A156309 -id:A156309 - cp's OEIS frontend

A273621 Decimal expansion of the solid angle (in steradians) subtended by a cone having the 'magic' angle A195696 as its polar angle.

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

Offset: 1

Stanislav Sykora, Aug 15 2016

An example of such a cone is the one circumscribed to a cube from one of its vertices. When expressed as a fraction of the full solid angle, this constant leads to A156309.

			2.65558657871115077573713025127469430382620630256473049081011931...

Stanislav Sykora, Table of n, a(n) for n = 1..2000

Cf. A000796, A156309, A195696, A210974.

First@RealDigits@N[2*Pi*(1 - Sqrt[1/3]), 25] (* G. C. Greubel, Aug 15 2016 *)

PARI
```
2*Pi*(1-sqrt(1/3))
```

Equals 2*Pi*(1-sqrt(1/3)) = 4*Pi*A156309 = 2*Pi*(1-cos(A210974)).

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

Original entry on oeis.org

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

Franck Maminirina Ramaharo, Oct 01 2018

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

			0.552284749830793398402251632279597438092895833835930...

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

Cf. A002193, A156309, A188582, A268683.

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

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

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

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

cp's OEIS Frontend

A273621 Decimal expansion of the solid angle (in steradians) subtended by a cone having the 'magic' angle A195696 as its polar angle.

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