Hello and welcome to the circle challenge!

Inspired by various reels of university math professors teaching how to correctly draw circles on a blackboards.

The rules are simple. Draw a moderately sized circle anywhere on screen with touch, stylus or mouse. (No cheating by drawing small circles!) The Algorithm^TM will then provide estimates of π based on various methods described below as well as calculate a "roundness value". Most round and best π wins!

Brought to you by vos ambassadeurs locaux des mathématiques

P.S.: Some nitty gritty details

• Circle-fitting

  On a cartesian plane, circles are described by the relation (x-a)² + (y-b)² = r² (with the center at (a,b) and radius r). If you've ever done any science assignment you would've heard of regression - and specifically a circle regression is done on the points along the curve. How regression works: For each point (x,y_original) the model may deviate and will produce a new point (x, y_new). The residuals given by y_new-y_original, specifically the sum of the squares of residuals, is something we want to minimise. But even this is too difficult to do exactly so we cheat and turn this into linear system of equations in a, b and r and basically say "its close enough". The details of how this works are given by this wiki article: Newton-Gauss Method.

• Calculating π

  Area method: Area of a circle should be given by the relation A = πr² where A and r are area and radius respectively, giving π = A/r². Substituting the area of the drawn shape and radius obtained from the circle regression into the expression gives a value for π.

  Circumference method: Circumference C of a circle should be given by the relation C = 2πr, giving π = C/2r. Substituting the perimeter of the drawn shape and radius obtained from the circle regression into the expression gives a value for π.

  Roundness method: The ratio between the area of a circle and the square of its circumferece should, from the above formulas, be 4π. (This is also the maximum value of this ratio for any shape in a flat 2D plane; that is, of all the shapes with a certain perimeter, the circle has the largest area! See the Isoperimetric Inequality Wikipedia page For why this is.). Finding this ratio between area and perimeter squared of the drawn shape, then dividing by four, gives a value of π if your circle is perfect.