Map projection process for three-dimensional point cloud

Question

I have some points given in ECEF coordinates (or LatLongAltitude based on WGS84, which is interchangable in my understanding). Currently I am working on a mapping software which allows to visualize these points in maps. The current software version orthographically projects the points onto a plane, but future versions should support a wider range of map projections like Mercator and others. (It is not yet decided which exactly but the software design should be open to extension.)

From what I have read so far, the usual process for creating maps is

• project the 3D points onto the chosen reference ellipsoid which yields (2D) LatLong coordinates (in case of WGS84 ellipsoid, I could use my current LatLong and "drop" the altitude information)

• transform the LatLong coordinates into map coordinates (depending on the chosen map projection)

This actually is different from what we are doing currently: By first projecting onto the reference ellipsoid, the altitude is "lost". If then the projected point is transformed (e.g. with an orthographic projection) to map coordinates, altitude has no effect. In contrast, if the 3d point is directly projected orthographically, "altitude" (wrt to any ellipsoid) has effect on the output coordinates.

I tried to illustrate this (based on the height definition in @Farid Cher's answer):

When Point P0 is directly projected (as in our case) it becomes P0p on the Projection plane, whereas by first projecting onto the ellipsoid (to P1) and then projecting on the Projection plane, it becomes P1p (which is not P0p).

My questions are these:

1.) Is there any (practically relevant) map projection which follows the approach that we have currently implemented or do they "all" adhere to the two step process described earlier? (I.e., does the Altitude have an influence on the map coordinates?)

2.) Is it safe to assume that all points with the same LatLong coordinates (wrt any practically used reference ellipsoid) are on a straight line? (I know that depending on the choice of geodetic or geocentric coordinates, these lines for different LatLong may have a common point or not, but that's not too important for me.)

Answer 1

To address your second question, I can say, that depends on how you project earth topography on your ellipsoid (how the straight line is defined). The straight line can be rhumb-line, a line vertical to the ellipsoid surface or a line that passes through the ellipsoid center. If you define this line, then you can say any LatLong on this line are equal.

So any point (LatLong) on earth surface has an equivalent point on the ellipsoid (e.g. wgs84). It is not like simply dropping the z (altitude) from 3D coordinate. This is usually the intersection of rhumb-line with the mathematical ellipsoid surface. When you transform a geographic coordinate system (WGS84) to a projected coordinate system, It doesn't mean you are losing (dropping) the Z (earth topography). The LatLong already contains the altitude implicitly (intersection of rumb-line with ellipsoid).

So to answer your first question, all map projections are doing the same process like your orthographic projection. But you should know how the ellipsoid X,Y (Phi,Lambda) are inferred from the location on earth surface. There are many ellipsoids or spheroids that estimate the earth surface. Each of them has a well defined mapping, from the earth topography on the mathematical earth surface (ellipsoid or spheroid).

I reccommend you to read esri online documentation about map projections: Projection basics the GIS professional needs to know

Update

In your example you are trying to project a simple 3D point (X,Y,Z) to a 2D space (not a map projection).

Why do we use map projections at all? With map projection we want to reach the 2D space from ellipsoidal (spherical) surface of the earth. It seems more complex than your simple projection. So we use approximation. For example with the popular TM (Transfer Mercator) projection, We fit a cylindrical shape to earth surface that the cylinder is longitudinal along the equator. Although with this specific projection, it may result in extreme stretching near poles.

Imagine you are projecting each point on estimated earth e.g. ellipsoid (LatLong) to this cylinder (This part is like your process of orthogonal projection). visualize you have cut the earth to equal slices; Then open the sliced earth to cover the cylinder.

Then suppose this cylinder is a paper and you cut the paper vertically. Now you have a 2D surface that contains all points from the complex 3D earth surface.

Answer 2

First, let me outline the two steps from the location on the physical Earth's surface to the projected (plane) coordinates since they are slightly different from what you described above:

1. From the physical location on Earth to the mathematical reference surface: when you capture coordinates with a GPS receiver, you get the ellipsoidal coordinates (lat, lon) in WGS84 reference system. WGS84 is the ellipsoid used by the GPS system which center coincides with the Earth's point of gravity. Note that vertical and horizontal (positional) datums are completely independent. Vertical datum represents the definition of the geoid surface which is not a mathematical surface. Also, I would avoid using "projecting" verb in this context, since projection is involved in the second step.

2. Second step represents a transition from the curved mathematical surface to the plane - flat surface: this step involves a cartographic projection. All projections involve distortions - you choose the projection that best fits your needs based on the area and other requirements. There is a fairly new research that studies dynamic projections, based on location: article

Answers to your questions:

1. Altitude influences the positional, ellipsoid's coordinates because ellipsoid's and geoid's tangent planes are in general not parallel at a certain location. In other words, the direction of the force of gravity is in general not parallel to the line perpendicular to the ellipsoid's surface.

2. I assume you mean multiple points with the same LatLons but different elevations. Those points are on the same line, which is also perpendicular to the ellipsoid's surface. However, the direction of the force of gravity would be different at each of those points, which may not be what you are concerned about - just wanted to point it out.

Also, you need to provide a clear answer to the question: what do your Z- coordinates represent? Ellipsoidal elevations or elevation from the geoid surface (above sea level)? That makes all the difference to the way how to treat them.