Abstract

An analysis is made of the degree to which a simple contorted mirror can reduce distortion in a projected image. Although it is impossible in general to remove the distortion completely by this means, significant reduction may be achieved. Equations are derived for the shape of a mirror which will give the maximum reduction possible. Equations are also developed for how to stress a planar mirror to create the desired optimal shape. Image blurring is estimated. An example is presented in which this technique is used to reduce the distortion which results when a projection lens appropriate for projection onto a flat screen is used instead to project onto a spherical surface from an oblique angle.

© 1985 Optical Society of America

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References

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  1. I. Whyte, A. W. Zeph, “Wide-Angle, Multiviewer, Infinity Display System,” AFHRL-TR-81-27(I);a report published by Air Force Human Resources Laboratory (1982).
  2. R. H. Freeman, J. E. Pearson, “Deformable Mirrors for All Seasons and Reasons,” Appl. Opt. 21, 580 (1982).
    [CrossRef] [PubMed]
  3. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  4. S. Timoshenko, Theory of Plates and Shells (McGraw-Hill, New York, 1940).

1982 (1)

Freeman, R. H.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

Pearson, J. E.

Timoshenko, S.

S. Timoshenko, Theory of Plates and Shells (McGraw-Hill, New York, 1940).

Whyte, I.

I. Whyte, A. W. Zeph, “Wide-Angle, Multiviewer, Infinity Display System,” AFHRL-TR-81-27(I);a report published by Air Force Human Resources Laboratory (1982).

Zeph, A. W.

I. Whyte, A. W. Zeph, “Wide-Angle, Multiviewer, Infinity Display System,” AFHRL-TR-81-27(I);a report published by Air Force Human Resources Laboratory (1982).

Appl. Opt. (1)

Other (3)

I. Whyte, A. W. Zeph, “Wide-Angle, Multiviewer, Infinity Display System,” AFHRL-TR-81-27(I);a report published by Air Force Human Resources Laboratory (1982).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

S. Timoshenko, Theory of Plates and Shells (McGraw-Hill, New York, 1940).

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Figures (14)

Fig. 1
Fig. 1

Additional deflection on reflection from a nonflat mirror.

Fig. 2
Fig. 2

Geometry for estimating deflection and blur. The mirror is located at a distance h from an intermediate image plane.

Fig. 3
Fig. 3

Normalized stress distribution (C′/D)p(x,y) as determined from p = (D/C′)∇2(∇ · Δ), Eq. (21), with ∇ · Δ of Eq. (33) for a projector at α = 25°.

Fig. 4
Fig. 4

Normalized moment distribution (C′/D)M(x,y) determined from M = −(D/C′) ∇ · Δ, Eq. (22), with ∇ · Δ of Eq. (33) for a projector at α = 25°.

Fig. 5
Fig. 5

Deformation w(x,y) of the correcting mirror, as given by Eq. (35) with C′ = 1, C1 = 0, α = 25°.

Fig. 6
Fig. 6

Distortion field Δ(x,y) = (Δxy) for a flat-screen projector at α = 25° on the dome surface, without the correction mirror. The solid grid pattern represents Δ(x,y) = 0.

Fig. 7
Fig. 7

Residual distortion field δ(x,y) = (δxy) for a flat-screen projector at α = 25° on the dome surface, with the correction mirror. The solid grid pattern represents δ(x,y) = 0.

Fig. 8
Fig. 8

Experimental setup.

Fig. 9
Fig. 9

Experiment: comparison of the desired image (dots) and uncorrected projected image in the lower-left quadrant.

Fig. 10
Fig. 10

Experiment: comparison of the desired image (dots) and corrected projected image in the lower-left quadrant.

Fig. 11
Fig. 11

Experiment: comparison of the desired image (dots) and uncorrected projected image in the upper-left quadrant.

Fig. 12
Fig. 12

Experiment: comparison of the desired image (dots) and corrected projected image in the upper-left quadrant.

Fig. 13
Fig. 13

Coordinate systems for describing the distorted image perceived by an observer at the center of a sphere, which results when a projector located on the sphere surface is used which would project an undistorted image only on a flat surface.

Fig. 14
Fig. 14

Relation of projector coordinates (up, υp)—dashed lines—and observer coordinates (uee)—solid line—as determined by Eq. (A7) for α = 25°.

