A346062 Decimal expansion of the minimum value of the area of a rhombus circumscribed around a cosine-shaped lens, whose vertices lie on coordinate axes.

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Offset: 1

Author

Gleb Koloskov, Jul 03 2021

Keywords

• nonn
• cons

Comments

Consider a lens-like shape S created by the curves cos(x) and -cos(x) for x in [-Pi/2,Pi/2] and a rhombus circumscribed around S, whose vertices lie on coordinate axes.

This constant represents the value of the minimum area of such a rhombus KLMN with vertices K(0,2v), L(-2u,0), M(0,-2v), N(2u,0).

The rhombus touches S at the midpoints of its sides, A(u,v), B(-u,v), C(-u,-v), D(u,-v) which define a rectangle ABCD of the maximum area, inscribed in S, whose sides are parallel to coordinate axes. The constant u can be found as a root of equation x=cot(x) and is known as A069855, and v=cos(u)=u/sqrt(1+u^2).

Examples

			4.4887707055283605403232300252898136708822792436449257365...
		

Links

• Gleb Koloskov, Geometric illustration

Crossrefs

Cf. A069855.

Programs

Mathematica

N[Minimize[{2 (x+Cot[x])^2 Sin[x],{x>0,x
				

PARI

u=solve(x=0.5,1,x-cotan(x));8*u^2/sqrt(1+u^2)

Formula

Equals 8*A069855^2/sqrt(1+A069855^2).