Abstract

Spatial vision relies on a spatial coding of object directions sensed by the retinal photoreceptors. An empirically measured constant error in this coding of perceived space is described classically as a barrel distortion involving long horizontal lines. A Cartesian retinal coordinate system is constructed to describe the transformation from object space to perceived space. This system establishes a subjective horizontal and vertical (isoelevation and isoazimuth) grid for each eye. The retinal entities that mediate subjective sense of constant elevation (horizontal) and constant azimuth (vertical) constitute the functional division of the retina, and they are described by a model with a single parameter zp, which determines the curvatures of a set of hyperbolic isopters. Empirical isoelevation positions were measured, which were described by this model. The best-fitting zp values appeared to be independent of viewing distance, similar for the two eyes of the same observer but differing significantly among observers. The functional-retinal-division model and horizontal retinal shear, as described by Helmholtz, were used to predict the spatial distributions of binocular corresponding points. The disparity field on a frontoparallel plane at a finite viewing distance showed marked asymmetry between the upper and the lower visual fields. The hyperbolic isoelevation and isoazimuth curves dramatically exaggerated this asymmetry. This theoretical disparity field provides a guideline for experiments that attempt to show the effect of binocular disparity in the periphery.

© 1998 Optical Society of America

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References

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  1. K. N. Ogle, Researches in Binocular Vision (Saunders, Philadelphia, Pa., 1950).
  2. H. v. Helmholtz, Helmholtz’s Treatise on Physiological Optics, J. P. C. Southall, transl. (Optical Society of America, New York, 1925).
  3. K. Nakayama, “Geometrical and physiological aspects of depth perception,” in Three-Dimensional Imaging, S. A. Benton, ed., Proc. SPIE120, 1–8 (1977).
  4. A. I. Cogan, “The relationship between the apparent vertical and the vertical horopter,” Vision Res. 19, 655–665 (1979).
    [CrossRef] [PubMed]
  5. J. J. Pollitt, Art and Experience in Classical Greece (Cambridge U. Press, Cambridge, UK, 1972).
  6. J. J. Coulton, Ancient Greek Architects at Work (Cornell U. Press, Ithaca, N.Y., 1977).
  7. K. G. Perls, Jean Fouquet (Hyperion, Paris, 1940), p. 144.
  8. A. Flocon, A. Barre, Curvilinear Perspective: From Visual Space to the Constructed Image (U. of California Press, Berkeley, Calif., 1987).
  9. A. von Tschermak-Seysenegg, Introduction to Physiological Optics, P. Boeder, Transl. (Charles C. Thomas, Springfield, Ill., 1952).
  10. E. Hering, Spatial Sense and Movements of the Eye, C. A. Radde, transl. (American Academy of Optometry, Baltimore, Md., 1942).
  11. Neither the eyeball nor the retina is a true sphere. The vertical and the transverse diameters of the eyeball are 24 and 23.5 mm, respectively.12 The thickest part of the sclera at the posterior pole is ∼1 mm. Therefore a sphere of an average diameter of 23 mm is an acceptable approximation. In most of our calculations it is the visual angle, not the linear distance, on the retinal surface that matters.
  12. H. Gray, Anatomy of the Human Body (Lea & Febiger, Philadelphia, Pa., 1973).
  13. C. M. Schor, J. S. Maxwell, S. B. Stevenson, “Isovergence surfaces: the conjugacy of vertical eye movements in tertiary positions of gaze,” Ophthalmic Physiol. Opt. 14, 279–286 (1994).
    [CrossRef] [PubMed]
  14. H. H. Emsley, Visual Optics, Vol. I. Optics of Vision, 5th ed. (Hatton, London, 1964).
  15. A directrix is a curve in space. A curve or a straight line (the generatrix) moves along the directrix to generate a three-dimensional surface. Therefore the directrix is always part of the surface generated. A cone is generated by a straight generatrix that moves along a directrix and pivots on a fixed point.
  16. K. Prazdny, “Stereoscopic matching, eye position, and absolute depth,” Perception 12, 151–160 (1983).
    [CrossRef] [PubMed]
  17. J. Porrill, J. E. W. Mayhew, J. P. Frisby, “Cyclotorsion, conformal invariance and induced effects in stereoscopic vision,” (Artificial Intelligence Vision Research Unit, Sheffield University, Sheffield, UK, 1985).
  18. T. Shipley, S. C. Rawlings, “The nonius horopter—I. History and theory,” Vision Res. 10, 1225–1262 (1970); II. An experimental report,” 1263–1299.
    [CrossRef] [PubMed]
  19. J. E. W. Mayhew, H. C. Longuet-Higgins, “A computational model of binocular depth perception,” Nature (London) 297, 376–378 (1982).
    [CrossRef]
  20. B. Gillam, B. Lawergren, “The induced effect, vertical disparity and stereoscopic theory,” Percept. Psychophys. 34, 121–130 (1983).
    [CrossRef] [PubMed]
  21. B. G. Cumming, E. B. Johnston, A. J. Parker, “Vertical disparities and perception of three-dimensional shape,” Nature (London) 349, 411–413 (1991).
    [CrossRef]
  22. B. J. Rogers, M. F. Bradshaw, “Vertical disparities, differential perspective and binocular stereopsis,” Nature (London) 361, 253–255 (1993).
    [CrossRef]
  23. C. W. Tyler, “Sensory processing of binocular disparity,” in Vergence Eye Movements: Basic and Clinical Aspects, C. M. Schor, K. Ciuffreda, eds. (Butterworth, Boston, 1983), pp. 199–295.
  24. R. A. Clement, Introduction to Vision Science (Erlbaum, Hillsdale, N.J., 1993).

