Abstract

First-order imaging properties are often represented in the form of a derivative matrix. This representation is not always expedient, however, since the elements of the matrix are not all independent; some elements can be written as functions of the others. Ideally, the first-order imaging properties should be represented without any redundant (and, therefore, possibly inconsistent) information. Further, it is convenient to characterize these properties in terms of entities with direct geometric interpretations. Hamilton’s methods are used here to determine a minimal set of geometric entities that is sufficient to characterize the first-order imaging properties of asymmetric systems. Although certain aspects of this problem have been discussed elsewhere, a particular facet has been consistently misinterpreted. This issue is resolved here by establishing that there is no unique first-order image plane for any optical system—regardless of symmetry.

© 1992 Optical Society of America

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References

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  1. See, for example, H. A. Buchdahl, An Introduction to Hamiltonian Optics (Cambridge U. Press, Cambridge, 1970), Sec. 10.
  2. J. Larmor, “The characteristics of an asymmetric optical combination,” Proc. Lond. Math. Soc. 20, 181–194 (1889).
  3. J. Larmor, “The simplest specification of a given optical path, and the observations required to determine it,” Proc. Lond. Math. Soc. 23, 165–173 (1892).
  4. R. K. Luneburg, Mathematical Theory of Optics (U. California Press, Los Angeles, Calif., 1964), Sec. 36.
  5. M. Herzberger, “Die Hauptsätze der Abbildung der Umgebung eines Strahls in allgemeinen optischen Systemen,” Zeits. Instrumentenk. 54, 336–350, 381–392, 429–441. (1934).
  6. M. Herzberger, “First-order laws in asymmetrical optical systems—part I. The image of a given congruence: fundamental conceptions,”J. Opt. Soc. Am. 26, 254–359 (1936).
  7. See Ref. 5 or M. Herzberger, “First-order laws in asymmetrical optical systems—II. The image congruences belonging to the rays emerging from a point in object and image space: fundamental forms,” J. Opt. Soc. Am. 26, 389–406 (1936).
    [Crossref]
  8. P. J. Sands, “First-order optics of the general optical system,”J. Opt. Soc. Am. 62, 369–372 (1972).
    [Crossref]
  9. A discussion of this problem and a list of references is given in P. J. Sands, “When is first-order optics meaningful?” J. Opt. Soc. Am. 58, 1365–1368 (1968). While Sands recognized the confusion, his resolution is incomplete.
    [Crossref]
  10. More precisely, this is a restricted mixed characteristic. For a description of restricted characteristic functions, see G. W. Forbes, “New class of characteristic functions in Hamiltonian optics,” J. Opt. Soc. Am. 72, 1698–1701 (1982). The convention is adopted here that all characteristic functions are denoted by C, and superscripts are used to distinguish one characteristic function from another. Specifically, the superscripts are related to the two arguments that appear in the characteristic function. If the first argument of the characteristic function is a position variable, then the first superscript is a 0, and if the first argument is a direction variable, the first superscript is a 1. Similarly for the second superscript and the second argument. See B. D. Stone, G. W. Forbes, “Foundations of first-order layout for asymmetric systems: an application of Hamilton’s methods,” J. Opt. Soc. Am. A 9, 96–109 (1992) for an explanation of the advantages of this notation. Note that in order for C01(y, p′) to exist, the anterior base plane cannot contain a front focal point.
    [Crossref]
  11. The exclusion of cases where ℳis singular precludes only those systems for which the derivative of the base ray configuration in image space with respect to the base ray configuration in object space is ill defined. Such cases occur, for example, when the base ray grazes an interface.
  12. G. W. Forbes, “New class of characteristic functions in Hamiltonian optics,”J. Opt. Soc. Am. 72, 1698–1701 (1982).
    [Crossref]
