Abstract:
A method for machining a face ( 1 ) of an optical object ( 6 ), includes providing a machine tool which itself includes a bed ( 1 ) for locating an object to be machined. The bed ( 1 ) has a receiving surface ( 3 ) that is angularly adjustable about an axis perpendicular to the receiving surface ( 3 ). A spindle ( 8 ) is suitable for rotating a machining tool ( 9 ) about an axis essentially parallel to the receiving surface ( 3 ) of the bed ( 1 ) and is suitable for moving the machining tool ( 9 ) translationally in a plane essentially parallel or perpendicular to the receiving surface ( 3 ) of the bed ( 1 ).

Description:
BACKGROUND OF THE INVENTION 
     1. Field of the Invention 
     The invention concerns the field of the fabrication of optical objects, such as ophthalmic lenses, molds or inserts, for example. 
     The invention more particularly concerns a method of machining one face of such an optical object. 
     2. Description of the Related Art 
     Machining optical objects generally necessitates particular attention as to the precision and the regularity of the machined shapes. In particular, machining defects linked to wear of the tool employed for this machining must be avoided. 
     Under these conditions, complex and costly machines necessitating delicate calibration are generally employed in this field. 
     For example, the document U.S. Pat. No. 5,231,587 describes a machine tool for lenses including a spherical tool mounted turning about its longitudinal axis, called the first axis, this tool moreover being orientable angularly by its pivoting about a second axis perpendicular to the first axis. A part-carrier, intended to support the lens, is arranged in a similar manner and enables rotation of the lens about a third axis, coplanar with the first axis, and enables angular orientation of the lens by its pivoting about a fourth axis perpendicular to the third axis. 
     There is also known from the document JP 2005 22 49 27 a machining method in the course of which a machining tool is positioned relative to a part to be machined so that the vector connecting a machining point and the center of the tool forms with the vector normal to the surface to be machined at said machining point a constant angle throughout the machining procedure. 
     SUMMARY OF THE INVENTION 
     The object of the invention is to improve the machining devices and methods the precision whereof is adapted to the machining of optical objects. 
     To this end, the invention is directed to a method of machining a face of an optical object, including a step of providing a machine tool that itself includes:
         a table for mounting an object to be machined, this table, which includes a receiving surface, being orientable angularly about an axis transverse to the receiving surface;   a spindle adapted to drive a machining tool in rotation about an axis substantially parallel to the receiving surface of the table and adapted to move this machining tool in translation in a plane substantially parallel or perpendicular to the receiving surface of the table;       

