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I've been reading Wikipedia and am having a bit of trouble.

https://en.wikipedia.org/wiki/Latitude

Geodetic latitude would clearly change if the Earth were squashed vertically.

But would places like London retain their original geocentric latitude if this happened?

Or would geocentric latitude change, too?

I made these two images.

The left image is of a perfect sphere. The right image is of an oblate ellipsoid with geodetic latitude lines. The height of the right ellipsoid is equal to sin(45) times the height of the left sphere.

The yellow dot on the left image marks Greenwich, England.

Where should the yellow dot go on the right image?

geocentric sphere

geodetic ellipsoid

If someone could place the dot roughly, then I can then work backwards to answer my question.

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  • Yes, as these two values depend on each other. There a formulas to calculate geocentric latitude from the geodetic latitude, see: en.wikipedia.org/wiki/Latitude – Zoltan Jul 22 '18 at 8:05
  • "Yes" to which question? – posfan12 Jul 22 '18 at 8:17
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    Yes, both values are changed. Have you checked the formulas? – Zoltan Jul 22 '18 at 12:48
  • Yes, I understand their relationship with respect /to each other/. What I can't quite grasp is their relationship with respect /to features on the Earth/. I will draw some pictures and post them. – posfan12 Jul 22 '18 at 14:23
  • I updated my post. – posfan12 Jul 22 '18 at 18:43
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There's no single "right answer" (to where the yellow dot goes). It all depends on what you want to preserve as you deform the ellipsoid.

Let's start with the continents drawn on the reference ellipsoid and you want to exaggerate the flattening of the earth to emphasize the ellipsoidal effects.

  • If you squash everything vertically, you are preserving the parametric latitude β.

  • If you squash radially (by a factor that depends on latitude), you are preserving the geocentric latitude θ.

  • If you wish to preserve shapes, you should hold the conformal latitude χ of points constant.

  • If you wish to preserve areas, you should hold the authalic latitude ξ of points constant. This has my vote because it leads to the most natural looking deformation.

  • If you wish to preserve meridional distances, you should hold the rectifying latitude μ of points constant.

  • Holding the geographic latitude φ constant is probably the worst choice because it leads to the bizarre result of everything being concentrated near the equator in the limit of extreme flattening.

ADDENDUM

To illustrate the various possibilities, I show here textured ellipsoids for a case of large flattening: b/a = 1/4, f = 3/4. In this sequence, the texture is mapped using φ, β, ξ, μ, χ, and θ, respectively:

geographicic geographic φ

parametric parametric β

authalic authalic ξ

rectifying rectifying μ

conformal conformal χ

geocentric geocentric θ

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  • I am trying to preserve the angle of the normal. I.e. the normals at each point on the sphere should have the same angle from the equator before and after the squashing. But the tricky part is altering the texture map of the sphere in this way, which is beyond the scope of this SE. – posfan12 Aug 24 '18 at 19:34
  • That corresponds to keeping the geographic latitude constant. of course. I think you'll find that, if you vary the flattening a lot, e.g., from 0 to 0.8, that you end up with everything too concentrated near the equator. – cffk Aug 24 '18 at 20:36
  • Yeah, I noticed that. And it probably is not important enough to simulate in the illustration I am creating. – posfan12 Aug 25 '18 at 2:28
  • @posfan12 I added figures to illustrate the effect of texture mapping with different auxiliary latitudes. – cffk Aug 25 '18 at 15:52
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To calculate the geodetic latitude of a point, we draw a line that passes through the point and is also perpendicular to the surface of the ellipsoid (called a Normal), and evaluate the angle it forms with the equatorial plane. enter image description here In this image from the Wikipedia article, you can see this relationship between the point and its latitude. The line PN (extended to C on the equatorial plane) is the normal used as a reference for the calculation. The geocentric latitude (which is much less common) is measured by drawing a line linking the point with the center , and then measuring the angle with the equatorial plane. If we squashed the Earth, all these angles would change indeed. If the Earth were a sphere, both geocentric and geodetic latitudes would be equal for any point, because the perpendicular line would always pass through the center. On the ellipsoid, they are equal only if the point is at the poles or on the equator.

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  • I have already created a texture for the graticule based on geodetic latitude. I now need to know what I should do with the texture lying underneath the graticule, of the continents, oceans, etc. I could simply squash or scale it vertically. Or I could UV map it using a method intended for spheres, but not for ellipsoids. Or I could... do something else? – posfan12 Jul 23 '18 at 22:51
  • Which software are you using? It looks to me like you might have trouble UV mapping your texture like this. All the examples of exaggerated ellipsoid that I know of seem to have only vertically flattened the texture along with the sphere. If you really need that much precision, I would just transform your image a bit and make sure it fits nicely (visually speaking) with the geodetic graticule. – FSimardGIS Jul 24 '18 at 12:29
  • I am using POV-Ray, which is very flexible. – posfan12 Jul 24 '18 at 21:40
  • I have settled on the fact that I need to deform the bitmap of the continents, oceans, etc. to match the geodetic graticule. That is beyond the scope of this discussion, so I will mark the question as answered. – posfan12 Jul 24 '18 at 21:51

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