Abstract

To help in the fabrication of off-axis conic sections, we present a method to approximate this off-axis section by an on-axis conic centered on the portion desired. This method is based on the method of continuum least squares to obtain the vertex’s curvature and conic constant of the fitted conic on-axis, given the curvature at the vertex and the conic constant of the parent conic from where we want the section and the size of that section. Simple analytic expressions for the curvature and conic constant are derived in terms of the parameters of the off-axis section.

© 1986 Optical Society of America

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References

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  1. G. LeMaitre, “Compensation des Aberrations par Elasticite,” Nouv. Rev. Opt. 5, 361 (1974).
    [CrossRef]
  2. J. Lubliner, J. E. Nelson, “Stressed Mirror Polishing. 1: A Technique for Producing Nonaxisymmetric Mirrors,” Appl. Opt. 19, 2332 (1980).
    [CrossRef] [PubMed]
  3. A. S. Leonard, “The New Science of Tilted-Mirror Optics and Its Application to High Performance Reflecting Telescopes,” presented at Western Amateur Astronomers and Association of Lunar and Planetary Observers, San Diego (1969), p. 31.
  4. O. N. Stravroudis, The Optics of Rays, Wavefronts and Caustics (Academic, New York, 1976), pp. 88 and 91.

1980

1974

G. LeMaitre, “Compensation des Aberrations par Elasticite,” Nouv. Rev. Opt. 5, 361 (1974).
[CrossRef]

LeMaitre, G.

G. LeMaitre, “Compensation des Aberrations par Elasticite,” Nouv. Rev. Opt. 5, 361 (1974).
[CrossRef]

Leonard, A. S.

A. S. Leonard, “The New Science of Tilted-Mirror Optics and Its Application to High Performance Reflecting Telescopes,” presented at Western Amateur Astronomers and Association of Lunar and Planetary Observers, San Diego (1969), p. 31.

Lubliner, J.

Nelson, J. E.

Stravroudis, O. N.

O. N. Stravroudis, The Optics of Rays, Wavefronts and Caustics (Academic, New York, 1976), pp. 88 and 91.

Appl. Opt.

Nouv. Rev. Opt.

G. LeMaitre, “Compensation des Aberrations par Elasticite,” Nouv. Rev. Opt. 5, 361 (1974).
[CrossRef]

Other

A. S. Leonard, “The New Science of Tilted-Mirror Optics and Its Application to High Performance Reflecting Telescopes,” presented at Western Amateur Astronomers and Association of Lunar and Planetary Observers, San Diego (1969), p. 31.

O. N. Stravroudis, The Optics of Rays, Wavefronts and Caustics (Academic, New York, 1976), pp. 88 and 91.

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Figures (2)

Fig. 1
Fig. 1

Local coordinates of the off-axis section centered at the point (x0, o, z0) of a conic with paraxial curvature c and conic constant k.

Fig. 2
Fig. 2

Difference W between the sagittas with respect to the local coordinates x and y (in parentheses) for a paraboloid with c = 2.5 × 10−4 cm−1, R = 70 cm, k = −1, and a segment in x0 = 100 cm. The conic constant and curvature of the fitting conic are k′ = −0.998126 and c′ = 2.498489 × 10−4 cm−1. Only positive y coordinates are depicted due to the symmetry of the problem.

Equations (24)

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z = c ρ 2 1 + 1 - ( k + 1 ) c 2 ρ 2 ,
x = x cos θ - z sin θ + x 0 , y = y , z = x sin θ + z cos θ + z 0 .
z 0 = c x 0 2 1 + 1 - ( k + 1 ) c 2 x 0 2 ;
N ¯ = - z x i + k ^ 1 + z x 2 .
tan θ = z x ,
z x = c x 0 1 - ( k + 1 ) c 2 x 0 2 .
z = γ β + β 2 - α γ ,
α = c ( 1 + k cos 2 θ ) β = 1 1 + k sin 2 θ - c k sin θ cos θ x , γ = c ( 1 + k sin 2 θ ) x 2 + c y 2 ,
c x 0 = sin θ 1 + k sin 2 θ .             1 - ( k + 1 ) c z 0 = cos θ 1 + k sin 2 θ .
z = β α ( 1 - 1 - α γ β 2 ) .
z 1 2 ( γ β + 1 4 α γ 2 β 3 + 3 8 α 2 γ 3 β 5 + ) ;
z ½ c δ ( δ 2 x 2 + y 2 ) + ½ c 2 δ 2 k sin θ cos θ x [ δ 2 x 2 + y 2 ] + c 3 δ 3 { [ 4 k 2 sin 2 θ cos 2 θ + δ 2 ( 1 + k cos 2 θ ) ] x 2 + ( 1 - k cos 2 θ ) y 2 } ( δ 2 k 2 + y 2 ) + .
z A ½ c ( x 2 + y 2 ) + c 3 ( k + 1 ) ( x 2 + y 2 ) 3 +
x = ρ cos ϕ ,             y = ρ sin ϕ ,
W = z A - z ,
W = α 20 ρ 2 + α 22 ρ 2 cos 2 ϕ + α 31 ρ 3 cos ϕ + α 33 ρ 3 cos 3 ϕ + α 40 ρ 4 + α 42 ρ 4 cos 2 ϕ + α 44 ρ 4 cos 4 ϕ + ,
α 20 = c 2 [ c c - 1 2 δ ( 1 + δ 2 ) ]             focus , α 22 = ¼ c δ ( 1 - δ 2 )             astigmatism , α 31 = - c 2 δ 2 k sin θ cos θ ( 1 + 3 δ 2 )             coma , α 32 = c 2 δ 2 k sin θ cos θ ( 1 + δ 2 ) , α 40 = c 3 8 { ( k + 1 ) c 3 - 1 8 δ 3 [ a ( 1 + 3 δ 2 ) + b ( 3 + δ 2 ) ] }             spherical aberration , α 42 = - / 16 1 c 3 δ 3 ( a δ 2 - b ) , α 44 = - / 64 1 c 3 δ 3 ( b - a ) ( 1 - δ 2 ) ,
b = 1 + k cos 2 θ a = 4 k 2 sin 2 θ cos 2 θ + b δ 2 .
E = 2 0 π 0 R W 2 ρ d ρ d ϕ ,
E = π R 6 ( α 20 2 + ½ α 20 α 40 R 2 + α 22 2 + ¼ α 22 α 40 R 2 + α 31 2 R 2 + α 33 2 R 2 + α 40 2 R 4 + / 10 1 α 42 2 R 4 + / 10 1 α 44 2 R 4 ) .
/ 160 3 ( k + 1 ) R 2 4 c 5 + ( k + 1 ) R 2 c 3 - / 64 3 { c δ ( 1 + δ 2 ) + / 20 1 c 3 δ 3 [ a ( 1 + 3 δ 2 ) + b ( 3 + δ 2 ) ] R 2 } ( k + 1 ) R 2 c 2 + c - / 12 1 { c δ ( 1 + δ 2 ) + / 64 3 c 3 δ 3 [ a ( 1 + 3 δ 2 ) + b ( 3 + δ 2 ) ] R 2 } = 0 ,
( k + 1 ) R 2 c 3 + 5 c - / 2 5 { c δ ( 1 + δ 2 ) + / 20 1 c 3 δ 3 [ a ( 1 + 3 δ 2 ) + b ( 3 + δ 2 ) ] R 2 } = 0.
c = 1 / 2 c δ ( 1 + δ 2 ) ,
k = [ a ( 1 + 3 δ 2 ) + b ( 3 + δ 2 ) ] / ( 1 + δ 2 ) 3 .

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