     this method being characterized in that it further includes the following steps: 
     a) fixing a support to the table so that this support projects transversely to the table; 
     b) fixing to the support of the optical object to be machined so that said face to be machined is disposed transversely to the receiving surface of the table; 
     c) machining of said face by the machining tool along a trajectory substantially parallel to the receiving surface of the table, the table being angularly oriented as the machining proceeds so that the machining tool is always in contact with said face on a predetermined same parallel and that a predetermined angle is maintained between the rotation axis of the machining tool and the normal to said face at the point of contact with the machining tool. 
     Such a method circumvents defects of machining tool shape error type. In the end it guarantees a better appearance of the machined surface and better durability of the machining tool. 
     The method circumvents the defects of the machining tool by ensuring that the point of contact between this tool and the face to be machined is always situated on a same parallel of the tool, and this on a machine having a rotating table and a machining tool mobile in translation. 
     This method further enables a trajectory of the machining tool that involves, in the first place, lower levels of acceleration and that, in the second place, is free of problems of reversing the trajectory. The spindles of the machine tool therefore do not need to be overspecified and wear of the tools is more regular. 
     For example, compared to a standard spiral machining trajectory, these advantages linked to the levels of acceleration and to reversing problems are complemented by the fact that, along the Cartesian trajectories enabled by the invention, there is no singular point at the center of the lens where, along a spiral trajectory, the rate of advance is zero at the center. Moreover, the machine tool of the invention enables machining of only the necessary portion of the lens. 
     According to preferred features, taken separately or in combination:
         the method further includes the following steps, after the step c):   moving the machining tool in translation in a direction substantially perpendicular to the receiving surface of the table;   where applicable, repetition of the step c);   the method further includes the following step, before the step c):   machining of said face by the machining tool along a trajectory substantially perpendicular to the receiving surface of the table, the table being angularly oriented as the machining proceeds so that the machining tool is always in contact with said face along a predetermined same parallel and that a predetermined angle is maintained between the rotation axis of the machining tool and the normal to said face at the point of contact with the machining tool;   the machining method further includes, before the step c), a step of plotting the dynamic contour of the machining tool;   the plotting of the dynamic contour of the machining tool is effected by driving the machining tool in front of means for plotting a profile;   the step of plotting the dynamic contour of the machining tool is followed by a step of selecting a predetermined parallel;   said predetermined parallel is selected from the planes perpendicular to the rotation axis of the machining tool and that intersect the dynamic contour of the machining tool;   the step of selecting a predetermined parallel is followed by a step of determining the dynamic center of the machining tool;   the step of determining the dynamic center is effected by determining the intersection between the normal to the dynamic contour of the machining tool at one of the points of intersection between the predetermined parallel and the contour of the machining tool, and the rotation axis of the machining tool;   the step c) is effected by angularly orienting the table as the machining proceeds so that the normal to said face to be machined at the point of contact between the machining tool and said face passes through the dynamic center of the machining tool;   the distance between the point of contact and the dynamic center is substantially equal to the dynamic radius of the machining tool;   the machining method further includes the following step:   machining of said face by the machining tool along a trajectory parallel to the receiving surface of the table and in the opposite direction to that of the step c), the machining tool turning in the same direction.       

     Another object of the invention is a machine tool adapted to the implementation of the method previously indicated, characterized in that it includes a rotating table having a receiving surface and a spindle adapted to drive a machining tool in rotation about an axis substantially parallel to the receiving surface of the rotating table and adapted to move this machining tool in translation in a plane substantially parallel to the receiving surface of the table, and a support fixed to the table so that this support projects transversely to the table, this support including means for holding the optical object so that the face to be machined of the optical object is disposed transversely to the receiving surface of the rotating table. 
     According to preferred features, taken separately or in combination:
         the spindle is also adapted to move the machining tool in translation in a direction substantially perpendicular to the receiving surface of the rotating table;   the machine further includes means for driving the machining tool in rotation disposed facing means for plotting a contour.       

    
    
     
       BRIEF DESCRIPTION OF THE DRAWING FIGURES 
       Other features and advantages of the invention become apparent in the light of the following description of a preferred embodiment given by way of nonlimiting example, which description is given with reference to the appended drawings, in which: 
         FIG. 1  is a diagrammatic view of the operative members of a machine tool of the invention; 
         FIG. 2  is a view of the face to be machined of an optical object on which the trajectory of the machining tool is represented diagrammatically; 
         FIG. 3  is a three-dimensional view illustrating the cooperation between the optical object and the machining tool; 
         FIGS. 4 and 5  are diagrammatic views illustrating the theoretical principle of machining along a predetermined same parallel; 
         FIGS. 6 and 7  are diagrammatic views illustrating the implementation of the principle illustrated in  FIGS. 3 and 4  by the  FIG. 1  machine; 
         FIG. 8A  is a three-dimensional view similar to  FIG. 3  illustrating in the form of an arrow the normal at the point of contact of the surface to be machined; 
         FIGS. 8B and 8C  are two-dimensional views of  FIG. 8A  respectively from above and from the front; 
         FIGS. 9A ,  9 B and  9 C are similar to  FIGS. 8A ,  8 B and  8 C, respectively, but for another point of contact between the optical object and the machining tool; 
         FIG. 10  is a view of an angular tool-part trajectory  11 ′ offset 90° relative to that of  FIG. 2 ; and 
         FIG. 11  is a view of a means for plotting a profile. 
     