1994 (1)

C. M. Schor, J. S. Maxwell, S. B. Stevenson, “Isovergence surfaces: the conjugacy of vertical eye movements in tertiary positions of gaze,” Ophthalmic Physiol. Opt. 14, 279–286 (1994).
[CrossRef] [PubMed]

1993 (1)

B. J. Rogers, M. F. Bradshaw, “Vertical disparities, differential perspective and binocular stereopsis,” Nature (London) 361, 253–255 (1993).
[CrossRef]

1991 (1)

B. G. Cumming, E. B. Johnston, A. J. Parker, “Vertical disparities and perception of three-dimensional shape,” Nature (London) 349, 411–413 (1991).
[CrossRef]

1983 (2)

B. Gillam, B. Lawergren, “The induced effect, vertical disparity and stereoscopic theory,” Percept. Psychophys. 34, 121–130 (1983).
[CrossRef] [PubMed]

K. Prazdny, “Stereoscopic matching, eye position, and absolute depth,” Perception 12, 151–160 (1983).
[CrossRef] [PubMed]

1982 (1)

J. E. W. Mayhew, H. C. Longuet-Higgins, “A computational model of binocular depth perception,” Nature (London) 297, 376–378 (1982).
[CrossRef]

1979 (1)

A. I. Cogan, “The relationship between the apparent vertical and the vertical horopter,” Vision Res. 19, 655–665 (1979).
[CrossRef] [PubMed]

1970 (1)

T. Shipley, S. C. Rawlings, “The nonius horopter—I. History and theory,” Vision Res. 10, 1225–1262 (1970); II. An experimental report,” 1263–1299.
[CrossRef] [PubMed]

Barre, A.

A. Flocon, A. Barre, Curvilinear Perspective: From Visual Space to the Constructed Image (U. of California Press, Berkeley, Calif., 1987).

Bradshaw, M. F.

B. J. Rogers, M. F. Bradshaw, “Vertical disparities, differential perspective and binocular stereopsis,” Nature (London) 361, 253–255 (1993).
[CrossRef]

Clement, R. A.

R. A. Clement, Introduction to Vision Science (Erlbaum, Hillsdale, N.J., 1993).

Cogan, A. I.

A. I. Cogan, “The relationship between the apparent vertical and the vertical horopter,” Vision Res. 19, 655–665 (1979).
[CrossRef] [PubMed]

Coulton, J. J.

J. J. Coulton, Ancient Greek Architects at Work (Cornell U. Press, Ithaca, N.Y., 1977).

Cumming, B. G.

B. G. Cumming, E. B. Johnston, A. J. Parker, “Vertical disparities and perception of three-dimensional shape,” Nature (London) 349, 411–413 (1991).
[CrossRef]

Emsley, H. H.

H. H. Emsley, Visual Optics, Vol. I. Optics of Vision, 5th ed. (Hatton, London, 1964).

Flocon, A.

A. Flocon, A. Barre, Curvilinear Perspective: From Visual Space to the Constructed Image (U. of California Press, Berkeley, Calif., 1987).