  13. It is commonly stated that ten, rather than eleven, entities are required for a minimal complete set. [This is done, for example, by T. Smith in “The primordial coefficients of asymmetrical lenses,” Trans. Opt. Soc. 29, 167–178 (1928)]. It appears that the ratio of indices of refraction in object and image space is generally ignored as a first-order imaging property. However, this ratio affects the imaging properties to all orders (including first order) and so is included here.
    [Crossref]
  14. B. D. Stone, G. W. Forbes, “Foundations of first-order layout for asymmetric systems: an application of Hamilton’s methods,” J. Opt. Soc. Am. A 9, 96–109 (1992). It is shown in this paper that, in general, three surfaces, separated by homogeneous media, are adequate for the design of a system that possesses any given set of these eleven first-order properties.
    [Crossref]
  15. The form of the characteristic function written in Eqs. (3.1) is a simple consequence of the axial symmetry about the base ray. The appearance of N in Eq. (3.1b) accounts for the form taken by the characteristic function (for an axially symmetric system) when the Xand the X′ axes are antiparallel, as opposed to parallel. With the conventions described here, the Xand X′ axes are antiparallel only when the base ray undergoes an odd number of reflections. For more details see, for example, Ref. 1, Secs. 10 and 14.
  16. As defined in Eq. (3.6), the magnification is independent of the number of reflecting surfaces present in the system. If the magnification is defined as the ratio between zd′ and z, the sign on the magnification depends on whether an even or an odd number of mirrors are present in the system. Similarly, the back focal length could be defined as the ratio between ydand βy′. However, if this is done, the definition of magnification would have to be changed to the alternative definition just mentioned; otherwise it would not be possible to determine the value of (−1)r. Also note that when the focal length is chosen to be negative 1 times the ratio between zdand βz′, the focal length of what is commonly referred to as a single, positive, refracting element is itself positive.
  17. The constant ais simply the tangent of the angle between the normal to the tilted plane and the base ray. The tilted plane intersects the plane x= 0 in a line, and ωrepresents the angle between the Yaxis and this line of intersection. The constants a′ and ω′ have similar interpretations. Note that a′ and ω′ are related to a′ of Eq. (3.9):a′2=a′2,tan(ω′)=(az′/ay′).
  18. Including the term sign(m) in this definition ensures that sign(m†) equals sign(m).
  19. For a description of the Scheimpflug condition see, for example, R. Kingslake, Optical System Design (Academic, Orlando, Fla., 1983), Chap. 5, Sec. VI. Briefly, the Scheimpflug condition states that an axially symmetric system forms the sharpest image with tilted object and image planes when the following two conditions hold:ω′-(-1)rω=π[1-sign(M)]/2,a′=∣M∣a.
  20. Keystone distortion is described by Luneburg in Sec. 34 of Ref. 4.
  21. See, for example, R. Kingslake, Lens Design Fundamentals, (Academic, Orlando, Fla., 1978), p. 185. Also see the references cited in Ref. 9.
  22. See, for example, Ref. 9. As another example, Buchdahl (Ref. 1, p. 29) states that “the angles between the focal lines and ℜ [the base ray] cannot be calculated on the basis of the quadratic terms of V[a characteristic function] alone.” This statement suggests that there are unique image lines but that to find the angle between these lines and the base ray requires higher-order terms. A similar discussion of focal lines in the context of first-order optics is presented in M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980), Sec. 4.6.1.
  23. The focal length, l1, associated with the plane x= Δ1can be defined analogously: l1=sign[Det(M˜)](μ/F¯˜11)(M˜112+M˜122)1/2.However, if this quantity is used in place of l2as the eleventh geometric entity, the procedure discussed in Subsection 4.B for measuring sign(−M˜22) {which is equal to sign[Det(M˜)]} less straightforward.