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
     In the  FIG. 1  diagrammatic view, the machine tool represented includes a rotating table  1  (seen from the side in this figure) of circular shape. This rotating table  1  can be oriented angularly about an axis perpendicular to its center in both directions (arrow  2  in  FIG. 1 ). 
     The rotating table  1  has a receiving surface  3  at the top. 
     A bracket  4  is fixed, for example screwed, to the receiving surface  3  so that a mounting surface  5  of the bracket  4  projects perpendicularly to the receiving surface  3 . 
     The bracket  4  includes jaws (not shown) adapted to hold an optical object, which is an ophthalmic lens  6  in the present example, so that a surface  7  to be machined of the ophthalmic lens  6  is disposed transversely to the receiving surface  3 . 
     This machine tool also includes a spindle  8  on which is mounted a machining tool  9  which in the present example is a grinding tool with a spherical bearing surface. The spindle  8  is adapted to drive the tool  9  in rotation as shown by the arrow  10  and to move this tool  9  in translation in the three directions X, Y and Z to enable the tool  9  to machine the entire surface  7  of the ophthalmic lens  6 . 
     Here the spindle  8  is parallel to the axis Z. 
     In a variant, the spindle  8  is inclined relative to the axis Z. 
     In another variant the movement of the tool  9  in the three directions X, Y and Z can be effected via a fixed spindle  8  and a rotating table  1  that is itself mobile in translation in the directions X, Y and Z. 
     Generally speaking, any combination of movements of the tool  9  and the rotating table  1  enabling such relative movement of the tool  9  and the rotating table  1  is an acceptable variant. 
     The surface  7  to be machined, which is seen from above in  FIG. 2 , is machined here along a fluted trajectory represented diagrammatically by the line  11 . Thus the machining is effected in the form of a series of passes of the tool  9  driven in rotation and moved along a trajectory parallel to the receiving surface  3 . 
     In this  FIG. 2 , the surface to be machined appears from the front as a disc, it being understood that the lens  6  is curved and that this surface  7  to be machined is therefore not plane. 
     The machining of the surface  7  of an ophthalmic lens  6  by the  FIG. 1  set-up proceeds in the manner described below. 
     The relative angular position of the surface  7  with respect to the tool  9  is effected along a predetermined same parallel.  FIG. 3  illustrates in three dimensions the tool-part relative positioning on a same parallel P of the tool  9 . 
     The principle of machining on a predetermined same parallel P of the tool  9  is illustrated theoretically in two dimensions in  FIGS. 4 and 5 . 
     Before being mounted on the spindle  8 , the tool  9  is mounted on equipment for determining its dynamic profile. This equipment is adapted to rotate the tool  9 . The dynamic profile of the tool is plotted, for example, by placing the tool  9  between a parallel light beam and a screen so that the shadow of the tool  9  projected onto the screen takes account of this dynamic profile  12 , or by filming the rotating tool  9  and displaying this image on a screen (see  FIG. 11 ). 
     The dynamic profile measuring equipment also enables manual or electronic manipulation of this image and measurement and tracing on this dynamic profile  12 . 
     For better precision, especially in the case where the tool  9  is a finishing tool, the tool can be trued and balanced directly on the spindle, after which its dynamic profile is measured. 
     There is then chosen a parallel P on this dynamic profile that appears in the figures in the form of a segment perpendicular to the rotation axis  13  of the tool  9  about which the dynamic profile  12  is symmetrical. 
     This parallel P is determined by the intersection of a plane perpendicular to the rotation axis  13  of the tool  9  and the dynamic profile  12  of the tool  9 . 
