Abstract

Rectilinear propagation of light rays in homogeneous isotropic media makes it possible for optical generation of ruled surfaces as the ray is deflected by a rotatable mirror. Scan patterns on a plane or curved surface are merely curves on the ruled surface. Based on this understanding, structures of the scan fields produced by mirror-scanning devices of different configurations are investigated in terms of differential geometry. Expressions of the first and second fundamental coefficients and the first and second Gauss differential forms are given for an investigation of the intrinsic properties of the optically generated ruled surfaces. The Plücker ruled conoid is then generalized for mathematical modeling of the scan fields produced by single-mirror scanning devices of different configurations. Part II will be devoted to a study of multi-mirror scanning systems for optical generation of well-known ruled surfaces such as helicoids and hyperbolic paraboloids.

© 2011 Optical Society of America

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  1. D. Hilbert and S. Cohn-Vossen, Geometry and the Imagination, (Chelsea, 1956), pp. 15–341.
  2. A. Gray, E. Abbena, and S. Salamon, Modern Differential Geometry of Curves and Surfaces with Mathematica, 3rd ed. (Chapman & Hall/CRC, 2006), Chap. 14, “Ruled Surfaces.”
  3. V. I. Smirnov, A Course of Higher Mathemantics II: Advanced Calculus, trans. by D. E. Brown, trans. ed. I. N. Sneddon (Pergamon, 1964), Chap. V, “Foundations of Differential Geometry.”
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  9. R. J. Sherman, “Polygon scanners: applications, performance and design,” in Optical Scanning, G.F.Marshall, ed. (Dekker, 1991), pp. 351–406.
  10. W. L. Edge, The Theory of Ruled Surfaces, 1st ed. (Cambridge University, 1931), Sec. 3.2.2.

2008 (2)

1997 (1)

1995 (2)

Abbena, E.

A. Gray, E. Abbena, and S. Salamon, Modern Differential Geometry of Curves and Surfaces with Mathematica, 3rd ed. (Chapman & Hall/CRC, 2006), Chap. 14, “Ruled Surfaces.”

Cohn-Vossen, S.

D. Hilbert and S. Cohn-Vossen, Geometry and the Imagination, (Chelsea, 1956), pp. 15–341.

Edge, W. L.

W. L. Edge, The Theory of Ruled Surfaces, 1st ed. (Cambridge University, 1931), Sec. 3.2.2.

Gray, A.

A. Gray, E. Abbena, and S. Salamon, Modern Differential Geometry of Curves and Surfaces with Mathematica, 3rd ed. (Chapman & Hall/CRC, 2006), Chap. 14, “Ruled Surfaces.”

Hilbert, D.

D. Hilbert and S. Cohn-Vossen, Geometry and the Imagination, (Chelsea, 1956), pp. 15–341.

Katz, J.

Li, Y.

Salamon, S.

A. Gray, E. Abbena, and S. Salamon, Modern Differential Geometry of Curves and Surfaces with Mathematica, 3rd ed. (Chapman & Hall/CRC, 2006), Chap. 14, “Ruled Surfaces.”

Sherman, R. J.

R. J. Sherman, “Polygon scanners: applications, performance and design,” in Optical Scanning, G.F.Marshall, ed. (Dekker, 1991), pp. 351–406.

Smirnov, V. I.

V. I. Smirnov, A Course of Higher Mathemantics II: Advanced Calculus, trans. by D. E. Brown, trans. ed. I. N. Sneddon (Pergamon, 1964), Chap. V, “Foundations of Differential Geometry.”

Appl. Opt. (5)

Other (5)

D. Hilbert and S. Cohn-Vossen, Geometry and the Imagination, (Chelsea, 1956), pp. 15–341.

A. Gray, E. Abbena, and S. Salamon, Modern Differential Geometry of Curves and Surfaces with Mathematica, 3rd ed. (Chapman & Hall/CRC, 2006), Chap. 14, “Ruled Surfaces.”

V. I. Smirnov, A Course of Higher Mathemantics II: Advanced Calculus, trans. by D. E. Brown, trans. ed. I. N. Sneddon (Pergamon, 1964), Chap. V, “Foundations of Differential Geometry.”

R. J. Sherman, “Polygon scanners: applications, performance and design,” in Optical Scanning, G.F.Marshall, ed. (Dekker, 1991), pp. 351–406.

W. L. Edge, The Theory of Ruled Surfaces, 1st ed. (Cambridge University, 1931), Sec. 3.2.2.

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