Abstract

We consider a surface (semitransparent or opaque) in space, viewed by orthogonal projection to a view plane that is rotating uniformly about an unknown axis (equivalently, a surface rotating about an unknown axis and viewed by orthogonal projection to a fixed view plane). We consider profiles of this surface (also known as apparent contours, occluding contours, and outlines), and we do not track marked points or curves nor assume that a correspondence problem has been solved. We show that, provided the angular speed is known, the location of the axis, and hence the surface, can be recovered from measurements on the profiles over an interval of time. If the angular speed is unknown, then there is a one-parameter family of solutions similar to the bas-relief ambiguity. The results are obtained by use of frontier points on the surface, which can also be viewed as points of epipolar tangency. Results of a numerical experiment showed that the performance was best with larger extents of rotation or when the axis was nearly perpendicular to the view direction.

© 1994 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. A. Blake, R. Cipolla, “Robust estimation of surface curvature from deformations of apparent contours,” Image Vis. Comput. 9, 107–112 (1991).
    [CrossRef]
  2. R. Cipolla, A. Blake, “The dynamic analysis of occluding contours,” in Proceedings of the Third International Conference on Computer Vision (Institute of Electrical and Electronics Engineers, New York, 1990), pp. 616–623.
    [CrossRef]
  3. R. Cipolla, A. Blake, “Surface shape from the deformation of apparent contours,” Internat. J. Computer Vis. 9, 83–112 (1992).
    [CrossRef]
  4. J. Callahan, R. S. Weiss, “A model for describing surface shape,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (Institute of Electrical and Electronics Engineers, New York, 1985), pp. 240–245.
  5. P. J. Giblin, R. S. Weiss, “Reconstruction of surfaces from profiles,” in Proceedings of the First International Conference on Computer VisionInstitute of Electrical and Electronics Engineers, New York, 1987), pp. 136–144.
  6. J. J. Koenderink, Solid Shape (MIT Press, Cambridge, Mass., 1990).
  7. J. J. Koenderink, A. J. van Doorn, “The singularities of the visual mapping,” Biol. Cybernet. 24, 51–59 (1976).
    [CrossRef]
  8. R. Vaillant, O. D. Faugeras, “Using extremal boundaries for 3–D object modelling,” IEEE Trans. Pat. Anal. Mach. Intell. 14, 157–172 (1992).
    [CrossRef]
  9. J. M. H. Beusmans, “Visual perception of solid shape from occluding contours,” Ph.D. dissertation (University of California, Irvine, Irvine, Calif., 1990)(also Tech. Rep. 90–40, Department of Information and Computer Science, University of California, Irvine, 1990).
  10. J. A. Webb, J. K. Aggarwal, “Structure from motion of rigid and jointed objects,” Artif. Intell. 19, 107–130 (1982).
    [CrossRef]
  11. B. M. Bennett, D. D. Hoffman, J. E. Nicola, C. Prakash, “Structure from two orthographic views of rigid motion,” J. Opt. Soc. Am. A 6, 1052–1069 (1989).
    [CrossRef] [PubMed]
  12. J. H. Rieger, “Three-dimensional motion from fixed points of a deforming profile curve,” Opt. Lett. 11, 123–135 (1986).
    [CrossRef] [PubMed]
  13. J. Porrill, S. Pollard, “Curve matching and stereo calibration,” Image Vis. Comput. 9, 45–50 (1991).
    [CrossRef]
  14. J. J. Koenderink, A. J. van Doorn, “Invariant properties of the motion parallax field due to the movement of rigid bodies relative to an observer,” Opt. Acta 22, 773–791 (1975).
    [CrossRef]
  15. T. S. Huang, C. H. Lee, “Motion and structure from orthographic projections,” IEEE Trans. Pat. Anal. Mach. Intell. 11, 536–540 (1989).
    [CrossRef]
  16. J. J. Koenderink, A. J. van Doorn, “Affine structure from motion,” J. Opt. Soc. Am. A 8, 377–386 (1991).
    [CrossRef] [PubMed]
  17. L. S. Shapiro, A. P. Zisserman, M. Brady, “Motion from point matches using affine epipolar geometry,” Int. J. Computer Vision (to be published).
  18. J. W. Bruce, P. J. Giblin, Curves and Singularities, 2nd ed. (Cambridge U. Press, Cambridge, 1992).
  19. A. K. Chhabra, T. A. Grogan, “Uniqueness, the minimum norm constraint and analog networks for optical flow along contours,” in Proceedings of the Third International Conference on Computer Vision (Institute of Electrical and Electronics Engineers, New York, 1990), pp. 80–84.
    [CrossRef]
  20. E. Hildreth, The Measurement of Visual Motion (MIT Press, Cambridge, Mass., 1984).
  21. A. Movshon, “Visual processing of moving images,” in Images and Understanding, H. Barlow, C. Blakemore, M. Weston-Smith, eds. (Cambridge U. Press, Cambridge, 1990).
  22. K. Nakayama, G. Silverman, “The aperture problem—I. Perception of nonrigidity and motion direction in translating sinusoidal fines,” Vision Res. 28, 739–746 (1988).
    [CrossRef]
  23. K. Nakayama, G. Silverman, “The aperture problem—II. Spatial integration of velocity information along contours,” Vision Res. 28, 747–753 (1988).
    [CrossRef]
  24. J. E. Rycroft, “A geometrical investigation into the projections of surfaces and space curves,” Ph.D. dissertation (University of Liverpool, Liverpool, UK, 1992).
  25. P. J. Giblin, J. E. Rycroft, F. E. Pollick, “Moving surfaces,” in Design and Applications of Curves and Surfaces, R. Fisher, ed., Institute of Mathematics and Its Applications Conference Proceedings Series (Oxford U. Press, Oxford, 1994), pp. 433–453.
  26. F. E. Pollick, S. Nishida, Y. Koike, M. Kawato, “Perceived motion in structure-from-motion: pointing responses to the axis of rotation,” Percept. Psychophys. (to be published).

