Pi Day once more!

In honour of Pi Day (March 14), the chart shows six ways of randomly selecting a point in a unit disc. Four of the methods are bad, for various reasons.

A. Midpoint of random p, q on circumference

p = (cos(𝜃₁), sin(𝜃₁)) is a point on the circumference

q = (cos(𝜃₂), sin(𝜃₂)) is another point on the circumference

x = ½ cos(𝜃₁) + ½ cos(𝜃₂) and

y = ½ sin(𝜃₁) + ½ sin(𝜃₂), for random 𝜃₁ and 𝜃₂, define their midpoint.

B. Random polar coordinates

x = r cos(𝜃)

and y = r sin(𝜃), for random angle 𝜃 and radius r ≤ 1. This gives choices biased towards the centre.

C. Random y, then restricted x

Random y, followed by random x in the range −√(1−y²) to √(1−y²). This gives choices biased towards the top and bottom.

D. Random point on chord in A

Similar to A, but x = a cos(𝜃₁) + (1−a) cos(𝜃₂)

and y = a sin(𝜃₁) + (1−a) sin(𝜃₂), for random 𝜃₁ and 𝜃₂ on the circumference of the circle and random a between 0 and 1. This gives choices biased towards the periphery.

E. Random polar with sqrt(r)

Similar to B, but x = √r cos(𝜃)

and y = √r sin(𝜃), for random angle 𝜃 and radius r. The square root operation makes the selection uniform across the disc.

F. Random x, y within disc

Random x and y, repeating the choice until x² + y² ≤ 1. This is uniform, and the selection condition restricts the final choice to the disc.

Oh, and here are some Pi Day activities.

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