you have a polar curve defined by

\[ r = f(\theta). \]

To find the area bound between the line \( \theta = \theta_1,\) the curve \( f(\theta),\) and the line \( \theta=\theta_2, \) you need to use the formula

\[ A = \int_{\theta_1}^{\theta_2} \frac{1}{2}\left[ f(\theta) \right] ^2 \, \mathrm{d}\theta.\]

For more information about this topic, take a peek at our article on the Area of Regions Bounded by Polar Curves.

If you instead need to find the length of the polar curve \( f(\theta),\) drawn between the angles \( \theta_1 \) and \( \theta_2,\) you will need to use the formula

\[ L = \int_{\theta_1}^{\theta_2} \sqrt{r^2+\left( \frac{\mathrm{d}r}{\mathrm{d}\theta} \right)^2}\,\mathrm{d}\theta.\]

Need more information on how to deal with this formula? See our article about the Arc Length in Polar Coordinates!

- Polar curves are defined by relations in terms of polar coordinates.
- Important types of polar curves include limaçons, cardioids, rose curves, Archimedean spirals, logarithmic spirals, Cassini ovals, and lemniscates.
- Three types of symmetry a polar curve can have are symmetry about the polar axis, symmetry about the pole, and symmetry about the vertical axis.
- To graph polar curves, know common types of polar curves and their properties, check for periodicity, check for symmetry, then find several points on the curve and connect them.

1. Eli Maor, *“e”: The Story of a Number*, 1994.

2. John W. Rutter, *Geometry of Curves*, 2000.

3. Guido Grandi, *Flores geometrici ex Rhodonearum, et Cloeliarum curvarum descriptione resultantes, …*,* *1728*.*

4. Ross L. Finney, Franklin D. Demana, Bert K. Waits, Daniel Kennedy, and David M. Bressoud, *Calculus: Graphical, Numerical, Algebraic*, 2016.

5. Benoit Mandelbrot, *Fractals and Chaos: Fractals and Beyond*, 2004.

6. James Stewart, *Calculus, Seventh Edition*, 2012.* *

A polar curve is a curve defined by a relation in terms of polar coordinates r and theta.

More about Polar Curves

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