[Proj] Equations for Apianus II, sinusoidal, cylindrical equidistant

[Michael Ossipoff]

Because Oscar asked “Where are the equations”, and so that that other person 
won’t think that I’m being like Fermat, I’m posting the equations for all 
four of the projections that I’ve recommended that have the linearly 
interpolable positions property.

I’ve already posted the equations for the graduated equidistant cylindrical, 
and in this posting are the equations for Apianus II (equally-spaced 
elliptical), the sinusoidal, and the ordinary equidistant cylindrical.

The other day, for the graduated equidistant cylindrical, I had the map 
co-ordinates’ origin, and 0 longitude, at the west edge of the projection, 
but for the remaining three, today, I’m saying it differently, in a way 
that’s more generally useful, so that I only have to say that part once, for 
all three projections.

The origin of the map co-ordinates is at the intersection of the equator and 
the central meridian.

“Lat” means latitude in degrees, positive if north, and negative if south.

“Lon” means longitude in degrees, with respect to the central meridian, 
positive if east of the central meridian, negative if west of the central 
meridian.

That applies for all of the following three projections.

Apianus II:

Y = R*Lat*(pi/180)

A = pi*R

X = sqr(A*A-4*Y*Y)*(lon/180)

Sinusoidal:

Y = R*Lat*(pi/180)

X = R*Lon*(pi/180)*cos(Lat)

Ordinary equidistant cylindrical:

Y = R*Lat*(pi/180)

X = R*Lon*(pi/180)

Michael Ossipoff