The final polar equation we will discuss is the Archimedes’ spiral, named for its discoverer, the Greek mathematician Archimedes (c. 287 BCE – c. 212 BCE), who is credited with numerous discoveries in the fields of geometry and mechanics.
Archimedes’ spiral
The formula that generates the graph of the Archimedes’ spiral is given by [latex]r=\theta[/latex] for [latex]\theta \ge 0[/latex]. As [latex]\theta[/latex] increases, [latex]r[/latex] increases at a constant rate in an ever-widening, never-ending, spiraling path.
How To: Given an Archimedes’ spiral over [latex]\left[0,2\pi \right][/latex], sketch the graph.
Make a table of values for [latex]r[/latex] and [latex]\theta[/latex] over the given domain.
Plot the points and sketch the graph.
Sketch the graph of [latex]r=\theta[/latex] over [latex]\left[0,2\pi \right][/latex].
As [latex]r[/latex] is equal to [latex]\theta[/latex], the plot of the Archimedes’ spiral begins at the pole at the point (0, 0). While the graph hints of symmetry, there is no formal symmetry with regard to passing the symmetry tests. Further, there is no maximum value, unless the domain is restricted.Create a table such as the one below.
[latex]\theta[/latex]
[latex]\frac{\pi }{4}[/latex]
[latex]\frac{\pi }{2}[/latex]
[latex]\pi[/latex]
[latex]\frac{3\pi }{2}[/latex]
[latex]\frac{7\pi }{4}[/latex]
[latex]2\pi[/latex]
[latex]r[/latex]
0.785
1.57
3.14
4.71
5.50
6.28
Notice that the r-values are just the decimal form of the angle measured in radians.
Analysis of the Solution
The domain of this polar curve is [latex]\left[0,2\pi \right][/latex]. In general, however, the domain of this function is [latex]\left(-\infty ,\infty \right)[/latex]. Graphing the equation of the Archimedes’ spiral is rather simple, although the image makes it seem like it would be complex.
Sketch the graph of [latex]r=-\theta[/latex] over the interval [latex]\left[0,4\pi \right][/latex].
![Two graphs side by side of Archimedes' spiral. (A) is r= theta, [0, 2pi]. (B) is r=theta, [0, 4pi]. Both start at origin and spiral out counterclockwise. The second has two spirals out while the first has one.](https://s3-us-west-2.amazonaws.com/courses-images/wp-content/uploads/sites/3675/2018/09/27165546/CNX_Precalc_Figure_08_04_020new.jpg)
![Graph of Archimedes' spiral r=theta over [0,2pi]. Starts at origin and spirals out in one loop counterclockwise. Points (pi/4, pi/4), (pi/2,pi/2), (pi,pi), (5pi/4, 5pi/4), (7pi/4, pi/4), and (2pi, 2pi) are marked.](https://s3-us-west-2.amazonaws.com/courses-images/wp-content/uploads/sites/3675/2018/09/27165549/CNX_Precalc_Figure_08_04_021F.jpg)
