login
A319905
Decimal expansion of 4*(sqrt(2) - 1)/3.
0
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
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.
Wikipedia, 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
KEYWORD
nonn,cons,easy
AUTHOR
STATUS
approved