Abstract

Reflection and refraction of light at internal and external faces of birefringent crystals are discussed using a method devised by Max Planck for treating a particular case. General cases where a plane-polarized incident beam gives rise to (i) two plane-polarized refracted beams and a single plane-polarized reflected beam at external face and (ii) one plane-polarized refracted beam and two plane-polarized reflected beams at internal face are studied. Every beam, whatever the azimuth of its plane of polarization, is considered as a whole, unlike the usual practice of splitting a beam into two components, one polarized in the plane of incidence and the other polarized in a plane perpendicular to the latter.

Study shows that when a plane-polarized incident beam gives rise to a single plane-polarized reflected beam and a single plane-polarized refracted beam, the magnetic fields of the three beams are coplanar and the plane of the magnetic fields make an angle Zˆb=tan1|csc(AˆBˆ)·csc(BˆCˆ)·sin2Bˆ·tanbˆ|, with the plane of polarization of the refracted beam, where Â, Bˆ, and Ĉ are, respectively, the angles of incidence refraction, and reflection and bˆ is the small angle between the refracted beam and its ray.

The main conclusions of the paper were verified with a cleaved rhomb of calcite using a spectrometer fitted with a polarizer and analyzer. The azimuths of the planes of polarization (i) αˆw and αˆt of the incident beam, (ii) γˆw and γˆt of the reflected beam, and (iii) βˆw and βˆt of the beams emerging through the rhomb, when there is only a single beam in the refracted light at the first surface are determined for a full range from +90° to −90° of the angle of incidence and for a fixed orientation of the rhomb.

The variation of γˆ, the azimuth of the plane of polarization of reflected beam, with the variation of αˆ, the azimuth of the plane of polarization of the incident beam at the first face of the rhomb is also determined. The coplanar law of the magnetic fields has also been verified for a case where both the incident and reflected beams were extra-ordinary beams inside calcite. Experimental results confirm all the theoretical conclusions. To aid the visualization of the variation of azimuths, graphs are drawn on spheres.

© 1967 Optical Society of America

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References

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  1. The incident beam is always named Ȧ, the refracted beam is named B˙, and the reflected beam is named Ċ.
  2. The subscript s indicates that the beam in the anisotropic medium is the slow beam for the given direction. The subscript attached to the associated beam, incident, reflected or refracted in an isotropic medium is w. Similarly, for the faster beam in an anisotropic medium the subscript added is f and the subscript t is added to the associated beam in the isotropic medium. When the incident beam is a slow beam in an anisotropic medium and it gives rise to a slow reflected beam, the subscript added is ss and when it gives rise to a fast reflected beam the subscript added is sf. In the same manner, the combinations of fs and ff are used. The same procedure is followed for the subscripts of directions, angles, magnetic fields, and electric fields. The extra subscript r is added for those connected with a ray.
  3. There are two different circumstances in which one of the internally reflected beams may be absent. In one case, the azimuths of the planes of polarization of the incident and reflected beams are such that all the reflected energy appears in one of the reflected beams and thus the intensity of illumination of the second reflected beam is zero. Such cases are treated in this paper. For a second case where one of the reflected beams is absent owing to “total refraction” of one component of the incident beam and where the corresponding refracted beam is elliptically polarized, see, V. C. Varghese, J. Opt. Soc. Am. 57, 90 (1967).
    [Crossref]
  4. Positive normal to a given face and for a given incidence is the direction of the incident beam when the corresponding reflected beam retraces the path of the incident beam. Thus, the direction of the positive normal depends on the side of the surface on which light is incident. All angles of incidence, refraction, and reflection are measured in the positive counterclockwise direction from the same normal.
  5. For experimental confirmation that the trajectory of the energy of light is the direction of the ray and not the direction of the beam, see, V. C. Varghese, J. Opt. Soc. Am. 57, 86 (1967).
    [Crossref]
  6. In Eq. (7d), when 90° > Â> 0°, both Ĉsf and Ĉff are between 90° and 180°. When 360° > Â> 270°, Ĉsf and Ĉff are between 180° and 270°.
  7. M. Planck, Theory of Light (Macmillan and Co., London, 1932), p. 165.
  8. I. Todhunter and J. G. Leathern, Spherical Trigonometry (Macmillan and Co., London, 1949), p. 111.