1992 (1)

1982 (2)

1972 (1)

1968 (1)

1936 (2)

M. Herzberger, “First-order laws in asymmetrical optical systems—part I. The image of a given congruence: fundamental conceptions,”J. Opt. Soc. Am. 26, 254–359 (1936).

See Ref. 5 or M. Herzberger, “First-order laws in asymmetrical optical systems—II. The image congruences belonging to the rays emerging from a point in object and image space: fundamental forms,” J. Opt. Soc. Am. 26, 389–406 (1936).
[Crossref]

1934 (1)

M. Herzberger, “Die Hauptsätze der Abbildung der Umgebung eines Strahls in allgemeinen optischen Systemen,” Zeits. Instrumentenk. 54, 336–350, 381–392, 429–441. (1934).

1928 (1)

It is commonly stated that ten, rather than eleven, entities are required for a minimal complete set. [This is done, for example, by T. Smith in “The primordial coefficients of asymmetrical lenses,” Trans. Opt. Soc. 29, 167–178 (1928)]. It appears that the ratio of indices of refraction in object and image space is generally ignored as a first-order imaging property. However, this ratio affects the imaging properties to all orders (including first order) and so is included here.
[Crossref]

1892 (1)

J. Larmor, “The simplest specification of a given optical path, and the observations required to determine it,” Proc. Lond. Math. Soc. 23, 165–173 (1892).

1889 (1)

J. Larmor, “The characteristics of an asymmetric optical combination,” Proc. Lond. Math. Soc. 20, 181–194 (1889).

Born, M.

See, for example, Ref. 9. As another example, Buchdahl (Ref. 1, p. 29) states that “the angles between the focal lines and ℜ [the base ray] cannot be calculated on the basis of the quadratic terms of V[a characteristic function] alone.” This statement suggests that there are unique image lines but that to find the angle between these lines and the base ray requires higher-order terms. A similar discussion of focal lines in the context of first-order optics is presented in M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980), Sec. 4.6.1.

Buchdahl, H. A.

See, for example, H. A. Buchdahl, An Introduction to Hamiltonian Optics (Cambridge U. Press, Cambridge, 1970), Sec. 10.

Forbes, G. W.

B. D. Stone, G. W. Forbes, “Foundations of first-order layout for asymmetric systems: an application of Hamilton’s methods,” J. Opt. Soc. Am. A 9, 96–109 (1992). It is shown in this paper that, in general, three surfaces, separated by homogeneous media, are adequate for the design of a system that possesses any given set of these eleven first-order properties.
[Crossref]

More precisely, this is a restricted mixed characteristic. For a description of restricted characteristic functions, see G. W. Forbes, “New class of characteristic functions in Hamiltonian optics,” J. Opt. Soc. Am. 72, 1698–1701 (1982). The convention is adopted here that all characteristic functions are denoted by C, and superscripts are used to distinguish one characteristic function from another. Specifically, the superscripts are related to the two arguments that appear in the characteristic function. If the first argument of the characteristic function is a position variable, then the first superscript is a 0, and if the first argument is a direction variable, the first superscript is a 1. Similarly for the second superscript and the second argument. See B. D. Stone, G. W. Forbes, “Foundations of first-order layout for asymmetric systems: an application of Hamilton’s methods,” J. Opt. Soc. Am. A 9, 96–109 (1992) for an explanation of the advantages of this notation. Note that in order for C01(y, p′) to exist, the anterior base plane cannot contain a front focal point.
[Crossref]

G. W. Forbes, “New class of characteristic functions in Hamiltonian optics,”J. Opt. Soc. Am. 72, 1698–1701 (1982).
[Crossref]

Herzberger, M.

M. Herzberger, “First-order laws in asymmetrical optical systems—part I. The image of a given congruence: fundamental conceptions,”J. Opt. Soc. Am. 26, 254–359 (1936).

See Ref. 5 or M. Herzberger, “First-order laws in asymmetrical optical systems—II. The image congruences belonging to the rays emerging from a point in object and image space: fundamental forms,” J. Opt. Soc. Am. 26, 389–406 (1936).
[Crossref]

M. Herzberger, “Die Hauptsätze der Abbildung der Umgebung eines Strahls in allgemeinen optischen Systemen,” Zeits. Instrumentenk. 54, 336–350, 381–392, 429–441. (1934).

Kingslake, R.

For a description of the Scheimpflug condition see, for example, R. Kingslake, Optical System Design (Academic, Orlando, Fla., 1983), Chap. 5, Sec. VI. Briefly, the Scheimpflug condition states that an axially symmetric system forms the sharpest image with tilted object and image planes when the following two conditions hold:ω′-(-1)rω=π[1-sign(M)]/2,a′=∣M∣a.

See, for example, R. Kingslake, Lens Design Fundamentals, (Academic, Orlando, Fla., 1978), p. 185. Also see the references cited in Ref. 9.

Larmor, J.

J. Larmor, “The simplest specification of a given optical path, and the observations required to determine it,” Proc. Lond. Math. Soc. 23, 165–173 (1892).

J. Larmor, “The characteristics of an asymmetric optical combination,” Proc. Lond. Math. Soc. 20, 181–194 (1889).

Luneburg, R. K.