Equations (71)

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x x C w .
x + Δ ( x , y ) x + Δ ( x , y ) C w ( x , y ) .
Δ ( x , y ) = × A ( x , y ) + B ( x , y ) .
δ dxdy [ ( Δ x C w / x ) 2 + ( Δ y C w / y ) 2 ] = 0 .
C 2 w ( x , y ) = Δ ( x , y ) .
C 2 w ( x , y ) = 2 B .
x + Δ x + × A + B B C w H = x + × A C w H ,
2 w H ( x , y ) = 0 .
δ dxdyP ( x , y ) ( C w Δ ) ( C w Δ ) = 0 .
C 2 w + C ln P w = Δ + Δ ln P .
Δ k ( k x , k y ) = exp ( i k x ) Δ ( x , y ) dxdy , k = ( k x , k y ) ,
k Δ k = i k ( k × A k ) + i ( 2 B k + k k B k ) ,
k × Δ k = i [ ( A k k ) k 2 A k ( k k ) A k + k ( k A k ) ] + i k B k × k .
M x = D ( 2 w x 2 + ν 2 w y 2 ) ,
M y = D ( 2 w y 2 + ν 2 w x 2 ) ,
M x y = M y x = D ( 1 ν ) 2 w x y ,
2 M x x 2 2 2 Mxy x y + 2 M y y 2 = p ,
D = E h 3 12 ( 1 ν 2 ) ,
ε x = σ x E ν σ y E , ε y = σ y E ν σ x E ,
Mxy = h / 2 h / 2 τ x y zdz ,
( 2 x 2 + 2 y 2 ) ( 2 w x 2 + 2 w y 2 ) = p D .
M = M x + M y 1 + ν ,
2 M x 2 + 2 M y 2 = p ,
2 w x 2 + 2 w y 2 = M D .
C 2 w ( x , y ) = Δ ( x , y ) .
p ( x , y ) = D C 2 [ Δ ( x , y ) ] ,
M ( x , y ) = D C Δ ( x , y ) .
deflection = h ( w / x ) .
ε = h I w x .
blur = h ( h β cos θ ) 2 w x 2 .
blur deflection ( h β cos θ ) ( 2 w x 2 / w x ) .
blur deflection h β I cos θ .
l s + l s = l f ,
β = O S = O f ( s s + 1 ) = O f ( I O + 1 ) .
blur deflection h f cos θ ( I O + 1 ) I O .
blur deflection ε O f cos θ w x ( I O + 1 ) .
Δ x = x [ C ( 1 + d ) cos α 2 y sin α 2 1 ] ,
Δ y = C ( 1 d ) sin α 2 + C y cos α 2 ( 1 + d ) cos α 2 y sin α 2 y ,
d = ( 1 + x 2 + y 2 ) 1 / 2 ,
Δ = G ( x , y ) [ ( 1 + d ) cos α 2 y sin α 2 ] 2 ,
G ( x , y ) = 4 cos 2 α 2 + C ( cos α 2 + 1 ) + d [ 4 cos 2 α 2 + C ( 1 + cos α 2 ) 2 C sin 2 α 2 ] 2 y 2 2 x 2 cos 2 α 2 + y ( 4 sin α 2 cos α 2 C sin α 2 ) 2 y C d sin α 2 cos α 2 + 4 y d sin α 2 cos α 2 + C [ y 2 d ( sin 2 α 2 cos 2 α 2 ) x 2 d cos α 2 ] .
w ( x , y ) = 1 4 π C d x d y Δ ( x , y ) × ln [ ( x x ) 2 + ( y y ) 2 ] .
δ x = Δ x w x ,
δ y = Δ y w x .
C 2 w ( x , y ) = Δ ( x , y )
p ( x , y ) = D C 2 [ Δ ( x , y ) ] ,
M ( x , y ) = D C Δ ( x , y ) .
k e = 1 1 + u e 2 + υ e 2 ,
l e = u e 1 + u e 2 + υ e 2 ,
m e = υ e 1 + u e 2 + υ e 2 .
x s = R k e ,
y s = R l e ,
z s = R m e .
x p = R cos α ,
y p = 0 ,
z p = R sin α .
k p = 1 b ( k e + cos α ) ,
l p = 1 b l e ,
m p = 1 b ( m e sin α ) ,
b = [ 2 ( 1 + k e cos α m e sin α ) ] 1 / 2 .
( k p l p m p ) = ( cos α / 2 0 sin α / 2 0 1 0 sin α / 2 0 cos α / 2 ) ( k p l p m p ) .
u p = l p k p ,
υ p = m p k p .
u p = u e ( 1 + d ) cos α 2 υ e sin α 2 ,
υ p = ( 1 d ) sin α 2 + υ e cos α 2 ( 1 + d ) cos α 2 υ e sin α 2 ,
d = 1 + u e 2 + υ e 2 .
Δ x = C u p u e ,
Δ y = C υ p υ e ,
Δ x = x [ C ( 1 + d ) cos α 2 y sin α 2 1 ] ,
Δ y = C ( 1 d ) sin α 2 + C y cos α 2 ( 1 + d ) cos α 2 y sin α 2 y ,
d = ( 1 + x 2 + y 2 ) 1 / 2 .

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