1967 (2)

J. Opt. Soc. Am. (2)

Other (6)

In Eq. (7d), when 90° > Â> 0°, both Ĉsf and Ĉff are between 90° and 180°. When 360° > Â> 270°, Ĉsf and Ĉff are between 180° and 270°.

M. Planck, Theory of Light (Macmillan and Co., London, 1932), p. 165.

I. Todhunter and J. G. Leathern, Spherical Trigonometry (Macmillan and Co., London, 1949), p. 111.

Positive normal to a given face and for a given incidence is the direction of the incident beam when the corresponding reflected beam retraces the path of the incident beam. Thus, the direction of the positive normal depends on the side of the surface on which light is incident. All angles of incidence, refraction, and reflection are measured in the positive counterclockwise direction from the same normal.

The incident beam is always named Ȧ, the refracted beam is named B˙, and the reflected beam is named Ċ.

The subscript s indicates that the beam in the anisotropic medium is the slow beam for the given direction. The subscript attached to the associated beam, incident, reflected or refracted in an isotropic medium is w. Similarly, for the faster beam in an anisotropic medium the subscript added is f and the subscript t is added to the associated beam in the isotropic medium. When the incident beam is a slow beam in an anisotropic medium and it gives rise to a slow reflected beam, the subscript added is ss and when it gives rise to a fast reflected beam the subscript added is sf. In the same manner, the combinations of fs and ff are used. The same procedure is followed for the subscripts of directions, angles, magnetic fields, and electric fields. The extra subscript r is added for those connected with a ray.

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Figures (10)

Fig. 1
Fig. 1

IJKL is a section of a cleaved rhomb of calcite by a plane perpendicular to all the four faces. N1 and N3 are the positive normals to the faces IJ and KL, respectively, and N2 is the outward drawn normal to the face JK. A, Bs, Bf, and C are the directions of the incident, refracted (slow), refracted (fast) and the reflected beam, respectively, at the first face IJ. The incident slow beam in the direction As(= Bs) gives rise to the reflected (slow), reflected (fast), and refracted beams in the directions Css, Csf, and Bw, respectively. Similarly, an incident fast beam in the direction Af(= Bf) gives rise to the reflected slow, reflected fast, and refracted beams, respectively, in the directions Cfs, Cff, and Bt. All the beams are plane polarized.

Fig. 2
Fig. 2

Stereographic projection showing the directions A, Bs, Bf, and C of the beams Ȧ, B ˙ s , B ˙ f, and C, respectively, shown in Fig. 1. N1 and N2 are the normals shown in Fig. 1, and N1N2 is the plane of projection. P is the pole of the plane of incidence and O is the optic axis of the crystal. Q is the direction common to the face IJ and the plane of incidence such that the arc N1Q = +90°. G is the first direction of intersection of the plane OP and the plane of incidence. Ha, Hbs, Hbf, and Hc are the positive directions of the oscillating magnetic fields of the beams Ȧ, Bs, B ˙ f, and Ċ, respectively. AHa, BsHbs, BfHbf, and CHc are, respectively, the planes of polarization of the same beams in the same order (since the cirection Hbf falls outside the area of the figure, direction Hbf opposite to that is shown in the figure) and their azimuths are, respectively, α ˆ , β ˆ s , β ˆ f, and γ ˆ. Ea, Ebs, Ebf and Ec are, respectively, the positive directions of the oscillating electric fields of the same set of beams in the same order. The azimuths of planes AEa, BsEbs, BfEbf, and CEc are, respectively, α ˆ , β ˆ s , β ˆ f , and γ ˆ . Brf is the direction of the ray of the beam B ˙ f and it is in the plane BfO. The angle between Bf and Brf is b ˆ f. In the Fig. 2, b ˆ f is negative.

