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

Author

Franck Maminirina Ramaharo, Oct 01 2018

Keywords

• nonn
• cons
• easy

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".

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

Examples

			0.552284749830793398402251632279597438092895833835930...
		

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.
• Adam G. Stanislav, Drawing a circle with Bézier Curves
• Wikipedia, Bézier curve
• Wikipedia, Composite Bézier curve

Crossrefs

Cf. A002193, A156309, A188582, A268683.

Programs

Maple

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

Mathematica

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

PARI

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

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