1992

R. Cipolla, A. Blake, “Surface shape from the deformation of apparent contours,” Internat. J. Computer Vis. 9, 83–112 (1992).
[CrossRef]

R. Vaillant, O. D. Faugeras, “Using extremal boundaries for 3–D object modelling,” IEEE Trans. Pat. Anal. Mach. Intell. 14, 157–172 (1992).
[CrossRef]

1991

J. Porrill, S. Pollard, “Curve matching and stereo calibration,” Image Vis. Comput. 9, 45–50 (1991).
[CrossRef]

A. Blake, R. Cipolla, “Robust estimation of surface curvature from deformations of apparent contours,” Image Vis. Comput. 9, 107–112 (1991).
[CrossRef]

J. J. Koenderink, A. J. van Doorn, “Affine structure from motion,” J. Opt. Soc. Am. A 8, 377–386 (1991).
[CrossRef] [PubMed]

1989

T. S. Huang, C. H. Lee, “Motion and structure from orthographic projections,” IEEE Trans. Pat. Anal. Mach. Intell. 11, 536–540 (1989).
[CrossRef]

B. M. Bennett, D. D. Hoffman, J. E. Nicola, C. Prakash, “Structure from two orthographic views of rigid motion,” J. Opt. Soc. Am. A 6, 1052–1069 (1989).
[CrossRef] [PubMed]

1988

K. Nakayama, G. Silverman, “The aperture problem—I. Perception of nonrigidity and motion direction in translating sinusoidal fines,” Vision Res. 28, 739–746 (1988).
[CrossRef]

K. Nakayama, G. Silverman, “The aperture problem—II. Spatial integration of velocity information along contours,” Vision Res. 28, 747–753 (1988).
[CrossRef]

1986

1982

J. A. Webb, J. K. Aggarwal, “Structure from motion of rigid and jointed objects,” Artif. Intell. 19, 107–130 (1982).
[CrossRef]

1976

J. J. Koenderink, A. J. van Doorn, “The singularities of the visual mapping,” Biol. Cybernet. 24, 51–59 (1976).
[CrossRef]

1975

J. J. Koenderink, A. J. van Doorn, “Invariant properties of the motion parallax field due to the movement of rigid bodies relative to an observer,” Opt. Acta 22, 773–791 (1975).
[CrossRef]

Aggarwal, J. K.

J. A. Webb, J. K. Aggarwal, “Structure from motion of rigid and jointed objects,” Artif. Intell. 19, 107–130 (1982).
[CrossRef]

Bennett, B. M.

