Abstract:
Computer-implemented method for virtually modifying a digital model of a dental restoration, wherein a user selects a point on the surface of the dental restoration, wherein a sphere having the selected point as its centre is automatically displayed, wherein the user may define to add material to or remove material from the dental restoration inside the sphere to generate a modified part of the dental restoration and leaving the rest of the dental restoration unmodified, wherein the step of adding material to or removing material from the dental restoration is automatically performed in real time on a display. Further, the invention is related to a computer-readable medium having stored thereon instructions which when executed by a processor perform the method steps of the inventive computer-implemented method.

Description:
FIELD OF THE INVENTION 
       [0001]    The invention relates to a computer-implemented method for virtually modifying a digital model of a dental restoration and a computer-readable medium. 
       BACKGROUND 
       [0002]    In the field of dentistry wax-knife tools and interactive editing tools are known for the design of virtual dental model. Virtual wax knife tools allow to add and to remove material from a given dental model. A virtual wax-knife tool allows refinement of the dental model in order to achieve e.g. a well-fitting preparation line. 
       SUMMARY OF THE INVENTION 
       [0003]    It is the object of the present invention to provide a method enabling a user to perform a virtual modelling of a digital model of a dental restoration. 
         [0004]    According to one embodiment, a computer-implemented method for virtually modifying a digital model of a dental restoration comprises steps of selecting, by a user, a point on the surface of the dental restoration, automatically displaying a sphere having the selected point as its centre, selecting, by the user, to add material to or remove material from the dental restoration inside the sphere to generate a modified part of the dental restoration and leaving the rest of the dental restoration unmodified, wherein the step of adding material to or removing material from the dental restoration is automatically performed in real time on a display. 
         [0005]    By selecting a point on the surface of the dental restoration the user can select a position on the surface around which he considers a modification to be relevant. The displayed sphere having the selected point as its centre indicates the volume inside which the modification of the dental restoration may take place. During a modification, the shape of the surface and accordingly the body (i.e. the volume and therewith the amount of material) of the dental restoration will change. 
         [0006]    A selection of a point on the surface of the dental restoration may be performed by a mouse click by the user on a desired position of the displayed surface of the dental restoration. 
         [0007]    Another possibility for selecting a point on the surface of the dental restoration may be given in that the user may select x-y-coordinates of points of the displayed dental restoration, wherein the x-y-coordinates e.g. may be listed in a list, displayed in a menu or the like. The user may also type in a desired x-y-coordinate, e.g. by means of a keyboard, at a designated part of the display of the displayed dental restoration or at an additionally provided designated part. 
         [0008]    By selecting a point either by means of a mouse click or by entering its x-y-coordinates, so far no information about the location of the point on the surface of the dental restoration is provided as the three-dimensional digital model of the dental restoration is displayed in only two dimensions. 
         [0009]    A buffer, a so-called z-buffer or depth buffer, may be provided wherein the z-coordinate of a pixel comprising the selected point may be stored and may represent the depth of the pixel. The depth defines which pixel is next to an observer of the display. The buffer may be arranged as a two-dimensional array (x-y) with one element for each pixel. 
         [0010]    Thus, by selecting a point in the two-dimensional representation and using the buffer information, i.e. the depth of a pixel, the point will be located on the surface of the dental restoration. This allows a quick result for automatically displaying a sphere having the selected point as its centre. 
         [0011]    In general, the volume of the sphere contains a part of the dental restoration which will consist of material, and further, said volume may contain a part where no part (i.e. no material) of the dental restoration is present. 
         [0012]    The radius of the sphere, i.e. the size of the sphere, may be defined by default or a user may define a value of the radius by its own. The default value of the radius may be 2.5 mm, 2 mm, 1.5 mm or may be larger or smaller and may also have values in between these value regions. 
         [0013]    Another option for defining the radius of the sphere is to automatically define a radius in case the default value of the radius or the value of a user defined radius would have such a size that e.g. the preparation line or some other region/volume excluded from modification would be influenced by the planned modification (i.e. when the default radius or the user defined radius would be taken for the radius of the sphere). The automatically defined radius may have a value such that the extent of the sphere would not reach into the region/volume excluded from modification. An alert may be indicated, for example, in a user interface, that the default value of the radius or the user defined radius have not been accepted but that an automatically defined radius value is used. 
