This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A319905 #20 Feb 25 2025 11:41:54
%S A319905 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,
%T A319905 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,
%U A319905 3,3,0,4,6,1,6,1,4,2,7,1,8,4,6,7,1,8,3
%N A319905 Decimal expansion of 4*(sqrt(2) - 1)/3.
%C A319905 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.
%C A319905 Riškus referred to this constant as "magic number".
%C A319905 The Bézier curve with control points {(1,0), (1,L), (L,1), (0,1)} has a minimum distance to the origin of 1 (at t in {0, 1/2, 1}), and it has a maximum distance to the origin of (1/3)*sqrt(71/6-2*sqrt(2)) = 1.00027253... at t in {(3 - sqrt(3))/6,(3 + sqrt(3))/6}. - _Peter Kagey_, Feb 21 2025
%H A319905 Tor Dokken, Morten Dæhlen, Tom Lyche and Knut Mørken, <a href="https://dx.doi.org/10.1016/0167-8396(90)90019-n">Good approximation of circles by curvature-continuous Bézier curves</a>, Computer Aided Geometric Design Vol. 7 (1990), 33-41.
%H A319905 Aleksas Riškus, <a href="http://www.itc.ktu.lt/index.php/ITC/article/view/11812">Approximation of a cubic Bézier curve by circular arcs and vice versa</a>, Information Technology And Control Vol. 35 (2006), 371-378.
%H A319905 Adam G. Stanislav, <a href="https://web.archive.org/web/20230616034149/http://whizkidtech.redprince.net/bezier/circle/">Drawing a circle with Bézier Curves</a>
%H A319905 Wikipedia, <a href="https://en.wikipedia.org/wiki/B%C3%A9zier_curve">Bézier curve</a>
%H A319905 Wikipedia, <a href="https://en.wikipedia.org/wiki/Composite_B%C3%A9zier_curve">Composite Bézier curve</a>
%F A319905 Equals (4/3)*tan(Pi/8).
%F A319905 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
%e A319905 0.552284749830793398402251632279597438092895833835930...
%p A319905 Digits:=1000; evalf(4*(sqrt(2) - 1)/3);
%t A319905 RealDigits[4*(Sqrt[2] - 1)/3, 10, 100][[1]]
%o A319905 (PARI) 4*(sqrt(2) - 1)/3
%Y A319905 Cf. A002193, A156309, A188582, A268683.
%K A319905 nonn,cons,easy
%O A319905 0,1
%A A319905 _Franck Maminirina Ramaharo_, Oct 01 2018