Beusmans, J. M. H.

J. M. H. Beusmans, “Visual perception of solid shape from occluding contours,” Ph.D. dissertation (University of California, Irvine, Irvine, Calif., 1990)(also Tech. Rep. 90–40, Department of Information and Computer Science, University of California, Irvine, 1990).

Blake, A.

R. Cipolla, A. Blake, “Surface shape from the deformation of apparent contours,” Internat. J. Computer Vis. 9, 83–112 (1992).
[CrossRef]

A. Blake, R. Cipolla, “Robust estimation of surface curvature from deformations of apparent contours,” Image Vis. Comput. 9, 107–112 (1991).
[CrossRef]

R. Cipolla, A. Blake, “The dynamic analysis of occluding contours,” in Proceedings of the Third International Conference on Computer Vision (Institute of Electrical and Electronics Engineers, New York, 1990), pp. 616–623.
[CrossRef]

Brady, M.

L. S. Shapiro, A. P. Zisserman, M. Brady, “Motion from point matches using affine epipolar geometry,” Int. J. Computer Vision (to be published).

Bruce, J. W.

J. W. Bruce, P. J. Giblin, Curves and Singularities, 2nd ed. (Cambridge U. Press, Cambridge, 1992).

Callahan, J.

J. Callahan, R. S. Weiss, “A model for describing surface shape,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (Institute of Electrical and Electronics Engineers, New York, 1985), pp. 240–245.

Chhabra, A. K.

A. K. Chhabra, T. A. Grogan, “Uniqueness, the minimum norm constraint and analog networks for optical flow along contours,” in Proceedings of the Third International Conference on Computer Vision (Institute of Electrical and Electronics Engineers, New York, 1990), pp. 80–84.
[CrossRef]

Cipolla, R.

R. Cipolla, A. Blake, “Surface shape from the deformation of apparent contours,” Internat. J. Computer Vis. 9, 83–112 (1992).
[CrossRef]

A. Blake, R. Cipolla, “Robust estimation of surface curvature from deformations of apparent contours,” Image Vis. Comput. 9, 107–112 (1991).
[CrossRef]

R. Cipolla, A. Blake, “The dynamic analysis of occluding contours,” in Proceedings of the Third International Conference on Computer Vision (Institute of Electrical and Electronics Engineers, New York, 1990), pp. 616–623.
[CrossRef]

Faugeras, O. D.

R. Vaillant, O. D. Faugeras, “Using extremal boundaries for 3–D object modelling,” IEEE Trans. Pat. Anal. Mach. Intell. 14, 157–172 (1992).
[CrossRef]

Giblin, P. J.

P. J. Giblin, R. S. Weiss, “Reconstruction of surfaces from profiles,” in Proceedings of the First International Conference on Computer VisionInstitute of Electrical and Electronics Engineers, New York, 1987), pp. 136–144.

J. W. Bruce, P. J. Giblin, Curves and Singularities, 2nd ed. (Cambridge U. Press, Cambridge, 1992).

P. J. Giblin, J. E. Rycroft, F. E. Pollick, “Moving surfaces,” in Design and Applications of Curves and Surfaces, R. Fisher, ed., Institute of Mathematics and Its Applications Conference Proceedings Series (Oxford U. Press, Oxford, 1994), pp. 433–453.

Grogan, T. A.

A. K. Chhabra, T. A. Grogan, “Uniqueness, the minimum norm constraint and analog networks for optical flow along contours,” in Proceedings of the Third International Conference on Computer Vision (Institute of Electrical and Electronics Engineers, New York, 1990), pp. 80–84.
[CrossRef]

Hildreth, E.

E. Hildreth, The Measurement of Visual Motion (MIT Press, Cambridge, Mass., 1984).

Hoffman, D. D.

Huang, T. S.

T. S. Huang, C. H. Lee, “Motion and structure from orthographic projections,” IEEE Trans. Pat. Anal. Mach. Intell. 11, 536–540 (1989).
[CrossRef]

Kawato, M.

F. E. Pollick, S. Nishida, Y. Koike, M. Kawato, “Perceived motion in structure-from-motion: pointing responses to the axis of rotation,” Percept. Psychophys. (to be published).

Koenderink, J. J.

