Monte Carlo Elliptic Integrals, Summaries of Geometry

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Monte Carlo Elliptic Integrals

Circumference of an ellipse? Can’t do it mate.

Josh Fogg (they/them)

Background

r

Circle Area = π r 2 Circumference = 2 π r

a

b

Ellipse Area = π ab Circumference = ???

Rough Approximations

The greats had a go:

1. Geometric Average s = 2 π

√ ab (Johannes Kepler, 1609)

2. Arithmetic Average

s = 2 π a^ + 2 b (Leonhard Euler, 1773)

3. Root Mean Square

s = 2 π

√ a^2 + b^2 2 (Leonhard Euler, 1773)

Exact Solutions

Colin Maclaurin (1742):

s = 2 π a

∑^ ∞

m=

(2m)! m!m!

k^2 m 16 m(1 − 2 m)

where k =

b^2 a^2

Leonhard Euler (1776):

s = π

2(a^2 + b^2 )

∑^ ∞

m=

δ 16

)m · (4m − 3)!! (m!)^2

where δ =

a^2 − b^2 a^2 + b^2

James Ivory (1796):

s = π (a + b)

∑^ ∞

m=

(2m)! m!m!

hm 16 m(1 − 2 m)^2

where h =

a − b a + b

Carl Gauss (1812) & Eduard Kummer (1836) showed these could be expressed in terms of the hypergeometric function:

2 π a · F

, 1 , k^2

= π · F

, 1 , δ

= π (a + b) · F

, 1 , h

Better Approximations

Srinivasa Ramanujan’s Approximations (1914)

He gave two:

s = π

3(a + b) −

(3a + b)(a + 3b)

s = π (a + b)

3 h 10 +

4 − 3 h

where h =

a − b a + b

David Cantrell’s Approximation (2001)

Dropped this equation on a Geometry Research Google group:

s = 4(a + b) − 2(4 − π ) ab Hp

where Hp =

ap^ + bp 2

) p^1

. (3)

The value of p can be optimized for the type of ellipse:

pround = 3 π − 8 8 − 2 π , plong = ln(2) ln(2 / (4 − π )) , pgeneral = 0_._ 825_._

Better Approximations

Parker’s Approximation (2020)

s = π

a +

b −

269 a^2 + 667ab + 371b^2

where a > b

Lazy Parker’s Approximation (2020)

s = π

a +

b

where a > b

Monte Carlo: Method

s ≈ 4 a ×

( π 2

× 1

×

#red #red + #blue

Monte Carlo: Convergence

Plots of sample count against absolute percentage error for varying a.

(a) a = 1 (b) a = 1_._ 01 (c) a = 1_._ 1

(d) a = 2 (e) a = 5 (f) a = 80

Conclusion

1. Don’t use Monte Carlo to do Elliptic Integrals in practice.
2. Like really don’t... 3.... but it is a cool exercise!

The theory of elliptic functions is the fairyland of mathematics. The mathematician who once gazes upon this enchanting and wondrous domain crowded with the most beautiful relations and concepts is forever captivated. -Richard Bellman

Figure: Ellipses with b = 1 and a = 1, 5, 20, and 80.

Figure: Ellipse with b = 1 and a = 3_._ 93; has the same eccentricity as Halley’s comet.

Figure: Ellipse with b = 1 and a = 76_._ 7, has the eccentricity of most eccentric comet.