## Abstract

We present a method to obtain the optimum surface shape for use as a starting point for the machining of aspherical surfaces of revolution. Applying this method, the volume that remains to be machined away can be set below an acceptable value. Subsequently, it is shown how this method can be applied for conic surfaces.

© 1997 Optical Society of America

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### Equations (11)

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(1)
$${f}_{t}\left(r\right)=\pm \left[h-{\left({R}^{2}-{r}^{2}\right)}^{1/2}\right],$$
(2)
$$h={\left({R}^{2}-{{r}_{o}}^{2}\right)}^{1/2}+\left|{f}_{a}\left({r}_{o}\right)\right|,$$
(3)
$$R=R\left({r}_{o}\right)={r}_{o}{\left[1+{\left(\partial {f}_{a}\left({r}_{o}\right)/\partial r\right)}^{-2}\right]}^{1/2},$$
(4)
$$\begin{array}{ll}\text{A})& {f}_{t}\left(r\right)\ge {f}_{a}\left(r\right)\\ \text{B})& {f}_{t}\left({r}_{o}\right)={f}_{a}\left({r}_{o}\right)\\ \text{C})& \text{if}{\text{f}}_{a}\left(r\right)\text{concave:}\text{\rho}\left({r}_{o}\right)\ge R\left({r}_{o}\right)\\ & \text{if}{\text{f}}_{a}\left(r\right)\text{convex:}\text{\rho}\left({r}_{o}\right)\le R\left({r}_{o}\right)\end{array}$$
(5)
$$\begin{array}{cc}\text{D})& \mathrm{lim}R\left({r}_{o}\right)=\text{\rho}\left(0\right)\\ & {r}_{o}\to 0\end{array}$$
(6)
$${f}_{s}\left({r}_{\mathrm{\Delta}}\right)=-{\left[\partial {f}_{a}\left({r}_{n}\right)/\partial r\right]}^{-1}\left({r}_{\mathrm{\Delta}}-{r}_{n}\right)+{f}_{a}\left({r}_{n}\right),$$
(7)
$$d={\left[{\left({r}_{n}-{r}_{\mathrm{\Delta}}\right)}^{2}+{\left({f}_{s}\left({r}_{\mathrm{\Delta}}\right)-{f}_{a}\left({r}_{n}\right)\right)}^{2}\right]}^{1/2}$$
(8)
$$V=\iint r\left({f}_{s}\left(r\right)-{f}_{a}\left(r\right)\right)drd\mathrm{\varphi}$$
(9)
$${f}_{c}\left(r\right)=\pm {r}^{2}/\left(p+{\left[{p}^{2}-{r}^{2}\left(k+1\right)\right]}^{1/2}\right)$$
(10)
$${{f}^{\prime}}_{c}\left(r\right)=\partial {f}_{c}\left(r\right)/\partial r=\pm r/{\left[{p}^{2}-{r}^{2}\left(k+1\right)\right]}^{1/2}$$
(11)
$${{f}^{\u2033}}_{c}\left(r\right)={\partial {f}^{\prime}}_{c}\left(r\right)/\partial r={p}^{2}/{\left[{p}^{2}-{r}^{2}\left(k+1\right)\right]}^{3/2}$$