         [0014]    Instead of automatically defining a new radius for a sphere in which a modification of the surface of a dental restoration may take place, the sphere (with the default radius or the user defined radius) extending into a region/volume excluded from modification may be coloured in red (or a colour being different from the colour used in normal operation) and/or highlighted to indicate that the default radius or the value of the user defined radius would have such a size that a region/volume excluded from modification would be influenced by the planned modification. By this indication the user is enabled to easily realise that a modification in the area defined by the sphere is not allowed. Then the user may change the position of the centre of the sphere and/or may change the radius of the sphere. 
         [0015]    The surface of the dental restoration of the digital model may be described by finite element triangles, wherein these triangles may additionally be used to indicate areas on the surface of the dental restoration which are allowed to be modified and such areas which are excluded from modification. Such an indication may be performed by displaying the triangles in different colours (the edges of the triangle and/or the whole triangle area) and/or different shading for allowed or excluded areas, respectively. It is possible to display the finite element triangles with said indications on the complete dental restoration or to display only those triangles that lie inside (the triangles do not have to lie completely within the sphere but at least with some part of the triangle) a sphere selected for virtually modifying a dental restoration. 
         [0016]    By automatically performing the step of adding material to or removing material from the dental restoration in real time on the display, time consuming waiting times may be prevented and the user can see immediately if the modifications of the dental restoration may result in a modified dental restoration which would fit required conditions. Such conditions may be providing an occlusal surface of the dental restoration comprising one or more fissures at desired positions on the surface. 
         [0017]    The method further comprises the steps of defining a radius of the sphere and a strength of adding material to or removing material from the dental restoration, wherein the definition is performed by the user. It is also possible that some default values are available that may be used for the radius and/or the strength, respectively. 
         [0018]    The radius of the sphere defines the extent of an area which results of the cross section of the sphere with the surface of the dental restoration. The border line of the area and thus the border line of the cross section are defined by the surface of the sphere. Depending on the shape of the surface of the dental restoration the projection of the cross section will in general not be circular as the surface of the dental restoration may be curved, possibly comprising flat portions. 
         [0019]    It is possible that an alert may be indicated, for example on a user interface and/or on an interface that depicts the digital model of the dental restoration to be modified, e.g. when the centre and/or the radius of a sphere are selected/defined in such a way that a region/volume of the dental restoration that is not allowed to be modified, e.g. the preparation line, would be influenced. 
         [0020]    The strength of adding material to or removing material from the dental restoration may be defined by the user or may be defined by default. E.g. a fixed value may be defined for all of the points on the surface of the dental restoration inside the sphere or some function for the strength may be defined. Such a function may be dependent on the distance of a point on the surface of the dental restoration inside the sphere and the centre of the sphere. This may result in a modification of the surface of the dental restoration that is stronger near the centre of the sphere than near the circumference of the sphere. 
         [0021]    Adding material to or removing material from the dental restoration may be performed for each point on the surface of the dental restoration inside the sphere as a function dependent on the distance of each point with respect to the centre of the sphere, wherein a shifting is performed along a direction defined by the normal vector of the surface of the dental restoration at the centre of the sphere, or wherein shifting is performed along a direction defined by the direction between the centre of the sphere and a point on the surface of the sphere, wherein said point is selected by the user. 
         [0022]    The adding and removing may be performed in such a way (e.g. by using a continuous or smooth distance depending function) that the resulting shape is smooth in the sense that no steps or angled surfaces are created. The adding or removing of material can be done such points intersected by the sphere are not shifted while points inside the sphere are shifted or the points intersected by the sphere are shifted only less than 1/100 or 1/1000 times the distance as the centre point of the sphere is shifted. 
