A228764 Decimal expansion of the arc length of Sylvester's Bicorn curve.

5, 0, 5, 6, 5, 3, 0, 0, 3, 2, 1, 2, 1, 2, 4, 4, 9, 7, 3, 2, 7, 0, 1, 6, 4, 8, 9, 6, 6, 6, 0, 4, 7, 4, 4, 6, 8, 7, 8, 5, 9, 0, 1, 0, 6, 5, 6, 5, 4, 3, 7, 5, 4, 9, 2, 0, 1, 3, 7, 4, 5, 8, 0, 2, 9, 8, 6, 5, 3, 3, 5, 7, 6, 9, 0, 4, 0, 7, 5, 4, 6, 0, 4, 3, 8, 4, 8, 9, 3, 9, 1, 4, 3, 6, 0, 2, 8, 4, 7, 1

Offset: 1

Author

Jean-François Alcover, Sep 03 2013

Keywords

• nonn
• cons

Comments

The Cartesian equation used here is y^2*(t^2-x^2) = (x^2+2*t*y-t^2)^2, with t=1. The arc length (perimeter) is proportional to the parameter t.

Examples

			5.056530032121244973270164896660474468785901065654375492013745802986533576904...
		

Links

• Eric Weisstein, Bicorn (MathWorld)
• Wikipedia, Bicorn

Programs

Mathematica

digits = 100; y1[x_] := (1 - x^2)/(2 - Sqrt[1 - x^2]); y2[x_] := (1 - x^2)/(2 + Sqrt[1 - x^2]); i1 = NIntegrate[Sqrt[1 + y1'[x]^2], {x, -1, 1}, WorkingPrecision -> digits+5]; i2 = NIntegrate[Sqrt[1 + y2'[x]^2], {x, -1, 1}, WorkingPrecision -> digits+5]; RealDigits[i1 + i2][[1]][[1 ;; digits]]