A319905 Decimal expansion of 4*(sqrt(2) - 1)/3.

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

Offset

0,1

Comments

A 90-degree unit-circular arc in the first quadrant can be approximated by a cubic Bézier curve. In this case, L = 4*(sqrt(2) - 1)/3 is the unit tangent vector scaling factor that minimizes the distance between the curve and the unit circle segment, provided its endpoints and midpoint are interpolated.

Riškus referred to this constant as "magic number".

Links

Table of n, a(n) for n=0..86.

Tor Dokken, Morten Dæhlen, Tom Lyche and Knut Mørken, Good approximation of circles by curvature-continuous Bézier curves, Computer Aided Geometric Design Vol. 7 (1990), 33-41.

Aleksas Riškus, Approximation of a cubic Bézier curve by circular arcs and vice versa, Information Technology And Control Vol. 35 (2006), 371-378.

Adam G. Stanislav, Drawing a circle with Bézier Curves

Wikipedia, Bézier curve

Wikipedia, Composite Bézier curve

Formula

Equals (4/3)*tan(Pi/8).

Irrational number represented by the periodic continued fraction [0; [1, 1, 4, 3]]; positive real root of 9*x^2 + 24*x - 16. - Peter Luschny, Oct 04 2018

Example

0.552284749830793398402251632279597438092895833835930...

Maple

Digits:=1000; evalf(4*(sqrt(2) - 1)/3);

Mathematica

RealDigits[4*(Sqrt[2] - 1)/3, 10, 100][[1]]

Prog

(PARI) 4*(sqrt(2) - 1)/3

Crossrefs

Cf. A002193, A156309, A188582, A268683.

Sequence in context: A011501 A319305 A196614 * A319593 A335321 A172125

Adjacent sequences: A319902 A319903 A319904 * A319906 A319907 A319908

Keyword

nonn,cons,easy

Author

Franck Maminirina Ramaharo, Oct 01 2018

Status

approved