A395716 Aspect ratio of the Goode homolosine map projection.

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, 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, 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

Offset

1,1

Comments

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.

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

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

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.

Links

Table of n, a(n) for n=1..105.

Formula

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).

Example

2.307602098422378543380170805130633908558516958648632484266618210...

Mathematica

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 *)

Prog

(Python)
import mpmath as mp
mp.mp.dps=1000
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))
ph=sol[0] # ph is the transition latitude in radians; ph*180/mp.pi is the transition latitude in degrees
th=sol[1] # th is auxiliary angle for Mollweide projection in radians
H=2*(ph + mp.sqrt(2)*(1-mp.sin(th)))/mp.pi
print(str(2/H)[0:200])

Crossrefs

Cf. A361260.

Sequence in context: A350463 A341339 A084257 * A365803 A378991 A185963

Adjacent sequences: A395713 A395714 A395715 * A395717 A395720 A395721

Keyword

nonn,cons

Author

Donghwi Park, May 04 2026

Status

approved