This Article 
   
 Share 
   
 Bibliographic References 
   
 Add to: 
 
Digg
Furl
Spurl
Blink
Simpy
Google
Del.icio.us
Y!MyWeb
 
 Search 
   
Line-Drawing Interpretation: Straight Lines and Conic Sections
July 1988 (vol. 10 no. 4)
pp. 514-529

Line drawings of man-made scenes often exhibit instances of straight lines and conic sections, i.e. ellipses, parabolas, and hyperbolas. Constraints imposed on the scene by such instances are investigated, under the assumption of general viewpoint, i.e. the mapping of the viewed surface onto the line drawing is stable under perturbation of the viewpoint within some open set. Both orthographic and perspective projection are considered. The viewed surfaces are assumed to be piecewise C/sup 3/. It is shown that straight lines and conic sections in line drawings are projections of scene edges which are also straight lines and conic sections, respectively. It is also shown that scene events which project onto straight lines or conic sections cannot be combinations of view-point-independent and viewpoint-dependent edges. Further, continuous-surface-normal depth discontinuities which project onto straight lines can be locally described by developable surfaces, and those which project onto conic sections can be locally described by nondevelopable quadric surfaces. Each of these quadric surfaces is determined up to four degrees-of-freedom by its projection.

[1] W. Blaschke,Vorlesungenüber Differential geometrie II. Berlin: Springer, 1923.
[2] E. A. Coddington,An Introduction to Ordinary Differential Equations. Englewood Cliffs, NJ: Prentice-Hall, 1961.
[3] D. Gans,Transformations and Geometries. New York: Appleton, 1969.
[4] V. Guillemin, and A. Pollack,Differential Topology. Englewood Cliffs, NJ: Prentice-Hall, 1974.
[5] D. Hilbert, and S. Cohn-Vossen,Geometry and the Imagination. New York: Chelsea, 1952.
[6] M. W. Hirsch,Differential Topology. New York: Springer-Verlag, 1976.
[7] J. J. Koenderink, "What does the occluding countour tell us about solid shape?"Perception, vol. 13, pp. 321-330, 1984.
[8] M. M. Lipschutz,Differential Geometry. New York: McGraw-Hill, 1969.
[9] H. Maschke, "On superosculating quadric surfaces,"Trans. Amer. Math. Soc., vol. 3, pp. 482-484, 1902.
[10] J. M. H. Olmsted,Solid Analytic Geometry. New York: Appleton, 1947.
[11] M. Spivak,Calculus on Manifolds. New York: Benjamin/Cummings, 1965.
[12] G. Strang,Linear Algebra and Its Applications. New York: Academic, 1976.
[13] H. Whitney, "On singularities of mappings of euclidean spaces. I. Mappings of the plane into the plane,"Ann. Math., vol. 62, no. 3, pp. 374-410, Nov. 1955.

Index Terms:
line drawing interpretation; picture processing; pattern recognition; straight lines; conic sections; ellipses; parabolas; hyperbolas; perturbation; line drawings; scene edges; quadric surfaces; pattern recognition; picture processing
Citation:
V.S. Nalwa, "Line-Drawing Interpretation: Straight Lines and Conic Sections," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 10, no. 4, pp. 514-529, July 1988, doi:10.1109/34.3914
Usage of this product signifies your acceptance of the Terms of Use.