Logarithmic Spiral

The logarithmic spiral has a polar equation of \[r = a e^{b\theta}\] where \(r\) is distance from the origin, \(\theta\) is the angle from the x-axis, and \(a\) and \(b\) are constants. Source

We can write the logarithmic spiral equation in Cartesian coordinates as: \[\begin{array}{c} x &= r \cos\theta = a \cos\theta e^{b\theta}\\y &= r \sin\theta = a \sin\theta e^{b\theta}\end{array}\]

We can implement this spiral in R by first defining it in polar coordinates and then converting the polar coordinates into a sequence of Cartesian points that should be connected by line segments.

# Define the angle of the spiral (polar coords)
# go around two full times (2*pi = one revolution)
theta <- seq(0, 4*pi, .01)
# Define the distance from the origin of the spiral
# Needs to have the same length as theta
r <- seq(0, 5, length.out = length(theta))
head(cbind(r, theta))

               r theta
[1,] 0.000000000  0.00
[2,] 0.003980892  0.01
[3,] 0.007961783  0.02
[4,] 0.011942675  0.03
[5,] 0.015923567  0.04
[6,] 0.019904459  0.05

While we could probably figure out how to make R plot in polar coordinates, it’s much easier to plot in Cartesian coordinates (because that’s what most plotting packages are set up to do). So we need to convert into Cartesian (XY) coordinates before we try to plot things.

Notice how my R code is in separate chunks, and between chunks everything is annotated?

# Now define x and y in cartesian coordinates
x <- r * cos(theta)
y <- r * sin(theta)
head(cbind(x, y))

               x            y
[1,] 0.000000000 0.000000e+00
[2,] 0.003980693 3.980825e-05
[3,] 0.007960191 1.592251e-04
[4,] 0.011937301 3.582265e-04
[5,] 0.015910830 6.367728e-04
[6,] 0.019879583 9.948083e-04

With x and y defined, we can create a line plot. By default, R will create a scatterplot, so we use type=l to tell R to create a line plot instead.

plot(x, y, type = "l")

With markdown, I can create nice formatting with simple text -

First header

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