## 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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### Equations (12)

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(1)
$$\mathbf{r}(\theta ,\rho )={x}_{0}(1,0,-\mathrm{cos}\theta )+\rho (\mathrm{cos}\theta ,\mathrm{sin}\theta ,0),$$
(2)
$$\mathbf{r}(\theta ,\rho )-({x}_{0},0,0)={x}_{0}(0,0,-\mathrm{cos}\theta )+\rho (\mathrm{cos}\theta ,\mathrm{sin}\theta ,0).$$
(3)
$$\mathbf{r}(\theta ,\rho )=\alpha (0,0,\theta )+\rho (\mathrm{cos}\theta ,\mathrm{sin}\theta ,0),$$
(4)
$$\mathbf{r}(\theta ,\rho )=2d(0,0,\mathrm{sin}q\theta )+\rho (\mathrm{cos}\theta ,\mathrm{sin}\theta ,0),$$
(5)
$${\widehat{\mathbf{n}}}_{A}=(\mathrm{cos}\theta ,0,-\mathrm{sin}\theta )=(\mathrm{cos}\omega t,0,-\mathrm{sin}\omega t),$$
(6)
$${\widehat{\mathbf{s}}}_{A}^{(i)}=(0,w-vt,0),$$
(7)
$$\mathbf{r}(\theta ,\rho )=(0,w-vt,0)+\rho (\mathrm{cos}2\theta ,0,-\mathrm{sin}2\theta ).$$
(8)
$$x=\rho \mathrm{cos}2\theta ,\phantom{\rule[-0.0ex]{2.0em}{0.0ex}}y=w-vt,\phantom{\rule[-0.0ex]{1.0em}{0.0ex}}\text{and}\phantom{\rule[-0.0ex]{1.0em}{0.0ex}}z=-\rho \mathrm{sin}2\theta .$$
(9)
$$\frac{z}{x}=-\mathrm{tan}2\theta =\mathrm{tan}[2(y-w)\frac{\omega}{v}]=\mathrm{tan}\left(2\frac{y-w}{OP}\right).$$
(10)
$$z=2\frac{x(y-w)}{OP},$$
(11)
$$\left(\frac{z}{OP}\right)=2\left(\frac{x}{OP}\right)\left(\frac{y-w}{OP}\right).$$
(12)
$$\left(\frac{z}{OP}\right)=(\frac{x}{OP}{)}^{2}-(\frac{y-w}{OP}{)}^{2}.$$