Frisby, J. P.

J. Porrill, J. E. W. Mayhew, J. P. Frisby, “Cyclotorsion, conformal invariance and induced effects in stereoscopic vision,” (Artificial Intelligence Vision Research Unit, Sheffield University, Sheffield, UK, 1985).

Gillam, B.

B. Gillam, B. Lawergren, “The induced effect, vertical disparity and stereoscopic theory,” Percept. Psychophys. 34, 121–130 (1983).
[CrossRef] [PubMed]

Gray, H.

H. Gray, Anatomy of the Human Body (Lea & Febiger, Philadelphia, Pa., 1973).

Helmholtz, H. v.

H. v. Helmholtz, Helmholtz’s Treatise on Physiological Optics, J. P. C. Southall, transl. (Optical Society of America, New York, 1925).

Hering, E.

E. Hering, Spatial Sense and Movements of the Eye, C. A. Radde, transl. (American Academy of Optometry, Baltimore, Md., 1942).

Johnston, E. B.

B. G. Cumming, E. B. Johnston, A. J. Parker, “Vertical disparities and perception of three-dimensional shape,” Nature (London) 349, 411–413 (1991).
[CrossRef]

Lawergren, B.

B. Gillam, B. Lawergren, “The induced effect, vertical disparity and stereoscopic theory,” Percept. Psychophys. 34, 121–130 (1983).
[CrossRef] [PubMed]

Longuet-Higgins, H. C.

J. E. W. Mayhew, H. C. Longuet-Higgins, “A computational model of binocular depth perception,” Nature (London) 297, 376–378 (1982).
[CrossRef]

Maxwell, J. S.

C. M. Schor, J. S. Maxwell, S. B. Stevenson, “Isovergence surfaces: the conjugacy of vertical eye movements in tertiary positions of gaze,” Ophthalmic Physiol. Opt. 14, 279–286 (1994).
[CrossRef] [PubMed]

Mayhew, J. E. W.

J. E. W. Mayhew, H. C. Longuet-Higgins, “A computational model of binocular depth perception,” Nature (London) 297, 376–378 (1982).
[CrossRef]

J. Porrill, J. E. W. Mayhew, J. P. Frisby, “Cyclotorsion, conformal invariance and induced effects in stereoscopic vision,” (Artificial Intelligence Vision Research Unit, Sheffield University, Sheffield, UK, 1985).

Nakayama, K.

K. Nakayama, “Geometrical and physiological aspects of depth perception,” in Three-Dimensional Imaging, S. A. Benton, ed., Proc. SPIE120, 1–8 (1977).

Ogle, K. N.

K. N. Ogle, Researches in Binocular Vision (Saunders, Philadelphia, Pa., 1950).

Parker, A. J.

B. G. Cumming, E. B. Johnston, A. J. Parker, “Vertical disparities and perception of three-dimensional shape,” Nature (London) 349, 411–413 (1991).
[CrossRef]

Perls, K. G.

K. G. Perls, Jean Fouquet (Hyperion, Paris, 1940), p. 144.

Pollitt, J. J.

J. J. Pollitt, Art and Experience in Classical Greece (Cambridge U. Press, Cambridge, UK, 1972).

Porrill, J.

J. Porrill, J. E. W. Mayhew, J. P. Frisby, “Cyclotorsion, conformal invariance and induced effects in stereoscopic vision,” (Artificial Intelligence Vision Research Unit, Sheffield University, Sheffield, UK, 1985).

Prazdny, K.

K. Prazdny, “Stereoscopic matching, eye position, and absolute depth,” Perception 12, 151–160 (1983).
[CrossRef] [PubMed]

Rawlings, S. C.

T. Shipley, S. C. Rawlings, “The nonius horopter—I. History and theory,” Vision Res. 10, 1225–1262 (1970); II. An experimental report,” 1263–1299.
[CrossRef] [PubMed]

Rogers, B. J.

B. J. Rogers, M. F. Bradshaw, “Vertical disparities, differential perspective and binocular stereopsis,” Nature (London) 361, 253–255 (1993).
[CrossRef]

Schor, C. M.

C. M. Schor, J. S. Maxwell, S. B. Stevenson, “Isovergence surfaces: the conjugacy of vertical eye movements in tertiary positions of gaze,” Ophthalmic Physiol. Opt. 14, 279–286 (1994).
[CrossRef] [PubMed]

Shipley, T.

