5

For simplification, let's have a source light for a map projection.

I know where to put the source light in order to get an orthographic, stereographic, gnomonic map projection, but I still don't understand, for example, what the difference geometrically (where you put the light and how to project) between the equal-area projection and conformal projection is?

In other words, what and how to make a projection equal-area and the other conformal?

  • Have you reviewed Snyder's book for explanations? (e.g. page 146) pubs.usgs.gov/pp/1395/report.pdf – Kirk Kuykendall Dec 4 '18 at 15:17
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    good question, but in practice I am afraid that most projections are not naturally attainable through "physical" projection from a source light. You need to dstort one property to make the other property constant. – radouxju Dec 4 '18 at 15:35
  • Even if I center the two projection at the same point if it's available, e.g LCC vs AEA and Transverse Mercator vs Transverse Cylindrical Equal Area? – user9322960 Dec 4 '18 at 19:47
3

Only a few map projections have a direct analogy to a simple model where a set of straight-line rays project from a globe and impinge on a developable surface (i.e., plane, cone, cylinder). They are the Azimuthal (or Zenithal), Conical and Cylindrical families of projections.

Most projections that are required to maintain important geometric qualities – these include the two you mention (Equal Area and Conformal) and Equidistant – are derived mathematically. They may resemble the above three families of projections but often have the prefix "Psuedo-" in their description. If you need to learn more about Equal Area and Conformal projections, try searching other questions on GIS.SE.

Having said all that, here's an example "exception" that exactly fits your question! It's the Lambert Cylindrical Equal Area projection. It is formed by projecting orthogonal rays onto a cylinder

enter image description here

  • The mercator projection is well known to be conformal and has it's pojection in the earth's center. To make a projection to a cylinder equal area, you need to place your light source into infinity. So i think Martin F gives the correct answer! – Andreas Müller Dec 5 '18 at 21:29
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    The Mercator projection is not a central perspective. That would be a Central cylindrical projection, and it is not conformal, in fact, it is very seldom used in practice. The Mercator projection is rather derived mathematically to maintain conformity (angles) at every point. – FSimardGIS Dec 5 '18 at 23:07
1

Map projections: A working manual, by John P. Snyder, has illustrations plus explanations of how they differ geometrically.

Stereographic

Gnomonic

Orthographic


Update Regarding how to make a projection equal-area and the other conformal? On page 28 Snyder says:

equations

So the equations don't tell you how to make a projection, only how to tell if the projection is either equal area or conformal.

Satisfy this equation for equal area: equal area

Satisfy this (Cauchy-Riemann) equation for conformal projections: Cauchy-Riemann

  • Thanks, I saw that in the book of Snyder, but my question is for e.g. let's take Lambert Azimuthal Equa Area as CRS for Europe, what does make this projection (geometrically, physically) equal-area and not conformal. the same for Conic Equal Area and Conic Conformal, that is what I don't understand – user9322960 Dec 4 '18 at 19:45
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    Nice graphics but not an answer to the question. – Martin F Dec 5 '18 at 1:33

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