     There is then determined on the profile  12  the tangent  14  to the contour of the dynamic profile at the point of intersection between one of the ends of the parallel P and the contour of the profile  12 . 
     The perpendicular  15  to the tangent  14  at the point C cuts the rotation axis  13  at a point R D  which is the dynamic radius of the tool  9 . This perpendicular  15  is therefore the normal to the dynamic profile  12  at the point C. 
     The machining is then carried out so that, in the first place, the tool  9  is always in contact with the surface to be machined at the point C, that is to say, the tool being a rotary tool, always on the same parallel P, and that, in the second place, the relative angular orientation between the tool and the surface to be machined is such that the normal N to the surface to be machined at the point of contact C passes through the point R D , in other words coincides with the perpendicular  15 . 
       FIG. 5  shows two possible positions of the tool  9  along a surface  7  to be machined conforming to the above principles. 
     In the  FIG. 1  machine, these principles are applied in accordance with  FIGS. 6 and 7 , which are views from above with respect to the  FIG. 1  representation. 
     When the tool  9  is moved up into contact with the surface  7 , as in  FIG. 6 , the rotating table  1  is angularly oriented so that the surface  7  is placed as shown in  FIG. 6 , i.e. so that the normal N to the surface  7  at the point of contact C passes through the center R D , which implies that the angle A between this normal N and the rotation axis  13  of the tool  9  is always preserved. 
     Localized-type machining is effected. This means that the same place on the spherical generatrix of the grinding tool is always used. All grinding tool/part points of contact will therefore form a circle lying in a plane orthogonal to the axis of the tool. The position of this plane relative to the center of the grinding tool is defined by the angle A. 
     The tool  9  is then moved along a trajectory parallel to the receiving surface  3  of the rotating table  1 , i.e. in the X, Z plane. 
       FIG. 7  shows another position of the tool  9  after movement. The rotating table  1  has been oriented angularly, as before, so that the normal N 2  at the point C 2  passes through the point R D . This angular orientation of the rotating table  1  is effected as the tool  9  travels over the surface  7  to be machined. Once this travel has been completed from one lateral extremity of the ophthalmic lens to the other, the tool  9  is moved in translation perpendicularly to the receiving surface  3 , i.e. along the axis Y, as shown in  FIG. 2 , after which a new pass in the X, Z plane is carried out in the same manner. These operations are repeated until the surface  7  has been machined completely. 
     It is therefore mandatory that the normal at the contact should coincide with the normal of the tool. This means that, the tool here being quasi-spherical, the normal to the part must pass through the center of the grinding tool. 
     Example of a Machining Configuration 
     The machining point C(X, Y, Z) part  and its normal              p (U, V, W) part  in the system of axes of the part are known.
     The grinding tool center point R D (X gt , Y gt , Z gt ) part  and its direction              p (U gt , V gt , W gt ) part  in the system of axes of the part are what is being looked for.
     Calculation of the Angle B 
     The grinding tool system of axes (             grinding tool ,            grinding tool ,            grinding tool ) is defined, which is a rectangular system of axes with its origin at the center of the grinding tool and colinear with the direction of the grinding tool.
     What is to be determined is the value of the rotation about the axis Y to be applied so that, at the point C, the normal to the surface passes through the generatrix of the cone whose apex is at the center of the grinding tool and whose cone angle is 
               π   2     -     A   .           
Let B denote this angle.
 