J. J. Koenderink, A. J. van Doorn, “Affine structure from motion,” J. Opt. Soc. Am. A 8, 377–386 (1991).
[CrossRef] [PubMed]

J. J. Koenderink, A. J. van Doorn, “The singularities of the visual mapping,” Biol. Cybernet. 24, 51–59 (1976).
[CrossRef]

J. J. Koenderink, A. J. van Doorn, “Invariant properties of the motion parallax field due to the movement of rigid bodies relative to an observer,” Opt. Acta 22, 773–791 (1975).
[CrossRef]

J. J. Koenderink, Solid Shape (MIT Press, Cambridge, Mass., 1990).

Koike, Y.

F. E. Pollick, S. Nishida, Y. Koike, M. Kawato, “Perceived motion in structure-from-motion: pointing responses to the axis of rotation,” Percept. Psychophys. (to be published).

Lee, C. H.

T. S. Huang, C. H. Lee, “Motion and structure from orthographic projections,” IEEE Trans. Pat. Anal. Mach. Intell. 11, 536–540 (1989).
[CrossRef]

Movshon, A.

A. Movshon, “Visual processing of moving images,” in Images and Understanding, H. Barlow, C. Blakemore, M. Weston-Smith, eds. (Cambridge U. Press, Cambridge, 1990).

Nakayama, K.

K. Nakayama, G. Silverman, “The aperture problem—I. Perception of nonrigidity and motion direction in translating sinusoidal fines,” Vision Res. 28, 739–746 (1988).
[CrossRef]

K. Nakayama, G. Silverman, “The aperture problem—II. Spatial integration of velocity information along contours,” Vision Res. 28, 747–753 (1988).
[CrossRef]

Nicola, J. E.

Nishida, S.

F. E. Pollick, S. Nishida, Y. Koike, M. Kawato, “Perceived motion in structure-from-motion: pointing responses to the axis of rotation,” Percept. Psychophys. (to be published).

Pollard, S.

J. Porrill, S. Pollard, “Curve matching and stereo calibration,” Image Vis. Comput. 9, 45–50 (1991).
[CrossRef]

Pollick, F. E.

P. J. Giblin, J. E. Rycroft, F. E. Pollick, “Moving surfaces,” in Design and Applications of Curves and Surfaces, R. Fisher, ed., Institute of Mathematics and Its Applications Conference Proceedings Series (Oxford U. Press, Oxford, 1994), pp. 433–453.

F. E. Pollick, S. Nishida, Y. Koike, M. Kawato, “Perceived motion in structure-from-motion: pointing responses to the axis of rotation,” Percept. Psychophys. (to be published).

Porrill, J.

J. Porrill, S. Pollard, “Curve matching and stereo calibration,” Image Vis. Comput. 9, 45–50 (1991).
[CrossRef]

Prakash, C.

Rieger, J. H.

Rycroft, J. E.

J. E. Rycroft, “A geometrical investigation into the projections of surfaces and space curves,” Ph.D. dissertation (University of Liverpool, Liverpool, UK, 1992).

P. J. Giblin, J. E. Rycroft, F. E. Pollick, “Moving surfaces,” in Design and Applications of Curves and Surfaces, R. Fisher, ed., Institute of Mathematics and Its Applications Conference Proceedings Series (Oxford U. Press, Oxford, 1994), pp. 433–453.

Shapiro, L. S.

L. S. Shapiro, A. P. Zisserman, M. Brady, “Motion from point matches using affine epipolar geometry,” Int. J. Computer Vision (to be published).

Silverman, G.

K. Nakayama, G. Silverman, “The aperture problem—I. Perception of nonrigidity and motion direction in translating sinusoidal fines,” Vision Res. 28, 739–746 (1988).
[CrossRef]

K. Nakayama, G. Silverman, “The aperture problem—II. Spatial integration of velocity information along contours,” Vision Res. 28, 747–753 (1988).
[CrossRef]

Vaillant, R.

R. Vaillant, O. D. Faugeras, “Using extremal boundaries for 3–D object modelling,” IEEE Trans. Pat. Anal. Mach. Intell. 14, 157–172 (1992).
[CrossRef]

van Doorn, A. J.