R. K. Luneburg, Mathematical Theory of Optics (U. California Press, Los Angeles, Calif., 1964), Sec. 36.

Sands, P. J.

Smith, T.

It is commonly stated that ten, rather than eleven, entities are required for a minimal complete set. [This is done, for example, by T. Smith in “The primordial coefficients of asymmetrical lenses,” Trans. Opt. Soc. 29, 167–178 (1928)]. It appears that the ratio of indices of refraction in object and image space is generally ignored as a first-order imaging property. However, this ratio affects the imaging properties to all orders (including first order) and so is included here.
[Crossref]

Stone, B. D.

Wolf, E.

See, for example, Ref. 9. As another example, Buchdahl (Ref. 1, p. 29) states that “the angles between the focal lines and ℜ [the base ray] cannot be calculated on the basis of the quadratic terms of V[a characteristic function] alone.” This statement suggests that there are unique image lines but that to find the angle between these lines and the base ray requires higher-order terms. A similar discussion of focal lines in the context of first-order optics is presented in M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980), Sec. 4.6.1.

J. Opt. Soc. Am. (6)

M. Herzberger, “First-order laws in asymmetrical optical systems—part I. The image of a given congruence: fundamental conceptions,”J. Opt. Soc. Am. 26, 254–359 (1936).

See Ref. 5 or M. Herzberger, “First-order laws in asymmetrical optical systems—II. The image congruences belonging to the rays emerging from a point in object and image space: fundamental forms,” J. Opt. Soc. Am. 26, 389–406 (1936).
[Crossref]

P. J. Sands, “First-order optics of the general optical system,”J. Opt. Soc. Am. 62, 369–372 (1972).
[Crossref]

A discussion of this problem and a list of references is given in P. J. Sands, “When is first-order optics meaningful?” J. Opt. Soc. Am. 58, 1365–1368 (1968). While Sands recognized the confusion, his resolution is incomplete.
[Crossref]

More precisely, this is a restricted mixed characteristic. For a description of restricted characteristic functions, see G. W. Forbes, “New class of characteristic functions in Hamiltonian optics,” J. Opt. Soc. Am. 72, 1698–1701 (1982). The convention is adopted here that all characteristic functions are denoted by C, and superscripts are used to distinguish one characteristic function from another. Specifically, the superscripts are related to the two arguments that appear in the characteristic function. If the first argument of the characteristic function is a position variable, then the first superscript is a 0, and if the first argument is a direction variable, the first superscript is a 1. Similarly for the second superscript and the second argument. See B. D. Stone, G. W. Forbes, “Foundations of first-order layout for asymmetric systems: an application of Hamilton’s methods,” J. Opt. Soc. Am. A 9, 96–109 (1992) for an explanation of the advantages of this notation. Note that in order for C01(y, p′) to exist, the anterior base plane cannot contain a front focal point.
[Crossref]

G. W. Forbes, “New class of characteristic functions in Hamiltonian optics,”J. Opt. Soc. Am. 72, 1698–1701 (1982).
[Crossref]

J. Opt. Soc. Am. A (1)

Proc. Lond. Math. Soc. (2)

J. Larmor, “The characteristics of an asymmetric optical combination,” Proc. Lond. Math. Soc. 20, 181–194 (1889).

J. Larmor, “The simplest specification of a given optical path, and the observations required to determine it,” Proc. Lond. Math. Soc. 23, 165–173 (1892).

Trans. Opt. Soc. (1)

It is commonly stated that ten, rather than eleven, entities are required for a minimal complete set. [This is done, for example, by T. Smith in “The primordial coefficients of asymmetrical lenses,” Trans. Opt. Soc. 29, 167–178 (1928)]. It appears that the ratio of indices of refraction in object and image space is generally ignored as a first-order imaging property. However, this ratio affects the imaging properties to all orders (including first order) and so is included here.
[Crossref]

Zeits. Instrumentenk. (1)

M. Herzberger, “Die Hauptsätze der Abbildung der Umgebung eines Strahls in allgemeinen optischen Systemen,” Zeits. Instrumentenk. 54, 336–350, 381–392, 429–441. (1934).