     The normal at the point C expressed in the part system of axes is such that:
 
           = U             p   +V             p   +W             p .

     After transposing the angle B into the system of axes of the grinding tool, we obtain:
 
           = U (           gt  cos  B −           gt  sin  B )+ V             gt   +W (           gt  sin  B+             gt  cos  B ).

     The coordinate of the vector             in the system of axes of the grinding tool after transposition is obtained in the form:
 
         =(− U  sin  B+W  cos  B )           gt   +V             gt +( U  cos  B+W  sin  B )           gt  

     What is required is for this “transposed” normal to form an angle of 
               π   2     -   A         
with the oriented axis of the grinding tool; we can therefore write that the scalar product of              grinding tool  by           is equal to the cosine of the angle of the cone formed by A.

     
       
         
           
             
               
                 gt 
               
               · 
             
             = 
             
               
                 cos 
                 ( 
                 
                   
                     π 
                     2 
                   
                   - 
                   A 
                 
                 ) 
               
               = 
               
                 sin 
                 ⁡ 
                 
                   ( 
                   A 
                   ) 
                 
               
             
           
         
       
     
     Which is written: 
                   -   U     ⁢           ⁢   sin   ⁢           ⁢   B     +     W   ⁢           ⁢   cos   ⁢           ⁢   B       =         sin   ⁢           ⁢   A     ⁢     
     -     sin   ⁢           ⁢   B     +       W   U     ⁢   cos   ⁢           ⁢   B       =       sin   ⁢           ⁢   A     W                       Setting   ⁢           ⁢     W   U       =     tan   ⁢           ⁢   t       ,         
the equation becomes:
 
     
       
         
           
             
               
                 
                   - 
                   sin 
                 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 B 
               
               + 
               
                 tan 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 t 
               
               + 
               
                 cos 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 B 
               
             
             = 
             
               
                 
                   
                     sin 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     A 
                   
                   W 
                 
                 ⁢ 
                 
                   
 
                 
                 - 
                 
                   cos 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   t 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   sin 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   B 
                 
                 + 
                 
                   sin 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   t 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   cos 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   B 
                 
               
               = 
               
                 
                   
                     sin 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     A 
                   
                   U 
                 
                 ⁢ 
                 cos 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 t 
               
             
           
         
       
     
     If the condition 
               -   1     ≤         sin   ⁢           ⁢   A     U     ⁢   cos   ⁢           ⁢   t     ≤   1         
is respected, we may set:
 
     
       
         
           
             
               
                 
                   sin 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   A 
                 
                 U 
               
               ⁢ 
               cos 
               ⁢ 
               
                   
               
               ⁢ 
               t 
             
             = 
             
               sin 
               ⁢ 
               
                   
               
               ⁢ 
               q 
             
           
         
       
     
     The equation then becomes: 
     
       
         
           
             
               
                 
                   - 
                   cos 
                 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 t 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 sin 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 B 
               
               + 
               
                 sin 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 t 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 cos 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 B 
               
             
             = 
             
               
                 
                   sin 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   A 
                 
                 U 
               
               ⁢ 
               cos 
               ⁢ 
               
                   
               
               ⁢ 
               t 
             
           
         
       
       
         
           
             
               sin 
               ⁢ 
               
                   
               
               ⁢ 
               
                 ( 
                 
                   t 
                   - 
                   B 
                 
                 ) 
               
             
             = 
             
               sin 
               ⁢ 
               
                   
               
               ⁢ 
               q 
             
           
         
       
     
     That is:
 
 t−B=q  or  t−B=π−q  
 
     Thus: 
     
       
         
           
             B 
             = 
             
               
                 - 
                 π 
               
               + 
               
                 arcsin 
                 ⁡ 
                 
                   ( 
                   
                     
                       
                         sin 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         A 
                       
                       U 
                     
                     ⁢ 
                     
                       cos 
                       ⁡ 
                       
                         ( 
                         
                           arc 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           tan 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             W 
                             U 
                           
                         
                         ) 
                       
                     
                   
                   ) 
                 
               
               + 
               
                 arc 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 tan 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   W 
                   U 
                 
               
             
           
         
       
       
         
           or 
         
       
       
         
           
             B 
             = 
             
               
                 - 
                 
                   arcsin 
                   ⁡ 
                   
                     ( 
                     
                       
                         
                           sin 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           A 
                         
                         U 
                       
                       ⁢ 
                       
                         cos 
                         ⁡ 
                         
                           ( 
                           
                             arc 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             tan 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               W 
                               U 
                             
                           
                           ) 
                         
                       
                     
                     ) 
                   
                 
               
               + 
               
                 arctan 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   W 
                   U 
                 
               
             
           
         
       
     
     We know that 
                 cos   ⁡     (     arctan   ⁢           ⁢     W   U       )       =     U         U   2     +     W   2             ,         
from which we deduce:
 
     
       
         
           
