Why do lines with slope further away from 0 bunch together?

Question

Here's an image taken from the article: Frequentism and Bayesianism IV: How to be a Bayesian in Python.

Since I can't add images, here's the link:

Image

It depicts lines generated with slopes between 0 and 10 in steps of 0.1

What accounts for the bunching of the line with higher slopes?

Background: In the article, the author provides this as a pedagogical example to not automatically consider flat priors.

Thank you.

Answer 1

I'll use $\alpha_n$ to mean the angle that the $n$-th line makes with the $x$-axis. The fact that the slopes are equally spaced from $0$ to $10$ means that the tangents of the angles are equally spaced, because the slope of the $n$-th line is $\tan\alpha_n$. But equally spaced tangents doesn't mean equally spaced angles. If you look at the graph of the tangent function, $x\mapsto\tan x$, you'll see that it gets extremely steep when $x$ is very slightly less than $\pi/2$. This means that, when the angles are just slightly below a right angle (i.e., when the lines are almost vertical), a very small change in angle can make a huge difference to the tangent of that angle. Equivalently, ordinary-sized (like $1$) changes in the tangent come from really tiny changes in the angle. That's what you're seeing in the picture. For large $n$, the difference of tangents, $\tan\alpha_{n+1}-\tan\alpha_n$, is simply $1$ but the corresponding difference of angles $\alpha_{n+1}-\alpha_n$ is tiny. So the lines look bunched together.

Answer 2

$\text{Slope}=\frac {\text {rise}} {\text {run}}=\frac{\Delta y}{\Delta x}$. Thus if you fix $\Delta x$ at $1$, $\Delta y$ will be the slope. Since you're increasing the slope by a constant amount, the $\Delta y$ also increases by a constant amount for $\Delta x=1$, which is why the lines which reach the right edge of your image are evenly spaced along that edge. If $\Delta y$ is fixed at 1, however, $\Delta x$ decreases as $\frac 1 {\text{slope}}$. Since $\frac 1 {a+1} - \frac 1 a$ gets small as $a$ gets large, the lines start to bunch as the slope increases.

Answer 3

They don't, in some sense. If you draw a vertical line through, say, $x = 1$ (the right edge of your picture; imagine extending the picture upwards until all lines are included all the way), and look at where your lines pass through it, they points are actually exactly evenly spaced.

However, they look more tightly spaced for two reasons. One: the steep lines disappear out of the picture before we can see the lines really separate. Two, even though the vertical spaces between the lines are equal, that's not what our human brain looks for when interpreting the distance between two lines. Our brains tend to look at the shortest distance between the lines, or the angle between them, and that does indeed shrink.