This is the wrong place to look for detailed mathematical explanations. Here is a hand-waving explanation:
Although it may appear that "map projections" and "camera models" should be different things, they are merely different terms to describe mathematical transformations. Objects in the universe are typically three-dimensional, while photographs are generally two-dimensional. The mathematics to represent/project/transform three-dimensional objects into two dimensions are map projections.
Transformations that preserve some of the original properties have different terms attached to them (adapted from Wikipedia):
azimuthal or zenithal: Preserve direction (possible only from one or two points to every other point)
conformal or orthomorphic: Preserve local shape
equal-area or equiareal or equivalent or authalic: Preserve area
equidistant: Preserve distance (possible only between one or two points and every other point)
gnomonic (also called rectilinear): This is the projection produced by pinhole cameras. It is the only projection that preserves the shortest route.
Jacek Turski, “Robotic Vision with the Conformal Camera: Modeling Perisaccadic Perception,” Journal of Robotics, vol. 2010, Article ID 130285, 16 pages, 2010. https://doi.org/10.1155/2010/130285.
Figure 3: The conformal camera. (a) Image projective transformations are generated by iterations of transformations covering translations “h” and rotations “k” of planar objects in the scene. (b) The 2D section of the conformal camera further explains how image projective transformations are generated and how the projective degrees of freedom are reduced in the camera; one image projective transformation in the conformal camera corresponds to different planar objects translations and rotations in the 3D world.
The mappings are conformal, that is, they preserve the oriented angles of two tangent vectors intersecting at a given point. Because of this property, the camera is called “conformal”. Although the conformal part of an image projective transformation can be removed with almost no computational cost, leaving only a perspective transformation of the image, the conformality provides an advantage in imaging because the conformal mappings rotate and dilate the image's infinitesimal neighborhoods, and, therefore, locally preserve the image pixels.