Elliptical stereographic

[Piotr]

The subject tells you everything. World in an ellipse, sphere is mapped to a pseudocylinder then unwrapped, parallels are spaced like stereographic. How would it look in:

1. 2:1 ratio
2. 1:1 ratio (circle)
3. ratio for correct scale at equator
4. ratio for correct scale at 45, 0
5. Bonus: Van der Grinten III parallel spacing (but still elliptical meridians) with 1., 2., 3. and 4.

[daan]

I don’t quite follow what you’re proposing. Let’s start here:

World in an ellipse,

Do you mean to use an ellipsoidal model of the earth before projecting?

— daan

[Piotr]

NO! I mean that the map is ellipse.

[daan]

Hm. Why would you propose to do this? What you describe is a pseudocylindric projection whose parallel spacing increases toward the poles instead of decreasing. Is my interpretation correct? That would cause magnification in the high latitudes, where there is no room for it, which forces massive compression in the low latitudes. Everything would be heavily distorted.

— daan

[RogerOwens]

The subject tells you everything. World in an ellipse, sphere is mapped to a pseudocylinder then unwrapped, parallels are spaced like stereographic. How would it look in:

1. 2:1 ratio
2. 1:1 ratio (circle)
3. ratio for correct scale at equator
4. ratio for correct scale at 45, 0
5. Bonus: Van der Grinten III parallel spacing (but still elliptical meridians) with 1., 2., 3. and 4.

I think you're asking the same question that I asked here some years ago. Daan was very helpful in answering it.

I asked how an elliptical-shaped conformal world map could be made. Daan's answer was there is such a map, but it requires partial differential equations, boundary-condition differential equations, which was a bit more than I wanted.

But, by the same technique that gives us August (from Lagrange), and Oblated Stereographic (for oblong continents), daan posted an image of an oval (but not elliptical) conformal world map.

That technique that I refer to consists of applying a function, usually a simple polynomial function, with only a few terms, to a complex variable.

A point on a flat 2-dimensional map can be represented by a complex number, x + iy, where x and y are the point's co-ordinates on the flat map.

"i" is the imaginary operator, the square-root of minus one.

A complex number can be treated as an ordinary real number, as the variable for a function. A function can be applied to a complex variable. The points on a flat 2-dimensional map can be regarded as values of the complex variable x + iy.

When a polynomial function is applied to all the values of x + iy belonging to the points on the map's graticule, and the shores of continents and boundaries of countries, it gives a new value of x + iy, the function's value, for each of the map's x +iy values that it's applied to. That new set of x +iy values amounts to a new map.

So, when you apply some simple polynomial function to the x +iy values of the old map's points on its graticule and boundaries of continents and countries, etc., you get a set of x + iy values that define a new map. The function's input is the old map, and its output is a new map.

And what makes that so useful is the fact that, if the function is a polynomial function, then, if the old map is conformal, then the new map will be conformal too.

So applying a polynomial function to a conformal map will give another conformal map.

In that way, Oblate Stereographic is made from ordinary Stereographic, August is made from Lagrange, and daan's oval conformal world map is made from Lagrange.

Different simple polynomial functions, of course, are used for those different transformations.

So, bottom-line:

There is an elliptical-shaped conformal world map, but its construction much more involved, complicated and advanced than most of us want.

But there's an oval conformal world map that only requires applying a simple polynomial function to the complex variable that represents the points on the Lagrange projection, to make the oval conformal world map from the Lagrange map.

Michael Ossipoff

[RogerOwens]

Admittedly it isn't very elliptical, but it's oval.

Alright, maybe its appearance isn't so very different from Lagrange.

Michael Ossipoff

[RogerOwens]

Here's the url for the thread in which that projection was discussed:

https://www.mapthematics.com/forums/vie ... 7&start=10

[Piotr]

I asked for a elliptical pseudocylindrical projection with stereographic spacing and possibly also Van der Grinten III spacing too. See the first post for aspect ratios.

[RogerOwens]

I asked for a elliptical pseudocylindrical projection with stereographic spacing and possibly also Van der Grinten III spacing too. See the first post for aspect ratios.

But the Stereographic is an azimuthal projection, with circular parallels. That's incompatible with a pseudocylindrical, which must have straight parallels.

Apianus II is an elliptical pseudocylindrical, with equally-spaced parallels. Of course Mollweide is the elliptical pseudocylindrical with parallels spaced for equal-area.

But there probably can't be any such thing as a conformal pseudocylindrical map (I'm using "pseudocylindrical" with a meaning that excludes "cylindrical")

Sometimes cylindrical maps are called "Pseudocylindrical", because a cylindrical map is just an extreme of a continuum of pseudocylindrical maps. Using that broader meaning of "pseudocylindrical" to encompass "cylindrical", then you could say that there' s conformal pseudocylindrical. But it certainly isn't elliptical.

Michael Ossipoff

[Piotr]

I didn't say "conformal" anywhere.

Also, I said that the spacing of parallels is like in Stereographic, Gall stereographic or Foucaut.