J. J. Koenderink, A. J. van Doorn, “Affine structure from motion,” J. Opt. Soc. Am. A 8, 377–386 (1991).
[CrossRef] [PubMed]

J. J. Koenderink, A. J. van Doorn, “The singularities of the visual mapping,” Biol. Cybernet. 24, 51–59 (1976).
[CrossRef]

J. J. Koenderink, A. J. van Doorn, “Invariant properties of the motion parallax field due to the movement of rigid bodies relative to an observer,” Opt. Acta 22, 773–791 (1975).
[CrossRef]

Webb, J. A.

J. A. Webb, J. K. Aggarwal, “Structure from motion of rigid and jointed objects,” Artif. Intell. 19, 107–130 (1982).
[CrossRef]

Weiss, R. S.

P. J. Giblin, R. S. Weiss, “Reconstruction of surfaces from profiles,” in Proceedings of the First International Conference on Computer VisionInstitute of Electrical and Electronics Engineers, New York, 1987), pp. 136–144.

J. Callahan, R. S. Weiss, “A model for describing surface shape,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (Institute of Electrical and Electronics Engineers, New York, 1985), pp. 240–245.

Zisserman, A. P.

L. S. Shapiro, A. P. Zisserman, M. Brady, “Motion from point matches using affine epipolar geometry,” Int. J. Computer Vision (to be published).

Artif. Intell.

J. A. Webb, J. K. Aggarwal, “Structure from motion of rigid and jointed objects,” Artif. Intell. 19, 107–130 (1982).
[CrossRef]

Biol. Cybernet.

J. J. Koenderink, A. J. van Doorn, “The singularities of the visual mapping,” Biol. Cybernet. 24, 51–59 (1976).
[CrossRef]

IEEE Trans. Pat. Anal. Mach. Intell.

R. Vaillant, O. D. Faugeras, “Using extremal boundaries for 3–D object modelling,” IEEE Trans. Pat. Anal. Mach. Intell. 14, 157–172 (1992).
[CrossRef]

T. S. Huang, C. H. Lee, “Motion and structure from orthographic projections,” IEEE Trans. Pat. Anal. Mach. Intell. 11, 536–540 (1989).
[CrossRef]

Image Vis. Comput.

J. Porrill, S. Pollard, “Curve matching and stereo calibration,” Image Vis. Comput. 9, 45–50 (1991).
[CrossRef]

A. Blake, R. Cipolla, “Robust estimation of surface curvature from deformations of apparent contours,” Image Vis. Comput. 9, 107–112 (1991).
[CrossRef]

Internat. J. Computer Vis.

R. Cipolla, A. Blake, “Surface shape from the deformation of apparent contours,” Internat. J. Computer Vis. 9, 83–112 (1992).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Acta

J. J. Koenderink, A. J. van Doorn, “Invariant properties of the motion parallax field due to the movement of rigid bodies relative to an observer,” Opt. Acta 22, 773–791 (1975).
[CrossRef]

Opt. Lett.

Vision Res.

K. Nakayama, G. Silverman, “The aperture problem—I. Perception of nonrigidity and motion direction in translating sinusoidal fines,” Vision Res. 28, 739–746 (1988).
[CrossRef]

K. Nakayama, G. Silverman, “The aperture problem—II. Spatial integration of velocity information along contours,” Vision Res. 28, 747–753 (1988).
[CrossRef]

Other

J. E. Rycroft, “A geometrical investigation into the projections of surfaces and space curves,” Ph.D. dissertation (University of Liverpool, Liverpool, UK, 1992).

P. J. Giblin, J. E. Rycroft, F. E. Pollick, “Moving surfaces,” in Design and Applications of Curves and Surfaces, R. Fisher, ed., Institute of Mathematics and Its Applications Conference Proceedings Series (Oxford U. Press, Oxford, 1994), pp. 433–453.

F. E. Pollick, S. Nishida, Y. Koike, M. Kawato, “Perceived motion in structure-from-motion: pointing responses to the axis of rotation,” Percept. Psychophys. (to be published).

L. S. Shapiro, A. P. Zisserman, M. Brady, “Motion from point matches using affine epipolar geometry,” Int. J. Computer Vision (to be published).

