A395716 - OEIS

%I #20 May 19 2026 18:08:51

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

%T 9,0,8,5,5,8,5,1,6,9,5,8,6,4,8,6,3,2,4,8,4,2,6,6,6,1,8,2,1,0,1,3,2,8,

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

%N Aspect ratio of the Goode homolosine map projection.

%C The Goode homolosine projection, introduced by John Paul Goode, is a map projection that uses the Mollweide projection above a latitude where the lengths of parallels match, and the sinusoidal projection below it.

%C For the Mollweide projection, 2*theta + sin(2*theta) = Pi*sin(phi), where theta is the auxiliary angle corresponding to latitude phi.

%C The condition Pi*cos(phi) = 2*sqrt(2)*cos(theta) matches the half-width of the sinusoidal and Mollweide projections at the transition latitude.

%C When the equatorial width is normalized to 2, the map height is H = 2*(phi + sqrt(2)*(1 - sin(theta)))/Pi, so the final aspect ratio is 2/H.

%F Equals Pi/(phi + sqrt(2)*(1 - sin(theta))) where Pi*sin(phi) = 2*theta + sin(2*theta) and Pi*cos(phi) = 2*sqrt(2)*cos(theta).

%e 2.307602098422378543380170805130633908558516958648632484266618210...

%t RealDigits[Pi/(phi + Sqrt[2]*(1-Sin[theta])) /. FindRoot[{Pi*Sin[phi] == 2*theta + Sin[2*theta], Pi*Cos[phi] == 2*Sqrt[2]*Cos[theta]}, {phi, 1/2}, {theta, 1/2}, WorkingPrecision -> 105]][[1]] (* _Amiram Eldar_, May 16 2026 *)

%o (Python)

%o import mpmath as mp

%o mp.mp.dps=1000

%o sol= mp.findroot(lambda ph,th: (2*th+mp.sin(2*th)-mp.pi*mp.sin(ph), mp.pi*mp.cos(ph)/(2*mp.sqrt(2)*mp.cos(th))-1), (0.71,0.57))

%o ph=sol[0] # ph is the transition latitude in radians; ph*180/mp.pi is the transition latitude in degrees

%o th=sol[1] # th is auxiliary angle for Mollweide projection in radians

%o H=2*(ph + mp.sqrt(2)*(1-mp.sin(th)))/mp.pi

%o print(str(2/H)[0:200])

%Y Cf. A361260.

%K nonn,cons

%O 1,1

%A _Donghwi Park_, May 04 2026