Abstract

People standing under a large spherical mirror see the world inverted, hanging above them as a real image. The shape of this image world depends upon where on the floor the observer stands. In this paper formulas for calculating image positions as a function of observer position are derived and depicted in diagrams of typical image worlds. The images formed by light reflecting several times in the mirror are also calculated. There are both an erect real image world and doubled rings of inverted images surrounding the single-reflection image world.

© 1971 Optical Society of America

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References

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  1. J. W. Klüver, Appl. Opt. 10, 2573 (1971).
    [CrossRef]

1971 (1)

J. W. Klüver, Appl. Opt. 10, 2573 (1971).
[CrossRef]

Klüver, J. W.

J. W. Klüver, Appl. Opt. 10, 2573 (1971).
[CrossRef]

Appl. Opt. (1)

J. W. Klüver, Appl. Opt. 10, 2573 (1971).
[CrossRef]

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Figures (7)

Fig. 1
Fig. 1

Diagram for calculating the optics of a sphere.

Fig. 2
Fig. 2

Diagram for calculating image position.

Fig. 3
Fig. 3

Images of numbers seen by observer 0, to scale and in the proper orientation. Only rays reflecting once from the mirror are considered. Images in the region outside the dashed line require two reflections.

Fig. 4
Fig. 4

Diagram for calculating two reflections.

Fig. 5
Fig. 5

Complete set of images seen by observer 0 near the center of the sphere.

Fig. 6
Fig. 6

Ray tracing for images from 3 seen in Fig. 5.

Fig. 7
Fig. 7

Images seen by observer 0 near the mirror. The unlabeled mirror regions near the horizons are two-reflection regions.

Equations (7)

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1 / f = ( 1 / s 1 ) + ( 1 / s 2 ) .
Σ = ( y m + y 0 ) / ( x m - x 0 )             and             Ψ = y m / x m .
x n = [ ( y m + y n ) / Θ ] - x m .
Σ = ( y m - y i ) / ( x m - x i )             and             Σ = Σ + d Σ = ( y m - y i ) / ( x m - x i ) ,
x i = x m + Σ [ ( d x m / d Σ ) - ( d y m / d Σ ) ] , y i = y m - Σ ( d y m / d Σ ) + Σ 2 ( d x m / d Σ ) ,
Γ = ( y s + y 0 ) / ( x s - x 0 )             and             Ψ = y s / x s .
Θ = ( Γ - Ψ ) / ( 1 + Ψ Γ )             and             Ψ = ( Θ - Ψ ) / ( 1 + Θ Ψ ) .

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