             B 
             = 
             
               
                 - 
                 π 
               
               + 
               
                 arcsin 
                 ( 
                 
                   
                     sin 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     A 
                   
                   
                     
                       
                         U 
                         2 
                       
                       + 
                       
                         W 
                         2 
                       
                     
                   
                 
                 ) 
               
               + 
               
                 arccos 
                 ( 
                 
                   U 
                   
                     
                       
                         U 
                         2 
                       
                       + 
                       
                         W 
                         2 
                       
                     
                   
                 
                 ) 
               
             
           
         
       
       
         
           
             B 
             = 
             
               
                 - 
                 
                   arcsin 
                   ( 
                   
                     
                       sin 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       A 
                     
                     
                       
                         
                           U 
                           2 
                         
                         + 
                         
                           W 
                           2 
                         
                       
                     
                   
                   ) 
                 
               
               + 
               
                 arccos 
                 ( 
                 
                   U 
                   
                     
                       
                         U 
                         2 
                       
                       + 
                       
                         W 
                         2 
                       
                     
                   
                 
                 ) 
               
             
           
         
       
     
     That is: 
     
       
         
           
             B 
             = 
             
               
                 - 
                 π 
               
               + 
               
                 arcsin 
                 ( 
                 
                   
                     sin 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     A 
                   
                   
                     
                       
                         U 
                         2 
                       
                       + 
                       
                         W 
                         2 
                       
                     
                   
                 
                 ) 
               
               + 
               
                 arcsin 
                 ( 
                 
                   W 
                   
                     
                       
                         U 
                         2 
                       
                       + 
                       
                         W 
                         2 
                       
                     
                   
                 
                 ) 
               
             
           
         
       
       
         
           or 
         
       
       
         
           
             B 
             = 
             
               
                 - 
                 
                   arcsin 
                   ( 
                   
                     
                       sin 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       A 
                     
                     
                       
                         
                           U 
                           2 
                         
                         + 
                         
                           W 
                           2 
                         
                       
                     
                   
                   ) 
                 
               
               + 
               
                 arcsin 
                 ( 
                 
                   W 
                   
                     
                       
                         U 
                         2 
                       
                       + 
                       
                         W 
                         2 
                       
                     
                   
                 
                 ) 
               
             
           
         
       
     
     It has been assumed that: 
     
       
         
           
             
               - 
               1 
             
             ≤ 
             
               
                 
                   sin 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   A 
                 
                 U 
               
               ⁢ 
               cos 
               ⁢ 
               
                   
               
               ⁢ 
               t 
             
             ≤ 
             
               1 
               ⁢ 
               
                 
 
               
               - 
               1 
             
             ≤ 
             
               ( 
               
                 
                   sin 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   A 
                 
                 
                   
                     
                       U 
                       2 
                     
                     + 
                     
                       W 
                       2 
                     
                   
                 
               
               ) 
             
             ≤ 
             1 
           
         
       
       
         
           
             
               
                 sin 
                 2 
               
               ⁢ 
               A 
             
             ≤ 
             
               
                 U 
                 2 
               
               + 
               
                 W 
                 2 
               
             
           
         
       
       
         
           
             
               
                 cos 
                 2 
               
               ⁢ 
               A 
             
             ≥ 
             
               V 
               2 
             
           
         
       
     
     The condition to be verified for the angle to be correct is cos 2  A≧V 2 . 
     We choose for B: 
     
       
         
           
             B 
             = 
             
               
                 - 
                 π 
               
               + 
               
                 Arc 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   sin 
                   ( 
                   
                     W 
                     
                       
                         
                           U 
                           2 
                         
                         + 
                         
                           W 
                           2 
                         
                       
                     
                   
                   ) 
                 
               
               + 
               
                 Arcsin 
                 ( 
                 
                   
                     sin 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     A 
                   
                   
                     
                       
                         U 
                         2 
                       
                       + 
                       
                         W 
                         2 
                       
                     
                   
                 
                 ) 
               
             
           
         
       
     
     with the following condition:
 
cos 2    A≧V   2  
 
     Calculation of the Direction of the Grinding Tool 
     The angle B being defined, the direction of the grinding tool            =(U gt , V gt , W gt ) part  in the part system of axes can be deduced therefrom.
     