Fig. 3
Fig. 3

Stereographic projection showing the normals N1 and N2 of the first and second faces of the rhomb of calcite and also showing the directions Aw, Bs, and Cw of the beams Ȧw, B ˙ s, and Ċw when only the ordinary beam B ˙ s is present in the light refracted at the first face IJ of the rhomb. N1N2 is the plane of incidence and it is also the plane of projection and P is the pole of that plane. O is the optic axis of that crystal. Haw, Hbs, and Hcw are the positive directions of the oscillating magnetic fields of the beams Ȧw, B ˙ s, and Ċw, respectively. The directions Haw and Hcw are in the plane of polarization BsHbs of the refracted beam B ˙ s.

Fig. 4
Fig. 4

Stereographic projection showing the normals N1 and N2 of the first and second faces, respectively, of the rhomb of calcite and also showing the directions At, Bf, and Ct of the beams Ȧt, B ˙ f, and Ċt when only the extra-ordinary beam B ˙ f is present in the light refracted at the first face IJ of rhomb. N1N2 is the plane of incidence and also the plane of projection. P is the pole of the plane of incidence and O is the optic axis of the crystal. Hat, Hbf, and Hct are the positive directions of, the oscillating magnetic fields, respectively, of the beams Ȧt, B ˙ f, and Ċt. The three directions Hat, Hbf, and Hct are coplanar and the plane makes an angle Z ˆ b f with the plane of polarization BfHbf of the beam B ˙ f. Z ˆ b f is given by Eq. (43).

Fig. 5
Fig. 5

Stereographic projection showing the normals N2 and N3 of the second and third faces of the rhomb, respectively, of calcite and also showing the different directions Asw, Bww, and Css of the beams Ȧsw, B ˙ w w, and Ċss when the beam Ċsf is absent from the reflected light at the third face. N2N3 is the plane of incidence and is also the plane of projection and P is its pole. O is the optic axis of the crystal. Hasw, Hbww, and Hcss are the positive directions of the oscillating magnetic fields of the beams Ȧsw, B ˙ w w, and Ċss, respectively. The directions Hasw and Hcss are in the plane of polarization BwwHbww of the refracted beam B ˙ w w.

Fig. 6
Fig. 6

Stereographic projection showing the normals N2 and N3 of the second and third faces, respectively, of the rhomb of calcite and also showing the different directions Aft, Btt, and Cff of the beams Ȧft, B ˙ t t, and Ċff, respectively, when the slow beam Ċfs is absent from the light reflected at the third face. N2N3 is the plane of incidence and it is also the plane of projection and P is its pole. O is the optic axis of the crystal. Haft, Hbtt, and Hcff are the positive directions of the oscillating magnetic fields of beams Ȧft, B ˙ t t, and Ċff, respectively. The directions Haft and Hcff are in the plane of polarization BttHbtt of the refracted beam B ˙ t t.

Fig. 7
Fig. 7

Stereographic projection of a sphere on which graphs are drawn. Longitude represents the angle of incidence A ˆ w and arcs measured from a fixed pole L represent the different azimuths α ˆ w , β ˆ w, and γ ˆ w of the planes of polarization of the beams Aw, Bw and Ċw.

Fig. 8
Fig. 8

Stereographic projection of a sphere on which graphs are drawn. Longitude represents the angle of incidence Ât and arcs measured from a fixed pole L represent the different azimuths α ˆ t , β ˆ t, and γ ˆ t of the planes of polarization of the beams Ȧt, B ˙ t, and Ċt.

Fig. 9
Fig. 9

Stereographic projection of a sphere on which a graph is drawn showing the relation between α ˆ and γ ˆ for an angle of incidence  = +64°07 and for the orientation shown in Figs. 1 and 2. Longitude on the sphere represents 2 α ˆ and arcs measured from a fixed pole L represents 2 γ ˆ. The graph is a double loop. The sphere is shown tilted by an angle (90° − arc LXc). The primitive of the projection is the circle with its center at Xc.