Other (12)

See, for example, H. A. Buchdahl, An Introduction to Hamiltonian Optics (Cambridge U. Press, Cambridge, 1970), Sec. 10.

R. K. Luneburg, Mathematical Theory of Optics (U. California Press, Los Angeles, Calif., 1964), Sec. 36.

The exclusion of cases where ℳis singular precludes only those systems for which the derivative of the base ray configuration in image space with respect to the base ray configuration in object space is ill defined. Such cases occur, for example, when the base ray grazes an interface.

The form of the characteristic function written in Eqs. (3.1) is a simple consequence of the axial symmetry about the base ray. The appearance of N in Eq. (3.1b) accounts for the form taken by the characteristic function (for an axially symmetric system) when the Xand the X′ axes are antiparallel, as opposed to parallel. With the conventions described here, the Xand X′ axes are antiparallel only when the base ray undergoes an odd number of reflections. For more details see, for example, Ref. 1, Secs. 10 and 14.

As defined in Eq. (3.6), the magnification is independent of the number of reflecting surfaces present in the system. If the magnification is defined as the ratio between zd′ and z, the sign on the magnification depends on whether an even or an odd number of mirrors are present in the system. Similarly, the back focal length could be defined as the ratio between ydand βy′. However, if this is done, the definition of magnification would have to be changed to the alternative definition just mentioned; otherwise it would not be possible to determine the value of (−1)r. Also note that when the focal length is chosen to be negative 1 times the ratio between zdand βz′, the focal length of what is commonly referred to as a single, positive, refracting element is itself positive.

The constant ais simply the tangent of the angle between the normal to the tilted plane and the base ray. The tilted plane intersects the plane x= 0 in a line, and ωrepresents the angle between the Yaxis and this line of intersection. The constants a′ and ω′ have similar interpretations. Note that a′ and ω′ are related to a′ of Eq. (3.9):a′2=a′2,tan(ω′)=(az′/ay′).

Including the term sign(m) in this definition ensures that sign(m†) equals sign(m).

For a description of the Scheimpflug condition see, for example, R. Kingslake, Optical System Design (Academic, Orlando, Fla., 1983), Chap. 5, Sec. VI. Briefly, the Scheimpflug condition states that an axially symmetric system forms the sharpest image with tilted object and image planes when the following two conditions hold:ω′-(-1)rω=π[1-sign(M)]/2,a′=∣M∣a.

Keystone distortion is described by Luneburg in Sec. 34 of Ref. 4.

See, for example, R. Kingslake, Lens Design Fundamentals, (Academic, Orlando, Fla., 1978), p. 185. Also see the references cited in Ref. 9.

See, for example, Ref. 9. As another example, Buchdahl (Ref. 1, p. 29) states that “the angles between the focal lines and ℜ [the base ray] cannot be calculated on the basis of the quadratic terms of V[a characteristic function] alone.” This statement suggests that there are unique image lines but that to find the angle between these lines and the base ray requires higher-order terms. A similar discussion of focal lines in the context of first-order optics is presented in M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980), Sec. 4.6.1.

The focal length, l1, associated with the plane x= Δ1can be defined analogously: l1=sign[Det(M˜)](μ/F¯˜11)(M˜112+M˜122)1/2.However, if this quantity is used in place of l2as the eleventh geometric entity, the procedure discussed in Subsection 4.B for measuring sign(−M˜22) {which is equal to sign[Det(M˜)]} less straightforward.

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Figures (3)

Fig. 1
Fig. 1

Schematic representation of the coordinate systems in both object and image space. A general ray is shown intersecting the anterior base plane at the point with y and z coordinates represented by y. The y- and z-direction cosines of the ray at this point are denoted by β. The point of intersection and the direction of this ray at the posterior base plane are written as y′ and β′. The direction cosines in object space (α, β) satisfy the relation (α2 + β2) = 1. A similar relation holds for (α′, β′).

Fig. 2
Fig. 2

Transfer of a ray from the plane x = Δ 1 to the plane x = ( Δ 1 + a T · y ). The vector β′ represents the y′- and z′-direction cosines of the ray. The transverse coordinate of the ray on the plane x = Δ 1 is y , while the ray intersects the plane x = ( Δ 1 + a T · y ) at the point with coordinates ( x rot , y rot ). The distance between the points ( Δ 1 , y ) and ( x rot , y rot ) is denoted by L.