       
         
           
             = 
             
               
                 ( 
                 
                   
                     
                       
                         
                           U 
                           gt 
                         
                         = 
                         
                           sin 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           B 
                         
                       
                     
                   
                   
                     
                       
                         
                           V 
                           gt 
                         
                         = 
                         0 
                       
                     
                   
                   
                     
                       
                         
                           W 
                           gt 
                         
                         = 
                         
                           cos 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           B 
                         
                       
                     
                   
                 
                 ) 
               
               
                 part 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 system 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 of 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 axes 
               
             
           
         
       
     
     Calculation of the Position of the Center of the Grinding Tool 
     Here it is a question of calculating the position to be imparted to the center of the grinding tool R D (X gt , Y gt , Z gt ) part  to machine the point C(X, Y, Z) part  with normal            (U, V, W) part  in the part system of axes.
     O: origin of the part system of axes. 
     C: machining point. 
     R D : center of the grinding tool. 
     We have:
 
 O               D   =O             +C             D  
 
 O             =X             p   =Y             p   +Z             p  
 
 C             D   =R   grinding tool           
 
 C             D =( R   grinding tool   U )           p +( R   grinding tool   V )           p +( R   grinding tool   W )           p  

     where R grinding tool  is the radius of the grinding tool. 
     Whence the position of the center of the grinding tool: 
     
       
         
           
             
               OR 
               D 
             
             = 
             
               
                 
                   ( 
                   
                     X 
                     + 
                     
                       
                         R 
                         
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     The machining can be carried out in two steps: 
     A first step in which the tool is positioned so that the normal of the point to be machined is “parallel to the surface of the cone”. 
     A second step in which the machining point is brought into contact with the point to be machined. 
     Thus, during machining, the tool is worn symmetrically on each side of the parallel P that has been chosen, which improves prediction and control of this wear. What is more, the tool  9  machines the surface  7  by attacking the material perpendicularly to the trajectory of movement of the tool  9 , which circumvents machining defects inherent to the machining mode in which the material is either “swallowed” or “pushed”, when the tool attacks the material parallel to its trajectory of movement. 
     The parallel P is chosen as a function of the shape of the surface  7  to be machined so that no portion of this surface  7  is inaccessible to this parallel P given the possible angular movements between the tool  9  and the rotating table  1  and taking into account the overall size of the spindle  8 . 
     The machining operations described with reference to  FIGS. 6 and 7  take place in three dimensions, of course, as  FIGS. 8A to 9C  illustrate.  FIGS. 8A to 8C  show the machining of the lens  6  by the tool  9  at a first point of contact C 1  (as in  FIG. 6 ), whereas  FIGS. 9A to 9C  show the machining of the lens  6  by the tool  9  at a second point of contact C 2  (as in  FIG. 7 ). 
     In each of these  FIGS. 8A to 9C  the normal N at the point of contact C of the surface  7  to be machined is represented. The passage from the point of contact C 1  in  FIGS. 8A to 8C  to the point of contact C 2  in  FIGS. 9A to 9C  shifts the normal N from its position N 1  to its position N 2 , of course. This normal N evolves as a function of the point of contact C within a conical volume. 
     Variants of the machining method and machine can be envisaged without departing from the scope of the invention. In particular, the machine tool can include two separate spindles, a first spindle for rough machining and a second spindle for finishing and semi-finishing of the optical object, such as an ophthalmic lens, a mold or an insert. The machine tool can advantageously further include a tool changer adapted to position a tool  9  on the spindle. 
     The above description relates to a tool-part trajectory conforming to  FIG. 2 , which has the advantage of machining without swallowing or pushing the material, although it is to be understood that the invention can equally well be implemented along an angular tool-part trajectory  11 ′ offset 90° relative to that of  FIG. 2  (see  FIG. 10 ).