T. Shipley, S. C. Rawlings, “The nonius horopter—I. History and theory,” Vision Res. 10, 1225–1262 (1970); II. An experimental report,” 1263–1299.
[CrossRef] [PubMed]

Stevenson, S. B.

C. M. Schor, J. S. Maxwell, S. B. Stevenson, “Isovergence surfaces: the conjugacy of vertical eye movements in tertiary positions of gaze,” Ophthalmic Physiol. Opt. 14, 279–286 (1994).
[CrossRef] [PubMed]

Tyler, C. W.

C. W. Tyler, “Sensory processing of binocular disparity,” in Vergence Eye Movements: Basic and Clinical Aspects, C. M. Schor, K. Ciuffreda, eds. (Butterworth, Boston, 1983), pp. 199–295.

von Tschermak-Seysenegg, A.

A. von Tschermak-Seysenegg, Introduction to Physiological Optics, P. Boeder, Transl. (Charles C. Thomas, Springfield, Ill., 1952).

Nature (London) (3)

J. E. W. Mayhew, H. C. Longuet-Higgins, “A computational model of binocular depth perception,” Nature (London) 297, 376–378 (1982).
[CrossRef]

B. G. Cumming, E. B. Johnston, A. J. Parker, “Vertical disparities and perception of three-dimensional shape,” Nature (London) 349, 411–413 (1991).
[CrossRef]

B. J. Rogers, M. F. Bradshaw, “Vertical disparities, differential perspective and binocular stereopsis,” Nature (London) 361, 253–255 (1993).
[CrossRef]

Ophthalmic Physiol. Opt. (1)

C. M. Schor, J. S. Maxwell, S. B. Stevenson, “Isovergence surfaces: the conjugacy of vertical eye movements in tertiary positions of gaze,” Ophthalmic Physiol. Opt. 14, 279–286 (1994).
[CrossRef] [PubMed]

Percept. Psychophys. (1)

B. Gillam, B. Lawergren, “The induced effect, vertical disparity and stereoscopic theory,” Percept. Psychophys. 34, 121–130 (1983).
[CrossRef] [PubMed]

Perception (1)

K. Prazdny, “Stereoscopic matching, eye position, and absolute depth,” Perception 12, 151–160 (1983).
[CrossRef] [PubMed]

Vision Res. (2)

A. I. Cogan, “The relationship between the apparent vertical and the vertical horopter,” Vision Res. 19, 655–665 (1979).
[CrossRef] [PubMed]

T. Shipley, S. C. Rawlings, “The nonius horopter—I. History and theory,” Vision Res. 10, 1225–1262 (1970); II. An experimental report,” 1263–1299.
[CrossRef] [PubMed]

Other (16)

C. W. Tyler, “Sensory processing of binocular disparity,” in Vergence Eye Movements: Basic and Clinical Aspects, C. M. Schor, K. Ciuffreda, eds. (Butterworth, Boston, 1983), pp. 199–295.

R. A. Clement, Introduction to Vision Science (Erlbaum, Hillsdale, N.J., 1993).

J. J. Pollitt, Art and Experience in Classical Greece (Cambridge U. Press, Cambridge, UK, 1972).

J. J. Coulton, Ancient Greek Architects at Work (Cornell U. Press, Ithaca, N.Y., 1977).

K. G. Perls, Jean Fouquet (Hyperion, Paris, 1940), p. 144.

A. Flocon, A. Barre, Curvilinear Perspective: From Visual Space to the Constructed Image (U. of California Press, Berkeley, Calif., 1987).

A. von Tschermak-Seysenegg, Introduction to Physiological Optics, P. Boeder, Transl. (Charles C. Thomas, Springfield, Ill., 1952).

E. Hering, Spatial Sense and Movements of the Eye, C. A. Radde, transl. (American Academy of Optometry, Baltimore, Md., 1942).

Neither the eyeball nor the retina is a true sphere. The vertical and the transverse diameters of the eyeball are 24 and 23.5 mm, respectively.12 The thickest part of the sclera at the posterior pole is ∼1 mm. Therefore a sphere of an average diameter of 23 mm is an acceptable approximation. In most of our calculations it is the visual angle, not the linear distance, on the retinal surface that matters.