Fig. 3
Fig. 3

Left, a test object that can be used to determine the direction of the positive Y ˜ axis, m1, m2, ζ1, and ζ2. It is shown here in one of the two orientations (which differ by π) where the lines on the test object are imaged sharply on the plane x = Δ 1 (shown at right).

Tables (1)

Tables Icon

Table 1 Expressions for Geometric Entities in Terms of μ and the Elements of F ¯ , B ¯,

Equations (75)

Equations on this page are rendered with MathJax. Learn more.

( y β ) = D b ( y β ) + O ( 2 ) ,
D b = [ y y y β β y β β ] | y = 0 , β = 0
p = n β ,
p = n β ,
C 01 ( y , p ) = c 0 + ½ y T F y + y T M p + ½ p T B p + O ( 3 ) ,
F = [ F 11 F 12 F 12 F 22 ] ,
M = [ M 11 M 12 M 21 M 22 ] ,
B = [ B 11 B 12 B 12 B 22 ] .
y = - ( C 01 p ) T = - ( M T y + B p ) + O ( 2 ) ,
p = - ( C 01 y ) T = ( - F y + M p ) + O ( 2 ) .
y = ( B M - 1 F - M T ) y + n B M - 1 β + O ( 2 ) ,
β = - 1 n M - 1 F y - n n M - 1 β + O ( 2 ) .
D b = [ ( B M - 1 F - M T ) n B M - 1 - 1 n M - 1 F - n n M - 1 ] .
F ¯ : = 1 n F ,
B ¯ : = n B ,
D b = 1 μ [ ( B ¯ M - 1 F ¯ - μ M T ) B ¯ M - 1 - M - 1 F ¯ - M - 1 ] ,
μ : = n / n .
y d ( Δ ; y , β ) = y + Δ β = 1 μ { [ ( B ¯ - Δ I ) M - 1 F ¯ - μ M T ] y + [ ( B ¯ - Δ I ) M - 1 ] β } ,
y d ( Δ ; y , β ) = y + Δ β = [ ( Δ F ¯ - I ) ( M T ) - 1 ] y + [ ( Δ F ¯ - I ) ( M T ) - 1 B ¯ - μ Δ M ] β .
F = F I ,
M = M N ,
B = B I ,
N : = [ 1 0 0 ( - 1 ) r ] .
y d ( Δ ; y , β ) = y + Δ β = N 1 μ M { [ ( B ¯ - Δ ) F ¯ - μ M 2 ] y + ( B ¯ - Δ ) β } ,
y d ( Δ ; y , β ) = y + Δ β = N 1 M { [ ( Δ F ¯ - 1 ) y + [ B ¯ ( Δ F ¯ - 1 ) - μ Δ M 2 ] β } .
Δ 1 = B ¯ .
Δ 1 = 1 / F ¯ .
m = - M .
l = ( - 1 ) r μ M F ¯ .
B ¯ = Δ 1 .
F ¯ = 1 Δ 1 ,
M = - m .
( - 1 ) r μ = - l 1 m Δ 1 .
x = ( a T · y + Δ 1 ) ,
( x rot y rot ) = [ Δ 1 + L ( 1 - β 2 ) 1 / 2 y + L β ] ,
L = a T · y ( 1 - β 2 ) 1 / 2 - a T · β .
( x rot y rot ) = ( Δ 1 + a T · y y ) + O ( 2 ) = [ Δ 1 - M ( a T N y ) - M N y ] + O ( 2 ) ,
h : = ( x rot 2 + y rot 2 ) 1 / 2 ,
h : = [ ( x rot - Δ 1 ) 2 + y rot 2 ] 1 / 2 .
