Abstract

The theory developed in Part I of this study [Y. Li, “Differential geometry of the ruled surfaces optically generated by mirror-scanning devices. I. Intrinsic and extrinsic properties of the scan field,” J. Opt. Soc. Am. A 28, 667 (2011)] for the ruled surfaces optically generated by single-mirror scanning devices is extended to multimirror scanning systems for an investigation of optical generation of the well-known ruled surfaces, such as helicoid, Plücker’s conoid, and hyperbolic paraboloid.

© 2011 Optical Society of America

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References

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  1. Y. Li, “Differential geometry of the ruled surfaces optically generated by mirror scanning devices: I. Intrinsic and extrinsic properties of the scan field,” J. Opt. Soc. Am. A 28, 667–674(2011).
    [CrossRef]
  2. E. W. Weisstein, CRC Concise Encyclopedia of Mathematics, 2nd ed. (Chapman & Hall/CRC, 2003).
  3. URL: http://en.wikipedia.org/wiki/Line_(geometry).
  4. URL: http://www.mathopenref.com/ray.html.
  5. A. Gray, E. Abbena, and S. Salamon, Modern Differential Geometry of Curves and Surfaces with Mathematica, 3rd ed. (Chapman & Hall/CRC, 2006), Chap. 14.
  6. URL: http://mathworld.wolfram.com/PlueckersConoid.html.
  7. D. Hilbert and S. Cohn-Vossen, Geometry and the Imagination (Chelsea, 1956), pp. 16, pp. 307–315.
  8. URL: http://en.wikipedia.org/wiki/Helicoid.
  9. URL: http://en.wikipedia.org/wiki/Ruled_surface.
  10. URL: http://mathworld.wolfram.com/HyperbolicParaboloid.html.

2011 (1)

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.

Cohn-Vossen, S.

D. Hilbert and S. Cohn-Vossen, Geometry and the Imagination (Chelsea, 1956), pp. 16, pp. 307–315.

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.

Hilbert, D.

D. Hilbert and S. Cohn-Vossen, Geometry and the Imagination (Chelsea, 1956), pp. 16, pp. 307–315.

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.

Weisstein, E. W.

E. W. Weisstein, CRC Concise Encyclopedia of Mathematics, 2nd ed. (Chapman & Hall/CRC, 2003).

J. Opt. Soc. Am. A (1)

Other (9)

E. W. Weisstein, CRC Concise Encyclopedia of Mathematics, 2nd ed. (Chapman & Hall/CRC, 2003).

URL: http://en.wikipedia.org/wiki/Line_(geometry).

URL: http://www.mathopenref.com/ray.html.

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

URL: http://mathworld.wolfram.com/PlueckersConoid.html.

D. Hilbert and S. Cohn-Vossen, Geometry and the Imagination (Chelsea, 1956), pp. 16, pp. 307–315.

URL: http://en.wikipedia.org/wiki/Helicoid.

URL: http://en.wikipedia.org/wiki/Ruled_surface.

URL: http://mathworld.wolfram.com/HyperbolicParaboloid.html.

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Figures (4)

Fig. 1
Fig. 1

(a) Plücker’s conoid of the one fold optically generated by a single-facet cantilevered scanner with a 45 ° titled mirror relative to the axis, the incident ray is parallel with the axis but displaced from the axis with a distance of x 0 . (b) Construction of the surface near the base curve. The process of construction starts with the creation of a pattern on a rectangular sheet of paper in the ( z , θ ) coordinates. The paper is then pasted around a carton cylinder; holes are drilled at the designated spots and nailed in holes with skewers.

Fig. 2
Fig. 2

(a) Plücker’s conoid of the one fold optically generated by scanning a straight line using the device shown in Fig. 3a. (b) The same as Fig. 1b.

Fig. 3
Fig. 3

(a) Generation of a scanning straight line that is infinitely long by a rotating pyramidal polygon with two 45 ° inclination mirrors to divide the incident narrow beam into two narrow beams R A and R B . (b) Generation of a helicoid with a constant pitch by using the proposed double- facet rotating pyramidal polygon moving along the axis of rotation with a constant sped. (c) Plücker’s conoids of two and three folds and the generation of a Plücker’s conoid by using the proposed double-facet rotating pyramidal polygon moving within a line segment on the axis of rotation.

Fig. 4
Fig. 4

(a) Schematic diagram showing the rulings on the surface of a hyperbolic paraboloid. (b) Scanning system for optical generation of the hyperbolic paraboloid. (c) Illustration of the notation.

Equations (12)

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r ( θ , ρ ) = x 0 ( 1 , 0 , cos θ ) + ρ ( cos θ , sin θ , 0 ) ,
r ( θ , ρ ) ( x 0 , 0 , 0 ) = x 0 ( 0 , 0 , cos θ ) + ρ ( cos θ , sin θ , 0 ) .
r ( θ , ρ ) = α ( 0 , 0 , θ ) + ρ ( cos θ , sin θ , 0 ) ,
r ( θ , ρ ) = 2 d ( 0 , 0 , sin q θ ) + ρ ( cos θ , sin θ , 0 ) ,
n ^ A = ( cos θ , 0 , sin θ ) = ( cos ω t , 0 , sin ω t ) ,
s ^ A ( i ) = ( 0 , w v t , 0 ) ,
r ( θ , ρ ) = ( 0 , w v t , 0 ) + ρ ( cos 2 θ , 0 , sin 2 θ ) .
x = ρ cos 2 θ , y = w v t , and z = ρ sin 2 θ .
z x = tan 2 θ = tan [ 2 ( y w ) ω v ] = tan ( 2 y w O P ) .
z = 2 x ( y w ) O P ,
( z O P ) = 2 ( x O P ) ( y w O P ) .
( z O P ) = ( x O P ) 2 ( y w O P ) 2 .

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