Fig. 10
Fig. 10

IJKLM, shows a section of a cleaved rhomb of calcite on which a new face MI has been prepared, with the normal to that face in the plane of the normals to the other four faces, and the normal to MI made a small angle with the normal to face IJ. A plane-polarized incident beam Ȧt, the azimuth of whose plane of polarization is α ˆ t, gives rise to a single refracted beam B ˆ f only. When incident on the third face KL the beam B ˆ f (then called beam Ȧf) gives rise to a refracted beam B ˆ t (in air) and a single reflected fast beam Ċff inside the crystal. Then Ât = +52°32 and α ˆ t = 330 ° 08 . Beams Ȧf and B ˆ t are then called Ȧft and B ˆ t t, respectively.

Tables (2)

Tables Icon

Table I A ˆ is the angle of incidence at the first face of a rhomb of calcite in the orientation shown in Figs. 1 and 2 and α ˆ w , γ ˆ w, and βw are the azimuths of the planes of polarization of the beams incident and reflected and of the beam emerging through the parallel face, respectively, when the refracted light at the first face contains only the ordinary beam plane-polarized in azimuth β ˆ s. Similarly, α ˆ t , γ ˆ t, and β ˆ t are the azimuths of the planes of polarization of the beams incident and reflected at the first face and of the beam emerging through the parallel face of the rhomb, respectively, when the light refracted at the first face contains only the extra-ordinary beam plane-polarized in azimuth β ˆ f.

Tables Icon

Table II Variation of the azimuth γ ˆ of the plane of polarization of light reflected from the front face of a crystal of calcite when the azimuth α ˆ of the plane of polarization of the incident beam is varied through 180°, for the orientation of the rhomb shown in Figs. 1 and 2 and when the angle of incidence A ˆ = + 64 ° 07 .

Equations (114)

