# An algorithm for creating a circle on a discrete plane and a limit for $\pi$

I know there is a well known algorithm which uses the circle equation to approximate it with pixels.

However, I wanted to approach this problem from the most basic principles.

So we start with a square with the side $a_0$ (all lengths are measured in pixels).

To make it more 'circular' we add several pixel rows on each side. The number of rows $k$ is defined to make the distance between the opposite sides as close as possible to the length of the diagonal.

$$(a_0+2k)^2=2a_0^2$$

$$\left[\frac{\sqrt{2}-1}{2} a_0 \right] \leq k \leq \left[\frac{\sqrt{2}-1}{2} a_0 \right]+1$$

Here $[]$ is the floor function.

Now we need to find the length of each row. Let's consider a rectangle formed by two opposite rows. Its diagonal should be as close as possible to the diagonal of the initial square.

$$a_n^2+(a_0+2n)^2=2a_0^2$$

$$\left[ \sqrt{a_0^2-4na_0-4n^2} \right] \leq a_n \leq \left[ \sqrt{a_0^2-4na_0-4n^2} \right]+1$$

Now if we use the 'exact' value of $k$, we get:

$$k=\frac{\sqrt{2}-1}{2} a_0~~~~\rightarrow~~a_k=0$$

Which means that the correct value for $k$ is:

$$k=\left[\frac{\sqrt{2}-1}{2} a_0 \right]$$

Value of $a_n$ is defined from symmetry considerations:

If $a_0$ is even, all $a_n$ should be even.

If $a_0$ is odd, all $a_n$ should be odd.

If we count the number of pixels in our circle (its discrete area), we get:

$$S_k=a_0^2+4 \sum_{n=1}^k a_n$$

$$\pi=\lim_{k \to \infty} \frac{2 S_k}{a_0^2}$$

And here are two examples:

$$a_0=100~~~~~~~~~k=20~~~~~~~~~S=15496~~~~~~~~~\pi = 3.0992$$

$$a_0=10000~~~~~~~~~k=2071~~~~~~~~~S=157059544~~~~~~~~~\pi = 3.14119$$

This sequence converges to $\pi$ from below. The convergence is stochastic and very slow. See the plot below for first $1000$ terms:

Is this algorithm correct (does it create a 'best' circle approximation for a given $a_0$)? Is there some other way to approximate $\pi$ using this algorithm?

If we just want to compute $\pi$ (without creating the circle) we can simplify the algorithm and get the following:

$$\pi=2+4 \lim_{p \to \infty} \frac{1}{p} \sum_{n=1}^k \sqrt{1-\frac{2n}{p}-\frac{n^2}{p^2}}$$

$$k=\left[(\sqrt{2}-1) p \right]$$

Have you seen this limit before? Maybe it's related to some known formula for $\pi$?

• The diagonals of the rectangles are approximate diameters of the circle, so half a diagonal is approximately the radius, which is $a_0/\sqrt2$ if you measure it according to the diagonal of the square of side $a_0$. This all seems a slightly roundabout way of solving equations $x^2+y^2=r^2$, though literally they are set up as $4x^2+4y^2=4r^2$. Whether it is "best" depends on what "best" is optimizing. It seems to be optimizing $x$-coordinates at a given $y$, so probably not optimizing the approximation of the area of the circle (but that's just a guess). – David K Mar 12 '16 at 2:12

You state that

$\pi=2+4 \lim_{p \to \infty} \frac{1}{p} \sum_{n=1}^k \sqrt{1-\frac{2n}{p}-\frac{n^2}{p^2}}$

This is a standard Riemann sum.

$\lim_{p \to \infty} \frac{1}{p} \sum_{n=1}^k \sqrt{1-\frac{2n}{p}-\frac{n^2}{p^2}} \to \int_0^{\sqrt{2}-1} \sqrt{1-2x-x^2} dx$

Since $\int \sqrt{1-2 x-x^2} dx = \frac12 \sqrt{-x^2-2 x+1} (x+1)+\sin^{-1}\frac{x+1}{\sqrt{2}}$, and $1-2x-x^2 = 0$ for $x = \sqrt{2}-1$ ($x^2+2x =3-2\sqrt{2}+2\sqrt{2}-2 = 1$),

$\begin{array}\\ \int_0^{\sqrt{2}-1} \sqrt{1-2 x-x^2} dx &= \left(\frac12 \sqrt{-x^2-2 x+1} (x+1)+\sin^{-1}\frac{x+1}{\sqrt{2}}\right)\big|_0^{\sqrt{2}-1}\\ &=(0+ \sin^{-1}\frac{(\sqrt{2}-1)+1}{\sqrt{2}})-(\frac12+\sin^{-1}\frac1{\sqrt{2}}) \\ &=\sin^{-1}(1)-(\frac12+\sin^{-1}\frac1{\sqrt{2}})\\ &=\frac{\pi}{2}-(\frac12+\frac{\pi}{4})\\ &=\frac{\pi}{4}-\frac12\\ \end{array}$