J. W. Bruce, P. J. Giblin, Curves and Singularities, 2nd ed. (Cambridge U. Press, Cambridge, 1992).

A. K. Chhabra, T. A. Grogan, “Uniqueness, the minimum norm constraint and analog networks for optical flow along contours,” in Proceedings of the Third International Conference on Computer Vision (Institute of Electrical and Electronics Engineers, New York, 1990), pp. 80–84.
[CrossRef]

E. Hildreth, The Measurement of Visual Motion (MIT Press, Cambridge, Mass., 1984).

A. Movshon, “Visual processing of moving images,” in Images and Understanding, H. Barlow, C. Blakemore, M. Weston-Smith, eds. (Cambridge U. Press, Cambridge, 1990).

J. Callahan, R. S. Weiss, “A model for describing surface shape,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (Institute of Electrical and Electronics Engineers, New York, 1985), pp. 240–245.

P. J. Giblin, R. S. Weiss, “Reconstruction of surfaces from profiles,” in Proceedings of the First International Conference on Computer VisionInstitute of Electrical and Electronics Engineers, New York, 1987), pp. 136–144.

J. J. Koenderink, Solid Shape (MIT Press, Cambridge, Mass., 1990).

R. Cipolla, A. Blake, “The dynamic analysis of occluding contours,” in Proceedings of the Third International Conference on Computer Vision (Institute of Electrical and Electronics Engineers, New York, 1990), pp. 616–623.
[CrossRef]

J. M. H. Beusmans, “Visual perception of solid shape from occluding contours,” Ph.D. dissertation (University of California, Irvine, Irvine, Calif., 1990)(also Tech. Rep. 90–40, Department of Information and Computer Science, University of California, Irvine, 1990).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (7)

Fig. 1
Fig. 1

Circular motion (orthogonal projection).

Fig. 2
Fig. 2

Orthogonal projection of a surface to a plane.

Fig. 3
Fig. 3

Gauss sphere: the part corresponding to the visible region and the frontier.

Fig. 4
Fig. 4

Critical sets along frontier F at (a) a nonparabolic point, (b) a parabolic point.

Fig. 5
Fig. 5

Tangents to a profile parallel to a given line: (a) full profile, (b) outer boundary of opaque profile.

Fig. 6
Fig. 6

Mean tilt recovered by the algorithm for the different conditions. Error bars indicate standard deviation.

Fig. 7
Fig. 7

Variability of the recovered tilts plotted versus angle β.

Equations (22)

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

w ( ϕ ) = ( c cos ϕ , c sin ϕ , s ) ,
f ( x , y , ϕ ) = h x cos ϕ + h y sin ϕ tan β .
F = { q M : n makes an angle ± β with the axis } .
n ( x 1 , x 2 ) · [ q ( x 1 , x 2 ) c ( t ) ] = 0 ,
e 1 ( ϕ ) = ( sin ϕ , cos ϕ , 0 ) , e 2 = ( s cos ϕ , s sin ϕ , c ) ,
V u = 0 ;
λ c V u = s ( u + V V u ) V ϕ .
d d ϕ ( V ( U ( ϕ ) , ϕ ) ) = V ϕ ( U ( ϕ ) , ϕ ) ,
F = ( ( x , y , h ( x , y ) ) : h x = cos ϕ tan β , h y = sin ϕ tan β ) .
u = x sin ϕ + y cos ϕ , υ = x s cos ϕ + y s sin ϕ c z ,
V t 1 u 1 d = V t 2 u 2 d = const . = ω sin β .
V t 1 V t 2 u 1 u 2 .
e 1 = c w s e 2 , e 2 = s e 1 w = c e 1 ( = d / d ϕ here )
0 = ( F / ϕ ) ( V u e 1 + e 2 ) = u ( s ) + V ϕ + V ( s V u ) + λ ( c V u ) .
u = u cos θ + υ sin θ , u = u sin θ + υ cos θ ,
d υ d ϕ 1 u
c k h ( x , y ) tan θ = ( x sin ϕ + y cos ϕ ) ( k s ) + tan θ ( s k 1 ) ( x cos ϕ + y sin ϕ ) .
H ( X , Y , ϕ ) = h ( X cos ϕ Y sin ϕ , X sin ϕ + Y cos ϕ ) .
Y H X X H Y + H ϕ = 0 .
k c tan θ H ( X , Y , ϕ ) = ( k s ) Y + tan θ ( k s 1 ) X
H X = tan β = p , say , H Y = tan θ sec β = q , say ,
H ( X , Y , ϕ ) = a X + b Y .

Metrics