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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References

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  1. P. R. Yoder, J. Opt. Soc. Am. 48, 496–499 (1958).
    [Crossref]
  2. K. N. Chandler and J. A. Reid, “Reflex reflectors,” (Her Majesty’s Stationery Office, London, 1958), p. 9.
  3. G. A. Van Lear, J. Opt. Soc. Am. 30, 464–465 (1940).
    [Crossref]
  4. See reference 2, p. 2.

1958 (1)

1940 (1)

G. A. Van Lear, J. Opt. Soc. Am. 30, 464–465 (1940).
[Crossref]

Chandler, K. N.

K. N. Chandler and J. A. Reid, “Reflex reflectors,” (Her Majesty’s Stationery Office, London, 1958), p. 9.

Reid, J. A.

K. N. Chandler and J. A. Reid, “Reflex reflectors,” (Her Majesty’s Stationery Office, London, 1958), p. 9.

Van Lear, G. A.

G. A. Van Lear, J. Opt. Soc. Am. 30, 464–465 (1940).
[Crossref]

Yoder, P. R.

J. Opt. Soc. Am. (2)

P. R. Yoder, J. Opt. Soc. Am. 48, 496–499 (1958).
[Crossref]

G. A. Van Lear, J. Opt. Soc. Am. 30, 464–465 (1940).
[Crossref]

Other (2)

See reference 2, p. 2.

K. N. Chandler and J. A. Reid, “Reflex reflectors,” (Her Majesty’s Stationery Office, London, 1958), p. 9.

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Figures (3)

Fig. 1
Fig. 1

View of a triple mirror system looking along an incident ray, which makes an angle of 30° with the symmetrical direction. The circled dots show the angular magnitudes (on the given scale) and directions, of the six reflected rays arising from errors in the angles of the corner-cubes of +4 min about OA, +5 min about OB, and −6 min about OC.

Fig. 2
Fig. 2

Unit sphere showing relation of external incident, refracted, reflected, and emergent rays, defined by the normals at p, q, t, u, for an inaccurate corner-cube reflector. PN/QN=μ=UN/TN. This shows that the radial and transverse components of the deviation pu of the reflected ray outside the reflector are μ cosr/cosi and μ times those of qt, provided qt is small.

Fig. 3
Fig. 3

Projection of the three-dimensional model O(ABC) onto the plane which is normal to the symmetrical direction; corresponding to a solid corner-cube reflector which deflects internal rays upwards or downwards by 0.34/μ deg (OD′ or OD″). The construction shows, from the position of H (OH=OD′), that α, β, γ in Eq. (8) should be 0, +0.14°, respectivly, relative to the mutally perpendicular axes OA, OB, OC.

Equations (15)

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t = q + 2 q × ( + α a - β b + γ c ) ,
a = b × c / Δ ,             b = c × a / Δ ,             c = a × b / Δ ,             Δ = [ abc ] ,
a · a = 1 ,             a · b = 0 ,             etc. , Δ 2 = 1 - ( b · c ) 2 - ( c · a ) 2 - ( a · b ) 2 + 2 b · c c · a a · b .
q 1 = q - 2 q · aa q 2 = q 1 - 2 q 1 · bb q 3 = q 2 - 2 q 2 · cc } ,
q 3 = q - 2 q · aa - 2 q · bb - 2 q · cc + 4 q · a a · bb + 4 q · a a · cc + 4 q · b b · cc - 8 q · a a · b b · cc .
a = a + a · bb + a · cc q = q · aa + q · bb + q · cc ,
q · aa + q · bb + q · cc = q + q · a a · bb + q · a a · cc + q · b b · aa + q · b b · c c + q · c c · aa + q · c c · bb .
q × a = ( - q · bc + q · cb ) / Δ ,
q 3 = - q + 2 Δ ( + b · ca - c · ab + a · bc ) × q + 2 q · b b · a ( a - a ) + 2 q · c c · a ( a - a ) + 2 q · c c · b ( b - b ) + 2 q · a a · b ( b - b ) + 2 q · a a · c ( c - c ) + 2 q · b b · c ( c - c ) - 8 q · a a · b b · cc .
b · c = cos ( 1 2 π - α ) = sin α .
t = q + 2 q × ( + α a - β b + γ c ) .
P N = μ Q N             and             U N = μ T N .
l u = L U = μ M T = μ m t
p l = P L / cos i = μ Q M / cos i = μ q m cos r / cos i .
2 β ( 2 3 ) 1 2 = angle represented by O H in the plane of Fig . 3 = angle represented by O D in the plane of Fig . 3 = 0.34 / μ deg .