Here's a question.
If the angle of the sun were a reflection from Earth, meaning you are looking at the sun from anywhere on Earth. Where would the initial light eminate from, assuming that energy was coming from Earth?
There are are some important factors you have left out, leaving us to make assumptions about the scenario.
Regardless of that, the short answer to your question is:
There is not any single place on earth the source can be to cause the Sun to appear in the sky in the location it appears, from any location on earth, after being reflected from some surface in the sky. Period.
Knowing you won't accept this simple answer, let's examine possibilities.
The analysis depends on an optics principle so well established that it's accepted as fact - angle of reflection (r) equals angle of incidence (i). If you're going to insist that this is not a criterion, then you're on your own. Assuming that this principle of optics
does apply, it's still not possible to formulate a meaningful answer without knowing:
The shape of whatever surface the Sun is being reflected from and whether the reflector is specular (like a mirror) or diffuse (like a wall or movie screen). Unless you give us something specific, we have to consider the obvious possibilities.
Specular reflections will be considered to be perfect - that is, no light is scattered or absorbed, only reflected according to i = r.
1) Specular hemispherical dome over a disc-shaped earth.
...1A) Source is located at the center of the disc, thus the center of the dome.
...
All the light will be reflected directly back to the source. It will not be seen anywhere else....1B) Source located at some other point on the disc.
...
An image[nb]The image won't be perfect because of spherical aberration, but we will ignore this secondary effect.[/nb] of the source will be formed on the surface of the disc at a point on the opposite side of center at the same distance from the center. The source will appear to fill the sky if you're within the image and not be visible at all elsewhere. It will also be
very hot within the image because all the energy from the source is concentrated into the image.
Other shapes for the dome (flatter section of a sphere, paraboloidal, etc.) will produce similar effects, but the image formed on the surface will be unfocused, and the entire sky will not be uniformly illuminated (but still won't look like the Sun) if you're standing in the unfocused image. If it becomes flatter and flatter, it will become more and more like scenario 2, below.
OK, that's not working. Let's try something else.
2) Specular plane reflector at some constant height above the disc.
...2A) Source is located at the center of the disc.
...
The reflection of the source will appear in the sky in the direction of the center of the disc from everywhere on the disc. The angle varies from straight up at the center, decreasing nonlinearly to 45° at a distance from the center twice the height of the plane reflector, and continuing to decrease more slowly as you get further away. If the center of the disc is the north pole, the Sun is always due north of you. The Sun does not appear to move in the sky unless you move; it's always due north, but its apparent height changes as E = tan
-1(2H/D) where E is the angle above the horizon, tan
-1 is the arc tangent function, H is the height of the reflector and D is your distance from the source (S and D must be in the same units). The apparent size of the source decreases as you get further away as a function of S
A = 2 tan
-1(S/(2(D
2 + (2H)
2)
1/2)). S
A is the apparent size (an angle), S is the linear size of the source (in miles, meters, furlongs, whatever), H and D as above (use the same units for S, H, and D).
...2A doesn't work.
...2B) Source located at some other point on the disc.
...
As above, but the reflection is not due north at all times. ......2Bi) The source moves around the disc at a constant distance from the center.
......
The reflection will appear to "orbit" in the sky, but will never reach the horizon from anywhere, and change size (smallest when lowest in the sky). If you're closer to the rim than the circle the source follows, its reflection will trace a sort of oval in the sky centered on the direction of the center of the disc. Also, wouldn't a lot of people see that sucker (and get out of its way!) as it moves in a circular path on the surface?
......2Bii) The source moves around the disc in a circle not a constant distance from the center.
......As above. If you're anywhere outside the circle the source is following, the reflection will stay in one side of the sky, centered over the direction to the center of whatever circle the source is following. Same problem with that "big, bright thing" following a path on the surface as above.
...OK, so it's not 2B.
3) Plane specular reflector tilted with respect to the disc. Source is located at the center of the disc. Reflector moves around the outside of the disc once per day.
...3A) Reflector at fixed angle
...
Reflection circles the sky at an elevation that depends on the angle of tilt and your distance from the reflector. Sun can skim horizon if reflector is vertical (and source on surface). Size changes as height changes. Math looks complicated; do it yourself if you care.
...3B) Reflector tilts up and down through the day.
...
Reflection circles the sky at an elevation that depends on the angle of tilt and your distance from the reflector. Size changes as height changes. You could probably devise a way for the reflection to trace the path of the actual Sun, including rising and setting, if you synchronize the tilt with azimuth,
but only for one point on the disc; everyone else is screwed. Math still looks complicated. Everything else (stars, the Moon) is on its own.
If anyone can think of other configurations, be my guest.
Since specular reflections seem to be a lost cause, let's consider a diffuse reflector.
4) Diffuse hemispherical dome over a disc-shaped earth.
...1A) Source is a projector located at the center of the disc, thus the center of the dome.
...This is essentially a planetarium. In a planetarium, you can project a starscape or the sun in a way that to varying degrees realistically mimics what we actually see in the sky, so it should be possible to do the sky itself this way
except the entire audience in a planetarium sees the same show at the same time - no sunset for some while the Sun is rising for others when it's noon here and the stars are out there (there's always something, isn't there).
Similar problems with a plane diffuse screen.
Unless you can think of a system not covered here, or elect to suspend the laws of Physics, we're left with the original short answer.
There is not any single place on earth the source can be to cause the Sun to appear in the sky in the location it appears, from any location on earth, after being reflected from some surface in the sky. Period.
Sorry.
I tried.