Π day

[Albert Chan]

(03-21-2022 07:24 AM)Thomas Klemm Wrote:  We end up with the same sequence as when using Vieta's formula.

Yes. For 2^n sided polygon, we have 2^(n+1) right triangles, with acute angle x = pi/2^n

area(inscribled triangle) ≤ area(sector) ≤ area(circumscribed triangle)
sin(x)*cos(x)/2 ≤ 1^2*x/2 ≤ 1*tan(x)/2
sin(2x)/(2x) ≤ 1 ≤ tan(x)/x

I(2^n)
= sin(2x)/(2x) * pi
= sin(pi/2^(n-1)) / (pi/2^(n-1)) * pi
= 2 / (cos(pi/4) * cos(pi/8) * ... * cos(pi/2^(n-1))       // Vieta's formula for pi

Of course, the formula does not required 2^n sided polygons
Example, we can start with hexagon: I(6) = 3/2*√3 ≈ 2.598, C(6) = 2*√3 ≈ 3.464

sin(2x)/(2x) = 1 - 2/3*x^2 + 2/15*x^4 + ...
tan(x) / x     = 1 + 1/3*x^2 + 2/15*x^4 + ...

Circumscribed polygon area about twice as accurate as inscribed. (opposite sign)
We can correct for this.

Code:

10 N=6 @ A=SIN(PI/N*2)*N/2 @ B=TAN(PI/N)*N
20 A=SQRT(A*B) @ B=B*A/((B+A)*.5) @ N=N+N
30 DISP N,A,B,(A+2*B)/3
40 IF N<100 THEN 20

Code:

>RUN
 12              3                    3.21539030918        3.14359353945
 24              3.10582854123        3.15965994211        3.14171614182
 48              3.13262861329        3.14608621514        3.14160034786
 96              3.13935020305        3.14271459965        3.14159313412
 192             3.14103195089        3.14187304998        3.14159268362

We can apply Richardson extrapolation, to remove O(x^4) error

>RES
 3.14159268362
>RES + (RES-3.14159313412)/(2^4-1)
 3.14159265359

For unit circle, circumscribed polygon perimeter = its area.
Inscribed n-sided polygon perimeter = Inscribed 2n-sided polygon area.
This make complicated polygon perimeter code un-necessary.