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p ˆ = arc O G = 153 ° 46 ,
g ˆ = arc N 1 G = 142 ° 32 ,
Z ˆ 0 = P N ˆ 1 O = 50 ° 57 .
arc λ ˆ b f = arc B f O = cos 1 | cos p ˆ · cos ( g ˆ B ˆ f ) | ,
arc λ ˆ c f f = cos 1 | cos p ˆ · cos ( g ˆ C ˆ f f ) | .
υ 0 = 0.60300 ,
υ e = 0.67276.
υ b f 2 = j ( k + cos 2 λ ˆ b f ) ,
υ c f f 2 = j ( k + cos 2 λ ˆ c f f ) ,
j = ( υ 0 2 υ e 2 ) / 2 = 0.04451 ,
k = ( υ 0 2 + υ e 2 ) / ( υ 0 2 υ e 2 ) = 9.16922.
b ˆ f = arc B f B r f = cot 1 | csc 2 λ ˆ b f · ( k + cos 2 λ ˆ b f ) | ,
c ˆ f f = arc C f f C r f f = cot 1 | csc 2 λ ˆ c f f · ( k + cos 2 λ ˆ c f f ) | .
υ r b f = υ b f · sec b ˆ f ,
υ r c f f = υ c f f · sec c ˆ f f .
A ˆ s = B ˆ s = sin 1 ( υ 0 · sin A ˆ ) ,
C ˆ s s = C ˆ f s = 180 ° A ˆ s ,
A ˆ f = B ˆ f = sin 1 ( υ b f · sin A ˆ ) ,
C ˆ s f = C ˆ f f = sin 1 ( υ c f f · sin A ˆ ) . 6
α ˆ s = β ˆ s = cot 1 | tan p ˆ · csc ( g ˆ B ˆ s ) | ,
γ ˆ f s = γ ˆ s s = cot 1 | tan p ˆ · csc ( g ˆ C ˆ s s ) | ,
α ˆ f = β ˆ f = cot 1 | + cot p ˆ · sin ( g ˆ B ˆ f ) | ,
γ ˆ s f = γ ˆ f f = cot 1 | + cot p ˆ · sin ( g ˆ C ˆ f f ) | .
α ˆ = α ˆ 90 ° ,
β ˆ s = β ˆ s 90 ° ,
β ˆ f = β ˆ f 90 ° ,
γ ˆ = γ ˆ 90 ° .
H ¯ c = r · H ¯ a ,
H ¯ b s = m s · H ¯ a ,
H ¯ b f = m f · H ¯ a .
E ¯ a = H ¯ a ,
E ¯ c = r · H ¯ a ,
E ¯ b s = ( υ b s = υ 0 ) · m s · H ¯ a ,
E ¯ b f = υ b f · m f · H ¯ a ,
cos A ˆ · sin α ˆ m s · cos B ˆ s · sin β ˆ s m f · cos B ˆ f · sin β ˆ f + r · cos C ˆ · sin γ ˆ = 0 ,
sin A ˆ · sin α ˆ m s · sin B ˆ s · sin β ˆ s m f · sin B ˆ f · sin β ˆ f + r · sin C ˆ · sin γ ˆ = 0 ,
cos α ˆ m s · cos β ˆ s m f · cos β ˆ f + r · cos γ ˆ = 0 ,
sin 2 A ˆ · cos α ˆ m s · sin 2 B ˆ s · cos β ˆ s m f · ( sin 2 B ˆ f · cos β ˆ f 2 sin 2 B ˆ f · tan b ˆ f ) + r · sin 2 C ˆ · cos γ ˆ = 0.
cot γ ˆ = sin A ˆ · | ( X n + X s + X f ) / ( Y n + Y s + Y f ) | ,
r 2 = | ( sin 2 A ˆ · sin α ˆ ) / ( cot α ˆ w cot α ˆ t ) | 2 . · | ( X n + X s + X f ) 2 + ( Y n + Y s + Y f ) 2 · csc 2 A ˆ | ,
m s = sin ( C ˆ A ˆ ) · csc ( B ˆ s C ˆ ) · sin α ˆ w · csc β ˆ s · sin ( α ˆ t α ˆ ) · csc ( α ˆ w α ˆ t ) ,
m f = sin ( C ˆ A ˆ ) · csc ( B ˆ f C ˆ ) · sin α ˆ t · csc β ˆ f · sin ( α ˆ α ˆ w ) · csc ( α ˆ w α ˆ t ) ,
cot α ˆ w = cos ( A ˆ B ˆ s ) · cot β ˆ s ,