H. Gray, Anatomy of the Human Body (Lea & Febiger, Philadelphia, Pa., 1973).

J. Porrill, J. E. W. Mayhew, J. P. Frisby, “Cyclotorsion, conformal invariance and induced effects in stereoscopic vision,” (Artificial Intelligence Vision Research Unit, Sheffield University, Sheffield, UK, 1985).

H. H. Emsley, Visual Optics, Vol. I. Optics of Vision, 5th ed. (Hatton, London, 1964).

A directrix is a curve in space. A curve or a straight line (the generatrix) moves along the directrix to generate a three-dimensional surface. Therefore the directrix is always part of the surface generated. A cone is generated by a straight generatrix that moves along a directrix and pivots on a fixed point.

K. N. Ogle, Researches in Binocular Vision (Saunders, Philadelphia, Pa., 1950).

H. v. Helmholtz, Helmholtz’s Treatise on Physiological Optics, J. P. C. Southall, transl. (Optical Society of America, New York, 1925).

K. Nakayama, “Geometrical and physiological aspects of depth perception,” in Three-Dimensional Imaging, S. A. Benton, ed., Proc. SPIE120, 1–8 (1977).

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Figures (11)

Fig. 1
Fig. 1

(a) Spatial distortion. When the horizontal line in the middle is fixated on, it appears straight; the lines below and above, however, appear concave toward the fixation point. (b) Helmholtz’s checkerboard. The checkerboard with a pincushion distortion appears rectilinear when viewed at a distance indicated by bar A. The pincushion distortion, which was derived from the so-called direction circle, was intended to compensate for the barrel distortion shown in (a). The pattern has to be so positioned that it is perpendicular to the visual axis of the viewing eye.

Fig. 2
Fig. 2

Functional division of the retina. Retinal division lines are modeled as the intersections of the retinal sphere and a set of planes (retinal division planes). The two insets show the side view and the back view of a set of such planes and the retinal sphere. Each retinal division plane, and thus the corresponding retinal division line, can be distinguished from others by angle α, which is the angle between this plane and the XZ plane. The retina and the external space are associated with a simple perspective projection through the projection center (the nodal point of the optics of the eye), which is zn mm in front of the bulbcenter.

Fig. 3
Fig. 3

Projection cone. A retinal division line (usually a minor circle of the retinal sphere) is projected to a screen in space through a projection cone whose vertex is at the projection center. One half of the projection cone passes through the retinal division line, and the other half extends to the space and intersects a screen. The intersection curve is the theoretical isoelevation curve corresponding to the retinal division line.

Fig. 4
Fig. 4

Effect of parameter zp. (a) Different retinal division schemes specified by different zp values. Only one retinal division line from each scheme is shown. These lines all pass through point T on the vertical meridian of the retinal sphere, and their projections on the screen all pass through point T* in (b). (b) Theoretical isoelevation curves corresponding to the retinal division lines shown in (a). The curves are in the lower visual field. The screen is 150 mm in front of the bulbcenter and is perpendicular to the visual axis of the eye. (c) The curvature of the theoretical isoelevation curves at T* changes with zp value. The zp values larger than 100–200 mm have little effect on the curvature of the isoelevation curve.

Fig. 5
Fig. 5

When a screen is set in front of the eye perpendicular to the visual axis of the eye (a), a checkerboard that would appear rectlinear on this screen (b) is not the symmetric pattern as portrayed by Helmholtz [Fig. 1(b)]. This is because while the isoelevation curves are symmetric (c), the isoazimuth curves are not (d), owing to the so-called horizontal shear. (b), (c), and (d) are illustrations for the left eye. Viewing distance D is 150 mm (the length of bar A), and zp=-40 mm. The angular separations between the isopters are 10°.

Fig. 6
Fig. 6

When the two eyes converge symmetrically at the center of a frontoparallel plane (a), the plane is not perpendicular to either eye’s visual axis. The pattern that would appear as a rectlinear checkerboard to the left eye (b) would have to undergo more distortion. The isoelevation curves are separated more on the nasal side of the field than on the temporal side, delineating a trapezoidal area (c). The separations between isoazimuth curves are larger on the nasal side of the field than on the temporal side (d). (b), (c), and (d) are illustrations for the left eye. Viewing distance Δ is 150 mm (the length of bar A). The interpupillary distance 2p=60 mm. zp=-40 mm. The angular separations between the isopters are 10°.

