Abstract

When a ray is reflected successively on three mirrors that are approximately mutually perpendicular, the ray is returned nearly parallel to the original direction. If the original and final directions are denoted by unit vectors q and −t, and if a, b, c are the normals to the three mirrors taken in order in a right-handed sense, then it is shown theoretically that

t=q+2q×(+αa-βb+γc),

to the first order in α, β, γ, where these are the small angles by which the angles between the three mirrors exceed right angles. A geometrical construction to realize this formula is described.

If the three reflecting faces are parts of the surface of a symmetrical solid tetrahedron, refraction at the fourth surface increases the angular deviation of the final ray from the incident ray by a factor of μ cosr/cosi for the component of the deviation in the plane defined by the incident ray and the normal, and by a factor of μ for the perpendicular component. (i is the angle of incidence, and μ sinr=sini.)

Finally, it is shown theoretically that this type of reflecting unit (with a curved refracting surface), when used at the rear of road vehicles, could have greater efficiency if one of the angles between the reflecting surfaces were 90.14 deg instead of 90 deg.

© 1960 Optical Society of America

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