I understand what a datum is: the Earth is not a sphere, so it can be better approximated using an oblate spheroid, so a datum is the specification of such a spheroid (usually relative to particular anchor points etc.)
What I find confusing is how to take this into account when projecting a map. Of course, many projections have ellipsoidal forms, and thus for a given set of source data and source datum, projecting this onto the 2D plane is simply a matter of using the appropriate ellipsoidal factor.
However, what if the projection is only defined for a sphere? It seems Geocart allows the datum to be specified for any map projection. But I'm not sure what this does for a projection that I imagine is only defined for the sphere (unless I'm wrong and in fact the ellipsoidal form has been derived for each and every one!) Of course, some projections (Hotine) only have an ellipsoidal form it seems.
If a particular projection does indeed only have a spherical formulation, does Geocart simply make an approximation and keep the source longitude and latitude as-is for such a projection? Or perhaps it always transforms to WGS-84, for consistency? Yes, I know WGS-84 is not a sphere, but I'm wondering if an approximation is made here particularly as some projections are only really used for small-scale mapping, so the flattening is practically indiscernible.
On the other hand, I'm aware that many projections can be derived from others, e.g. conformal mappings can be derived via the stereographic projection (which has an ellipsoidal formulation), so on second thoughts maybe Geocart does indeed have ellipsoidal forms for more projections than I realise.
Thanks!
Projections and Datums
Re: Projections and Datums
Hello jasondavies.
Geocart has two ways to project the ellipsoid: 1) Directly using explicit ellipsoidal formulations for a few projections; or 2) Using auxiliary latitudes for any projection. In the second case, Geocart first projects the ellipsoid to a sphere and then uses the regular spherical development from there. When projecting to the sphere, you decide what property you want to preserve: conformality, area, or the length of any meridian that would project with constant scale from sphere to plane. No matter which you choose, you can achieve it by adjusting only the latitudes (and possibly the sphere’s radius); hence “auxiliary latitude”.
The reason explicit ellipsoidal formulas are needed sometimes is because you might want to preserve some other property. In normal usage, for example, the transverse Mercator requires constant scale along the central meridian. You can’t achieve that using the auxiliary latitudes, so Geocart has explicit formulas for that. You can tell if Geocart has an explicit ellipsoidal development or not when you select the datum. If the datum dialog forces you to select an auxiliary latitude when you have chosen an ellipsoidal datum, then Geocart has no explicit ellipsoidal formulation.
Does that help?
Happy mapping!
— daan
Geocart has two ways to project the ellipsoid: 1) Directly using explicit ellipsoidal formulations for a few projections; or 2) Using auxiliary latitudes for any projection. In the second case, Geocart first projects the ellipsoid to a sphere and then uses the regular spherical development from there. When projecting to the sphere, you decide what property you want to preserve: conformality, area, or the length of any meridian that would project with constant scale from sphere to plane. No matter which you choose, you can achieve it by adjusting only the latitudes (and possibly the sphere’s radius); hence “auxiliary latitude”.
The reason explicit ellipsoidal formulas are needed sometimes is because you might want to preserve some other property. In normal usage, for example, the transverse Mercator requires constant scale along the central meridian. You can’t achieve that using the auxiliary latitudes, so Geocart has explicit formulas for that. You can tell if Geocart has an explicit ellipsoidal development or not when you select the datum. If the datum dialog forces you to select an auxiliary latitude when you have chosen an ellipsoidal datum, then Geocart has no explicit ellipsoidal formulation.
Does that help?
Happy mapping!
— daan
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jasondavies
- Posts: 11
- Joined: Tue Feb 12, 2013 1:40 pm
Re: Projections and Datums
Fantastically helpful, thank you!