m : = sign ( m ) h h | y rot = 0 ,
sign ( ξ ) : = { 1 ξ > 0 - 1 ξ < 0 .
m ( ζ ) = - M [ 1 + a 2 cos 2 ( ζ - ω ) 1 + a 2 cos 2 ( ζ - ω ) ] 1 / 2 .
y = R ( φ ) y ˜ ,
R ( φ ) : = [ cos ( φ ) - sin ( φ ) sin ( φ ) cos ( φ ) ] .
F ˜ = R T ( φ ) F R ( φ ) .
M ˜ = R T ( φ ) M R ( φ ) .
y d ( Δ ; 0 , β ) = 1 μ [ ( B ¯ - Δ I ) M - 1 ] β .
Δ = 1 2 { ( B ¯ 11 + B ¯ 22 ) ± [ ( B ¯ 11 - B ¯ 22 ) 2 + 4 B ¯ 12 2 ] 1 / 2 } .
φ = ½ tan - 1 ( 2 B 12 B 11 - B 22 ) .
Δ 1 = B ¯ ˜ 11 ,
Δ 2 = B ¯ ˜ 22 ,
y ˜ d ( Δ 1 ; y ˜ , β ˜ ) = { - ( M ˜ 11 , M ˜ 21 ) · y ˜ ( B ¯ ˜ 22 - B ¯ ˜ 11 ) μ det ( M ˜ ) [ ( - M ˜ 21 F ¯ ˜ 11 , M ˜ 11 F ¯ ˜ 22 ) · y ˜ + ( - M ˜ 21 , M ˜ 11 ) · β ˜ ] - ( M ˜ 12 , M ˜ 22 ) · y ˜ } .
( x ˜ rot , 1 y ˜ rot , 1 ) = [ Δ 1 + a ˜ T · y ˜ d ( Δ 1 ; y ˜ , β ˜ ) y ˜ d ( Δ 1 ; y ˜ , β ˜ ) ] + O ( 2 ) ,
φ = 1 2 tan - 1 ( 2 F 12 F 11 - F 22 ) ,
y ˜ d ( Δ ; y ˜ , 0 ) = [ ( Δ F ¯ ˜ - I ) ( M ˜ T ) - 1 ] y ˜ .
Δ 1 = 1 / F ¯ ˜ 11 ,
Δ 2 = 1 / F ¯ ˜ 22 .
d 1 = a [ M ˜ 11 cos ( ζ ) + M ˜ 21 sin ( ζ ) ] .
ζ 1 = tan - 1 ( M ˜ 21 / M ˜ 11 ) ,
m 1 = ( M ˜ 21 2 + M ˜ 11 2 ) 1 / 2 .
ζ 2 = tan - 1 ( M ˜ 22 / M ˜ 12 ) ,
m 2 = ( M ˜ 22 2 + M ˜ 12 2 ) 1 / 2 .
d 2 = ( μ / F ¯ ˜ 22 ) a [ M ˜ 21 cos ( ζ ) + M ˜ 22 sin ( ζ ) ] .
l 2 = sign [ Det ( M ˜ ) ] μ F ¯ ˜ 22 ( M ˜ 21 2 + M ˜ 22 2 ) 1 / 2 .
F ¯ = R ( φ ) [ Δ 1 0 0 Δ 2 ] - 1 R T ( φ ) ,
B ¯ = R ( φ ) [ Δ 1 0 0 Δ 2 ] R T ( φ ) ,
M = - R ( φ ) [ m 1 cos ( ζ 1 ) sign [ l 2 Δ 2 sin ( ζ 2 ) ] m 2 cos ( ζ 2 ) m 1 sin ( ζ 1 ) sign ( l 2 Δ 2 ) m 2 sin ( ζ 2 ) ] × R T ( φ ) ,
μ = | l 2 Δ 2 | 1 [ m 1 2 sin 2 ( ζ 1 ) + m 2 2 sin 2 ( ζ 2 ) ] 1 / 2 ,
y d { Δ 1 ; [ a cos ( ζ ) , a sin ( ζ ) ] , β } = - a [ M 11 cos ( ζ ) + M 21 sin ( ζ ) M 12 cos ( ζ ) + M 22 sin ( ζ ) ] .
φ = 1 2 tan - 1 [ 2 ( M 11 M 12 + M 21 M 22 ) M 11 2 + M 21 2 - M 12 2 - M 22 2 ] .
M 11 = M 22 ,
M 12 = - sign ( M 11 M 22 ) M 21 .
tan ( φ ) = - sign ( M 22 M 21 ) M 21 / M 22 .
a2=a2,tan(ω)=(az/ay).
l1=sign[Det(M˜)](μ/F¯˜11)(M˜112+M˜122)1/2.

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