### Geometric Representation of Aspherical Surface

An aspheric surface is a rotationally symmetrical surface that deviates from a spherical surface in shape. A spherical or aspherical surface with a vertex at the origin and an optical axis along the *Z-*axis can be expressed as

$$zleft( h right) = frac{{ch^{2} }}{{1 + sqrt {1 – (1 + k)c^{2} h^{2} } }} + sum {A_{m} h^{2m} } ,$$

(1)

where (c = {1 mathord{left/ {vphantom {1 R}} right. kern-nulldelimiterspace} R}_{0}) is the vertex curvature; *R*_{0} is the vertex curvature radius; *h* is the radial distance from the optical axis (Figure 1); *k* is the conic constant that determines the shape of the surface; (A_{m} h^{2m}) represents the high-order terms of the aspherical surface, where *m* is an integer.

Eq. (1) is reduced to a quadratic aspheric surface if higher-order terms are absent. It can be rewritten as

$$frac{{left( {z – frac{1}{{left( {1 + k} right)c}}} right)^{2} }}{{left( {frac{1}{{left( {1 + k} right)c}}} right)^{2} }} + frac{{h^{2} }}{{left( {frac{1}{{sqrt {1 + k} c}}} right)^{2} }} = 1.$$

(2)

For a point *P* (*x*, *y*, *z*) on a quadratic aspherical surface Figure 2), its coordinates can be expressed as a parametric equation, i.e.,

$$left{ {begin{array}{*{20}l} {x = hcos theta ,} hfill \ {y = hsin theta ,} hfill \ {z = frac{{ch^{2} }}{{1 + sqrt {1 – (1 + k)c^{2} h^{2} } }},} hfill \ end{array} } right.$$

(3)

where *θ* is the angle between the positive direction of the *X*-axis and the projected vector of (overrightarrow {OP}) on the *XOY* plane, as shown in Figure 2.

### Geometric Representation of Arc Grinding Wheel

A parallel grinding wheel with a circular arc was used for the arc envelope grinding of aspheric optical elements. The coordinate system of the grinding wheel is illustrated in Figure 3. The *Y*′-axis was along the wheel axis, and the origin *O*′ was located at the intersection of the *Y*′-axis and the axial symmetrical surface of the wheel. The wheel radius in this symmetry plane, i.e., *X*′*O*′*Z*′ coordinate plane, was *R*. The wheel surface with abrasive grains was rotationally symmetrical along the *Y*′-axis. Its generatrix is an arc with a radius *QP*′ = *r*. The distance from the arc center *Q* to the wheel axis is (a = R – r). The distance from point *P*′ on the wheel surface to the *Y*′-axis is denoted as *l*. The angle between the line *P*′*O*′ and *X*′*O*′*Y*′ plane is *θ*′ (clockwise), which ranges from 0° to 360°. The angle between the radius *QP*′ and the *X*′*O*′*Z*′ plane is *γ*′.

In the *O′X′Y′Z′* coordinate system, the wheel surface can be described by the following equations:

$$begin{gathered} left{ begin{gathered} x^{prime} = lcos theta^{prime}, hfill \ y^{prime} = rsin gamma^{prime}, hfill \ z^{prime} = lsin theta^{prime}, hfill \ end{gathered} right. hfill \ l = a + rcos gamma^{prime}, hfill \ end{gathered}$$

(4)

where the range of *γ′* is (- arcsin left( {{W mathord{left/ {vphantom {W {2r}}} right. kern-nulldelimiterspace} {2r}}} right) le gamma^{prime} le arcsin left( {{W mathord{left/ {vphantom {W {2r}}} right. kern-nulldelimiterspace} {2r}}} right)), where *W* is the width of the grinding wheel.

### Modeling of Aspherical Surface Generation

During the modeling of the generation process of the aspherical surface, the *X-*, *Y-*, and *Z-*axes of the workpiece coordinate system are assumed to be parallel to the *X*′-, *Y*′*–*, and *Z*′-axes of the wheel coordinate system, respectively, as shown in Figure 4. The workpiece surface was tangential to the wheel surface during grinding at the grinding point. The positions of the grinding point on both the wheel and workpiece changed during grinding. The coordinates in the wheel coordinate system can be transformed into the workpiece coordinate system via the following translation transformation:

$$left[ {begin{array}{*{20}c} x \ y \ z \ end{array} } right] = left[ {begin{array}{*{20}c} {x^{prime}} \ {y^{prime}} \ {z^{prime}} \ end{array} } right] + {varvec{B}},$$

(5)

where ** B** = (overrightarrow {{OO^{prime}}}) is the translation vector, which corresponds to the coordinates of

*O′*in the workpiece coordinate system.

*O′*is the programming point and its trajectory is the grinding path.

$${varvec{B}} = left[ {begin{array}{*{20}c} {X_{s} } \ {Y_{s} } \ {Z_{s} } \ end{array} } right].$$

(6)

In the initial grinding state, the *X-* and *Y-*components of *B*_{0} represent the wheel setting errors along the tangential and axial directions of the wheel, respectively.

