When Pi doesn’t equal 3.14…
And Other Properties of Circles in the Lᵖ Norm
What is a Circle?
A circle is simply the set of points that are an equal distance, equidistant, from a certain central point:
If we set the origin (0,0) as our center point, we can find the distance between it and another point on a flat plane via:
Where x and y are the coordinates of the point, and d is the point’s distance from the origin.
This idea of distance extends to any number of dimensions (circles →spheres →4D hyperspheres → …) and we can accommodate for it by simply adding another squared term for every new dimension:
In the above definition we have some point, or vector, called x. The vector’s length, or magnitude, is denoted ‖x‖. The components of the vector are denoted x₁ to xₙ where n is the vector’s dimensionality.
Taxicab Metric
But there’s a big assumption in our definition of the circle: distance. While we usually use what’s called the Euclidean norm (the formula used above), there are other valid notions of distance we could have used instead.
Take, for example, the taxicab metric:
Notice the little 1 underneath the magnitude of x on the left side. This is because this isn’t the standard definition of distance. This new notation will be explained further down below.
The equation above describes distance the way you’d encounter it in a big city, like Manhattan (in fact this metric is also referred to as Manhattan distance). You can’t move through buildings, so you move across avenues (x axis) and streets (y axis):
The green line represents the standard Euclidean distance, while the blue, red, and yellow lines all represent the taxicab distance.
Notice that there are multiple paths that give the same minimum distance in the taxicab metric (yellow, red and blue all have the same length), while the Euclidean metric has only one shortest path.
Lᵖ Metric
As it turns out, the taxicab metric and the Euclidean metric can both be generalized into a new type of metric, the Lᵖ-norm:
Plugging a number in for p (where p≥1) would give us a new formula for calculating distance. The magnitude of a vector in the L²-norm (p=2), for example, would look like this:
Well would you look at that! The Euclidean metric is the same thing as the L² norm. (Note that since the euclidean norm is the most common notion of distance, we often omit the 2 when using it.)
The same can be done with the taxicab, or L¹, metric:
Circles with the Lᵖ-norm
Going back to our definition of a circle, if we graph all the points that are equidistant from each other using these new metrics (in 2D), we get some interesting results:
L∞ Norm?
As shown in the graph above, as p grows larger, the unit circle created by its corresponding Lᵖ-norm looks more and more like a square. Formally speaking, the L∞-norm is the limit as p approaches infinity of the Lᵖ-norm:
What is the limit of the Lᵖ-norm as p approaches infinity? Well actually, it turns out to be a very simple expression:
To find the L∞ distance between a point and the origin, simply pick the component with the largest absolute value (the max function just returns the highest input it’s given).
It seems strange how such an expression could fall out of our previous definition of Lᵖ-norms, but if we take a look at the L∞ unit circle we find that this max distance definition makes a lot of sense. All points on the square are equidistant because they all share at least one component with a value of 1.
Concavity & Fractional Lᵖ
As we graph circles of higher and higher Lᵖ-norms, we find that they appear more and more convex, approaching a square.
What’s also interesting, but should make some intuitive sense, is that if we graph circles of fractional Lᵖ-“norms” (0<p<1), we get shapes that are more and more concave:
I put “norms” in quotation marks because the formal definition of a norm is only satisfied by the Lᵖ-norm when p≥1.
π ≠ π?
We usually define π as the ratio of a circle’s circumference to its diameter but, this only applies in euclidean space (i.e the L²-norm). When it comes to Lᵖ circles in general, the ratio between the circle’s circumference and diameter doesn’t have to be π.
Take p=1 for example. It’s circumference is just 4 times the length of its sides. So in this case the circle constant is just 4s/s which equals 4. The same is true for p=∞. The circumference is still 4s and so the circle constant is again 4.
I used 1 and infinity as examples because they could be done with simple math. You can still find the circumference of the other Lᵖ unit circles, you just need to know a bit of integral calculus.
Above I’ve linked a demo that calculates the circle constant for different values of p in the Lᵖ-norm.
Why 2?
Seeing all these different definitions of distance makes one wonder, why is it that we use the L²-norm? Well it’s quite simply because nature, that is the real world, uses it. That’s just how distance works. While this is an unsatisfactory answer we can still examine some special characteristics of the L²-norm that make it stand out from the rest.
Rotational Invariance
The euclidean circle we are all familiar with is symmetrical on all sides, that is to say it has total rotational symmetry. The circles of the other norms do NOT have this total symmetry. Our definition of rotation is to move points around in a circle but, since the definition of a circle has changed in these norms, these rotations change the shape of the thing being rotated. Below we can see what it looks like to rotate a square in the L¹ norm:
Notice that, for the L¹ norm, the only true symmetries the square can have are those around 90°, 180°, 270°, and 360°. Unlike the Euclidean norm which can have symmetries around any angle.
Minimum Circle Constant
Another interesting point to note is that when we graph all the possible circle constants of different Lᵖ spaces, we find that the circle constant at L² (3.14….) is actually the lowest value of all (that is, it’s the global minimum of the function):
I’ve also linked an interactive graph of the circle constants in the Lᵖ norms above.
