Are Tissot's indicatrices enough to tell whether or a not a projection will keep 3D lines also straight in the projected 2D space

Question

I have more of a math question, but one for which I want to know how it applies in the particular case of GIS, especially to map projections.

Can a map projection keep any 3D straight line (e.g., think of a laser beam directed anywhere you want -- making abstraction of the Earth itself if necessary -- to infinity, and for which we imaginatively replace the atmosphere by the vacuum, thus taking into consideration only the mathematical definition of space and no physical perturbations) also straight in the resulting 2D space?

Obviously not, because map projections are supposed to transform a curved surface into a flat one. Ok, but... maybe there are special cases where they can! So how could you find these out?

After some thoughts, I finally came to the (probably wrong) conclusion that if Tissot's indicatrices can always be drawn as circles (e.g. in conformal projections**) on the projection plane, in other words, that the projected space itself is perfectly isotropic, then any 3D line will stay as a straight line when projected in this 2D space.

But because Tissot's indicatrix is a way to describe how a set of 3D circles drawn "on the globe" are projected, I'm not 100% sure that these 3D circles are actually carrying all the 3D information to verify this assumption.

Finally, I also remembered old linear algebra courses which taught me that a transformation is linear if it keeps a line straight and if it doesn't move the origin. Therefore, there is probably no single map projection that is linear (by definition, as they are all meant to "flatten" a spherical shape) and the answer is simply; "no, 3D lines can never be preserved as straight lines when transformed to a 2D space using a cartographic projection".

(Another difficulty here is that most map projections transform 2D surfaces into 2D surfaces, thus not really taking into account the 3rd dimension I'm concerned with in my questioning. And also that if a 3D->2D projection exists, since it loses a spatial dimension -- which also means it has a determinant of 0 -- I know that the reverse transformation would lead to a plane. In other words, if a 3D line collapses into a 2D line, it will never be the only one; there will be an infinite number of lines describing a plane which would collapse to the same 2D line. But I don't really care about that here.)

So, there is this duality in my mind right now, hence my question: can Tissot's indicatrices tell whether or a not a (cartographic) projection will always keep 3D lines also straight in the projected 2D space? (which implies: does it exist at least one, or one set of projection(s) for which 3D lines are preserved as lines when projected in 2D?)

** E.g.: RGF93 / Lambert-93 (EPSG:2154)

or WGS 84 / Pseudo-Mercator (EPSG:3857)

Maps drawn with ♥ using the QGIS Indicatrix mapper plugin: https://plugins.qgis.org/plugins/tiss/

Answer 1

A cartographic projection is a nonlinear transformation in two dimensions.

So in the first place you must reduce the three-dimensional line to the mathematical surface on which the cartographic projection will be applied (a sphere, in this answer).

This reduction always projects the three-dimensional straight line into what is called a great circle (the intersection between the sphere and a plane passing through its center). And yes, all three-dimensional straight lines that belong to that plane will be projected onto the same great circle.

A great circle is projected (cartographically) to the plane as a straight line on the gnomonic map projection (geometric one with the point of view at the center of the sphere, and not a conformal projection).

Now, the Tissot indicatrices indicate the direction of the principal tangents (maximum and minimum deformation) and their modules, for a particular map projection and a particular point on the Earth.

Can we deduce a particular projection knowing the direction and modules of principal tangents at any point on the Earth? I think not, as I think we don't need it either.