For parallel grinding, the wheel moves along the *Y-* and *Z-*axes, and the velocity directions of the grinding wheel and workpiece are parallel at the grinding point, as shown in Figure 5a. Therefore, *X*_{s} in vector ** B** remains constant and is the lateral wheel setting error. For cross grinding, the wheel moves along the

*X-*and

*Z-*axes, and the velocity directions of the wheel and workpiece are perpendicular to each other at the grinding point, as shown in Figure 5b. Hence,

*Y*

_{s}remains constant during grinding and is the lateral wheel setting error.

The grinding point is the intersection point between the workpiece and wheel surfaces. Combining Eqs. (3)‒(5) yields

$$left[ {begin{array}{*{20}c} {hcos theta } \ {hsin theta } \ {frac{{ch^{2} }}{{1 + sqrt {1 – (1 + k)c^{2} h^{2} } }}} \ end{array} } right] = left[ {begin{array}{*{20}c} {(a + rcos gamma^{prime})cos theta^{prime}} \ {rsin gamma^{prime}} \ {(a + rcos gamma^{prime})sin theta^{prime}} \ end{array} } right] + left[ {begin{array}{*{20}c} {X_{s} } \ {Y_{s} } \ {Z_{s} } \ end{array} } right].$$

(7)

In addition, the workpiece surface is tangential to the wheel surface at the grinding point. Therefore, the normals of the workpiece and wheel surfaces are collinear at the grinding point. The normal vector at point *P*(*h*, *θ*) on the aspherical surface can be expressed as

$$vec{n}_{s} = left[ {begin{array}{*{20}c} {frac{partial (y,z)}{{partial (h,theta )}}} \ {frac{partial (z,x)}{{partial (h,theta )}}} \ {frac{partial (x,y)}{{partial (h,theta )}}} \ end{array} } right] = left[ {begin{array}{*{20}c} { – frac{{{text{d}}z}}{{{text{d}}h}}hcos theta } \ { – frac{{{text{d}}z}}{{{text{d}}h}}hsin theta } \ h \ end{array} } right],$$

(8)

where

$$begin{aligned} frac{{{text{d}}z}}{{{text{d}}h}} = z^{prime}left( h right) = frac{ch}{{1 + sqrt {1 – left( {k + 1} right)c^{2} h^{2} } }} hfill \ left( {2 + frac{{left( {k + 1} right)c^{2} h^{2} }}{{left( {1 + sqrt {1 – left( {k + 1} right)c^{2} h^{2} } } right)sqrt {1 – left( {k + 1} right)c^{2} h^{2} } }}} right). hfill \ end{aligned}$$

(9)

Similarly, the normal vector at point *P′*(*γ′*, *θ′*) on the grinding wheel can be written as

$$vec{n}_{w} = left[ {begin{array}{*{20}c} {frac{partial (y,z)}{{partial (gamma^{prime},theta^{prime})}}} \ {frac{partial (z,x)}{{partial (gamma^{prime},theta^{prime})}}} \ {frac{partial (x,y)}{{partial (gamma^{prime},theta^{prime})}}} \ end{array} } right] = left[ {begin{array}{*{20}c} {rleft( {a + rcos gamma^{prime}} right)cos gamma^{prime}cos theta^{prime}} \ {rleft( {a + rcos gamma^{prime}} right)sin gamma^{prime}} \ {rleft( {a + rcos gamma^{prime}} right)sin theta^{prime}cos gamma^{prime}} \ end{array} } right].$$

(10)

The normals of the workpiece and wheel surfaces at the grinding point are collinear. Therefore, the following equations can be obtained:

$$begin{aligned} frac{{ – frac{{{text{d}}z}}{{{text{d}}h}}hcos theta }}{{rleft( {a + rcos gamma^{prime}} right)cos gamma^{prime}cos theta^{prime}}} &= frac{{ – frac{{{text{d}}z}}{{text{d}}h}hsin theta }}{{rleft( {a + rcos gamma^{prime}} right)sin gamma^{prime}}} hfill \ &= frac{h}{{rleft( {a + rcos gamma^{prime}} right)sin theta^{prime}cos gamma^{prime}}}. hfill \ end{aligned}$$

(11)

The coordinates of the grinding point can be obtained by solving Eqs. (7) and (11). Subsequently, the corresponding grinding path *Z*_{s} = *f*(*X*_{s}) or *Z*_{s} = *f*(*Y*_{s}) can be obtained as follows:

$$left[ {begin{array}{*{20}c} {hcos theta } \ {hsin theta } \ {zleft( h right)} \ end{array} } right] = left[ {begin{array}{*{20}c} {(a + rcos gamma^{prime})cos theta^{prime}} \ {rsin gamma^{prime}} \ {(a + rcos gamma^{prime})sin theta^{prime}} \ end{array} } right] + left[ {begin{array}{*{20}c} {X_{s} } \ {Y_{s} } \ {Z_{s} } \ end{array} } right].$$

(12)

If the grinding path is known, then the expression of (zleft( h right)) can be obtained by solving Eqs. (11) and (12).

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