It took me a while to derive the angle of incidence and then the rotation of the ellipse (Figure 3).
For the geometric optician: Consider isosceles triangle (-1,-2,0)(-1,-1,0)(0,0,0) formed by the lightpath. It is bisected by (-1,-1,0)(-1/2,-1,-1/2) into two 30-60-90 triangles. The bisector is orthogonal to the prism face, hence angle of incidence = 60*. Project the lightpath (-1,-1,0)(0,0,0) parallel to the bisector onto the prism face (-1,-1,0)(0,0,1)(-1,1,0), yielding (-1,-1,0)(-1/2,0,1/2). This is the orthogonal projection of the bending lightpath with respect to the reflective surface; i.e. the bending lightpath appears straight. Interestingly, the angle formed with the prism edge (which is parallel to the prism roof line) is alpha/2, half the cross-section apex angle of the prism (alpha = 2*arctan 1/sqrt 2).
In Figure 3, which shows the prism faces unfolded like a book opened flat, light travels parallel to the main diagonal line when viewed from above. When the faces are folded back up until they are orthogonal, and then tilted up 45*, the lightpath will bend 120* on each face and 90* in combination, with a fully inverted image and no image rotation. The view through an eyepiece will be Right Angle Correct Image.
To make a mirror version, I will attach Newtonian elliptical secondary mirrors to the inside of an aluminum angle, positioned according to Figure 3. Figure 4 shows "end caps" for the aluminum angle casing (also made from aluminum angle) with eyepiece barrel and helical focuser at the two ends. The helical focuser will be slightly repositionable to allow centering with a hollographic collimator.
The mirrors are respectively 2.6" and 2.3" minor axis, or 1.9" and 1.6" effective entrance/exit clear aperture for the 1:2 aspect-ratio ellipse. A single unit makes a full-aperture 2" RACI diagonal, which is not available on the market except from EMS for around US$1000 (each). A pair would make a binoscope back. Unlike the EMS 60*-elbow-pair design, interocular distance adjustment and co-collimation are left to the mount.
Another design, shown in Figure 5, simply uses collimatable Newtonian secondary fixtures with the mirror holders cut at 30* rather than 45*. Definitely not rocket science once the geometry and measurements are known.
Finally, HomeDepot sells 60* pipe elbows. It is a fun puzzle to give to someone: fit two 60* elbows together so the openings are perpendicular (hint below).
-- William Chang (Saratoga, CA)
(spoiler)
Hint: use the reversability principle -- in/out openings are symmetric -- so counter-rotate them in equal amounts until done.