cot α ˆ t = cos ( A ˆ B ˆ f ) · cot β ˆ f csc ( A ˆ + B ˆ f ) · sin 2 B ˆ f · tan b ˆ f · csc β ˆ f ,
X n = csc ( A ˆ C ˆ ) · ( cot α ˆ w cot α ˆ t ) · cot α ˆ ,
X s = csc ( B ˆ s C ˆ ) · ( cot α ˆ t cot α ˆ ) · cot β ˆ s ,
X f = csc ( B ˆ f C ˆ ) · ( cot α ˆ cot α ˆ w ) · cot β ˆ f ,
Y n = csc ( A ˆ C ˆ ) · ( cot α ˆ w cot α ˆ t ) · sin A ˆ ,
Y s = csc ( B ˆ s C ˆ ) · ( cot α ˆ t cot α ˆ ) · sin B ˆ s ,
Y f = csc ( B ˆ f C ˆ ) · ( cot α ˆ cot α ˆ w ) · sin B ˆ f .
cot γ ˆ w = cos ( B ˆ s C ˆ ) · cot β ˆ s ,
cot γ ˆ t = cos ( B ˆ f C ˆ ) · cot β ˆ f csc ( B ˆ f + C ˆ ) · sin 2 B ˆ f · tan b ˆ f · csc β ˆ f ,
r w = sin ( A ˆ B ˆ s ) · csc ( B ˆ s C ˆ ) · sin α ˆ w · csc γ ˆ w ,
r t = sin ( A ˆ B ˆ f ) · csc ( B ˆ f C ˆ ) · sin α ˆ t · csc γ ˆ t ,
m s w = sin ( C ˆ A ˆ ) · csc ( B ˆ s C ˆ ) · sin α ˆ w · csc β ˆ s ,
m f t = sin ( C ˆ A ˆ ) · csc ( B ˆ f C ˆ ) · sin α ˆ t · csc β ˆ f .
γ ˆ w p = γ ˆ t p = R ˆ / 2 .
cos ( B ˆ s p + A ˆ p ) · cot β ˆ s p = cos ( B ˆ f p + A ˆ p ) · cot β ˆ f p + csc ( B ˆ f p A ˆ p ) · sin 2 B ˆ f p · tan b ˆ f p · csc β ˆ f p ,
cot β ˆ w = sin B ˆ w · | ( X s n + X s s + X s f ) / ( Y s n + Y s s + Y s f ) | ,
m w 2 = | sin ( A ˆ s B ˆ w ) · sin α ˆ s / ( K s s K s f ) | 2 · | ( X s n + X s s + X s f ) 2 + ( Y s n + Y s s + Y s f ) 2 · csc 2 B ˆ w | ,
r s s = sin ( A ˆ s B ˆ w ) · csc ( B ˆ w C ˆ s s ) · sin α ˆ s · csc γ ˆ s s · | ( K s n K s f ) / ( K s s K s f ) | ,
r s f = sin ( A ˆ s B ˆ w ) · csc ( B ˆ w C ˆ s f ) · sin α ˆ s · csc γ ˆ s f · | ( K s s K s n ) / ( K s s K s f ) | ,
X s n = ( K s s K s f ) · csc ( B ˆ w A ˆ s ) · cot α ˆ s ,
X s s = ( K s f K s n ) · csc ( B ˆ w C ˆ s s ) · cot γ ˆ s s ,
X s f = ( K s n K s s ) · csc ( B ˆ w C ˆ s f ) · cot γ ˆ s f ,
Y s n = ( K s s K s f ) · csc ( B ˆ w A ˆ s ) · sin A ˆ s ,
Y s s = ( K s f K s n ) · csc ( B ˆ w C ˆ s s ) · sin C ˆ s s ,
Y s f = ( K s n K s s ) · csc ( B ˆ w C ˆ s f ) · sin C ˆ s f ,
K s n = cos ( B ˆ w + A ˆ s ) · cot s α ˆ ,
K s s = cos ( B ˆ w + C ˆ s s ) · cot γ ˆ s s ,
K s f = cos ( B ˆ w + C ˆ s f ) · cot γ ˆ s f + csc ( B ˆ w C ˆ c f ) · sin 2 C ˆ s f · tan c ˆ s f · csc γ ˆ s f ,
cot β ˆ t = sin B ˆ t · | ( X f n + X f s + X f f ) / ( Y f n + Y f s + Y f f ) | ,