Fig. 7
Fig. 7

Apparatus. A set of binocular fixation targets was presented at the center of the screen to maintain fixation. A reference dot above or below the fixation and a test dot at various horizontal eccentricities were flashed at 200-ms duration. The task of the observer was to adjust the vertical position of the test dot to match the apparent height of the reference dot. A set of polarizers on the screen and the polarizer glasses worn by the observer determined which eye saw which dot. A 30°-by-50° visual field was explored. The vertical positions of the reference dot and the horizontal eccentricities of the test dot are shown in the inset.

Fig. 8
Fig. 8

Monocular isoelevation tests. (a) Stimulus configurations of the monocular tests. The reference dot and the test dot were seen by the same eye. Fixation targets were always visible to both eyes. Nonius lines were used to monitor convergence. (b) Results of the monocular tests for three observers. Stars are the positions of the reference dot on the screen. Open circles and open squares are the apparent-equal-height positions for the right eye and the left eye, respectively. Each symbol represents the average of 10–30 adjustments. The error bars are smaller than the size of the symbol. Two of the observers also conducted the experiment at a viewing distance of 300 mm. The continuous curves in the graphs are the best-fitting theoretical isoelevation curves, which are computed with equations (9), (14), and (15). zp is the single parameter that was varied to achieve the least-squares fitting. The best-fitting zp values and the minimum sum of squared errors are summarized in Table 1.

Fig. 9
Fig. 9

Dichoptic isoelevation tests. (a) Stimulus configurations of the dichoptic tests. The reference dot and the test dot were seen by different eyes. When the test dot was in the left eye, only the left half of the field was tested. The opposite was true for the right eye. Fixation targets were always visible to both eyes. Nonius lines were used to monitor convergence. (b) Results of the dichoptic tests for three observers. Stars are the positions of the reference dot on the screen. Solid triangles are dichoptic equal-height positions. Open squares and open circles are monocular left-eye and right-eye equal-height positions, respectively. Each symbol represents the average of 10–30 adjustments.

Fig. 10
Fig. 10

Field of geometric disparity. The dark-gray and the white isopter grids are for the left eye and the right eye, respectively. The black arrows connecting the points of the same nominal azimuth and elevation values represent the spatial offsets between corresponding retinal points.

Fig. 11
Fig. 11

Helmholtz’s direction circle.

Tables (1)

Tables Icon

Table 1 Best-Fitting zp (ΣE2)

Equations (36)

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

x2+y2+z2=r2.
y=(z+zp)tan α.
tan α=hzp+(r-b),
tan θ=hzn+(r-b),
rsin θ=znsin ξ,
q=hr=sin θ cos ξ+cos θ sin ξ=sin θ1-zn2r2sin2 θ1/2+znrcos θ sin θ.
tan α=zn+(r2-h2)1/2zp+(r2-h2)1/2tan θ=zn+r(1-h2/r2)1/2zp+r(1-h2/r2)1/2tan θ=zn+r(1-q2)1/2zp+r(1-q2)1/2tan θ.
x=(z+zp)tan α.
xt2+yt2+-zn+z+znt2=r2,
yt=-zn+z+znt+zptan α
ax2+by2+cy(z+zn)+d(z+zn)2=0.
bx2+ay2+cx(z+zn)+d(z+zn)2=0,
a=tan2(α)(zp-zn)2,
b=tan2(α)(zp-zn)2+zn2-r2,
c=2×tan(α)(r2-zpzn),
d=tan2(α)(zp2-r2).
ax2+by2+c(zn-D)y+d(zn-D)2=0,
bx2+ay2+c(zn-D)x+d(zn-D)2=0.
z=-x tan(Φ/2)-vd,
z=x tan(Φ/2)-vd,
F(x, y, z)=0,
G(x, y, z)=0,
F(x1, y1, z1)=0,
G(x1, y1,z1)=0.
x=x0+t(x1-x0),
y=y0+t(y1-y0),
z=z0+t(z1-z0).
x1=x0+1t(x-x0),
y1=y0+1t(y-y0),
z1=z0+1t(z-z0).
Fx0+x-x0t, y0+y-y0t, z0+z-z0t=0,
Gx0+x-x0t, y0+y-y0t, z0+z-z0t=0.
x2+y2+z2=R2,
y=tan(θ/2)(z-R),
tan2(θ/2)x2+[tan2(θ/2)-1]y2+2 tan(θ/2)yz=0.
y=tan(θ/2)(z+R).

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