m t 2 = | sin ( A ˆ f B ˆ f ) · sin α ˆ f / ( K f s K f f ) | 2 · | ( X f n + X f s + X f f ) 2 + ( Y f n + Y f s + Y f f ) 2 · csc 2 B ˆ t | ,
r f s = sin ( A ˆ f B ˆ t ) · csc ( B ˆ t C ˆ f s ) · sin α ˆ f · csc γ ˆ f s · | ( K f n K f f ) / ( K f s K f f ) | ,
r f f = sin ( A ˆ f B ˆ t ) · csc ( B ˆ t C ˆ f f ) · sin α ˆ f · csc γ ˆ f f · | ( K f s K f n ) / ( K f s K f f ) | ,
X f n = ( K f s K f f ) · csc ( B ˆ t A ˆ f ) · cot α ˆ f ,
X f s = ( K f f K f n ) · csc ( B ˆ t C ˆ f s ) · cot γ ˆ f s ,
X f f = ( K f n K f s ) · csc ( B ˆ t C ˆ f f ) · cot γ ˆ f f ,
Y f n = ( K f s K f f ) · csc ( B ˆ t A ˆ f ) · sin A ˆ f ,
Y f s = ( K f f K f n ) · csc ( B ˆ t C ˆ f s ) · sin C ˆ f s ,
Y f f = ( K f n K f s ) · csc ( B ˆ t C ˆ f f ) · sin C ˆ f f ,
K f n = cos ( B ˆ t + A ˆ f ) · cot α ˆ f + csc ( B ˆ t A ˆ f ) · sin 2 A ˆ f · tan α ˆ f · csc α ˆ f ,
K f s = cos ( B ˆ t + C ˆ f s ) · cot γ ˆ f s ,
K f f = cos ( B ˆ t + C ˆ f f ) · cot γ ˆ f f + csc ( B ˆ t C ˆ f f ) · sin 2 C ˆ f f · tan c ˆ f f · csc γ ˆ f f ,
sin ( B ˆ s C ˆ ) · cot α ˆ w + sin ( C ˆ A ˆ ) · cot β ˆ s + sin ( A ˆ B ˆ s ) · cot γ ˆ w = 0 ,
sin ( B ˆ f C ˆ ) · cot α ˆ t + sin ( C ˆ A ˆ ) · cot β ˆ f + sin ( A ˆ B ˆ f ) · cot γ ˆ t = 0 ,
Z ˆ b f = tan 1 | csc ( A ˆ B ˆ f ) · csc ( B ˆ f C ˆ ) · sin 2 B ˆ f · tan b ˆ f | ,
K s n = K s s .
sin ( B ˆ w w C ˆ s s ) · cot α ˆ s w + sin ( C ˆ s s A ˆ s w ) · cot β ˆ w w + sin ( A ˆ s w B ˆ w w ) · cot γ ˆ s s = 0 ,
B ˆ w w = 90 ° g ˆ = 52 ° 32 .
D ˆ = 90 ° g ˆ = B ˆ w w .
A ˆ s w = 331 ° 24 ,
C ˆ s s = 208 ° 36 ,
α ˆ s w = 17 ° 23 ,
β ˆ w w = 15 ° 18 ,
γ ˆ s s = 298 ° 20 .
K f n = K f f .
sin ( B ˆ t t C ˆ f f ) · cot α ˆ f t + sin ( C ˆ f f A ˆ f t ) · cot β ˆ t t + sin ( A ˆ f t B ˆ t t ) · cot γ ˆ f f = 0 ,
B ˆ t t = g ˆ 90 ° = + 52 ° 32 ,
A ˆ f t = 31 ° 55 ,
C ˆ f f = 150 ° 38 ,
α ˆ f t = 152 ° 14 ,
β ˆ t t = 153 ° 46 ,
γ ˆ f f = 74 ° 06 .
Z ˆ b = tan 1 | csc ( A ˆ B ˆ ) · csc ( B ˆ C ˆ ) · sin 2 B ˆ · tan b ˆ | ,
P ˆ υ = P ˆ z Z ˆ 0 ,
P ˆ 1 P ˆ υ = α ˆ w or α ˆ t ,
Y ˆ 1 Y ˆ υ = β ˆ w or β ˆ t .
P ˆ 2 P ˆ υ = α ˆ t or α ˆ w ,
Y ˆ 2 Y ˆ υ = β ˆ t or β ˆ w .
R ˆ 1 Y ˆ υ = γ ˆ w or γ ˆ t ,
R ˆ 2 Y ˆ υ = γ ˆ t or γ ˆ w .
P ˆ 3 P ˆ υ = α ˆ w .
P ˆ 4 P ˆ υ = α ˆ t .