         [0023]    When the shifting is performed along a direction defined by the direction between the centre of the sphere and a point on the surface of the sphere, wherein said point is selected by the user and when a plurality of points on the surface of the sphere are selected by the user, e.g. by selecting one point and then dragging a line along a path defined by the user, the method may enable displaying in real time, for each of the plurality of selected points, a respective shifting of each point on the surface of the dental restoration inside the sphere. 
         [0024]    This enables the user to see immediately the result of his planned modification. For example by switching between two points on the surface of the sphere, i.e. alternately selecting of these two points, the user can easily compare two modifications which can result in an optimized and thus well fitting dental restoration. 
         [0025]    Another possibility to define a direction along which a shifting may be performed is to calculate an average value of all the normal vectors of the surface points of the dental restoration which are inside a second sphere, for example inside a so-called double sphere having a radius twice the size of the radius of the sphere inside which the modifying of the digital model of the dental restoration should be performed. 
         [0026]    Instead of using a double sphere, it is also possible to use a second sphere having a radius which is larger or smaller than the one of the double sphere. 
         [0027]    Thus, the radius of the second sphere may have a size in between the radius of the double sphere and the radius of the normal (single radius) sphere (i.e. the sphere inside which the modification of the surface of the dental restoration will be performed). Further, it is also possible that the radius of the second sphere may have the size of the radius of the normal (single radius) sphere (i.e. the second sphere is identical to the normal (single radius) sphere). Moreover, it is possible that the radius of the second sphere has a size smaller than the size of the radius of the normal (single radius) sphere. 
         [0028]    In another embodiment, it is possible that the shifting of each point on the surface of the model inside the sphere may also be performed along the respective direction defined by the normal vector of the surface at each point, i.e. one of each of the points has its own direction for shifting, wherein this direction is given by the normal vector of the surface at the position of this one point. 
         [0029]    A function for the shifting may be given by: 
         [0000]        s=d/|d|*f* exp(−( b−a ) 2 )/(| r|/ 2) 2 ),
 
         [0000]    wherein b is a three dimensional vector defining the centre of the sphere, f is the strength of the adding material to or removing material from the dental restoration, a is a three dimensional vector defining a point on the surface of the dental restoration inside the sphere, r is a three dimensional vector defining the radius of the sphere and d is a three dimensional vector defining the vector which is defining a three dimensional direction of shifting (“*” stands for the mathematical operation of a multiplication). 
         [0030]    By using this function, the user can define a direction in which points on the surface of the dental restoration should be shifted. This may be advantageous in case a user sees that a shifting of the points in a direction to a left upper corner of the dental restoration may result in an optimised dental restoration. Otherwise, it may be possible that the direction of shifting may be given by the normal vector of the surface of the dental restoration at the centre of the sphere. This normal vector may be calculated automatically. 
         [0031]    The method further comprises that an amount of material to be added to or to be removed from the dental restoration may be automatically calculated and indicated, wherein the indication may be given on a user interface and/or on an interface that depicts the digital model of the dental restoration to be modified. 
         [0032]    The step of automatically performing the step of adding material to or removing material from the dental restoration in real time on a display further may comprise storing x and y coordinates of a pixel of the display and a z coordinate of said pixel, representing the depth of said pixel, in one buffer. 
         [0033]    The z coordinate defines which pixel is in front or behind another pixel. Thus the two dimensional display has some three dimensional information. 
         [0034]    The sphere may be displayed in a transparent manner or in a colour shade different from a colour of the dental restoration and/or the digital model, thus that the shape of the dental restoration is still visible in total and that the user is also provided with the information on which part of the dental restoration his changes will have influence. 
         [0035]    Further, the invention is related to a computer-readable medium having stored thereon instructions which when executed by a processor performs the above-identified the method steps. 
         [0036]    A high flexibility can be achieved as all modifying operations may be cancelled or made again in real time at each possible step of the modifying process. Thus, it is possible to easily perform alternative modelling drafts, access a draft at a later state and change a current daft quickly. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0037]    Preferred embodiments of the invention will be illustrated with reference to the enclosed figures. In the figures: 
           [0038]      FIG. 1  shows a schematic view of a display displaying a digital model of a dental restoration and a user interface; 
           [0039]      FIG. 2   a  shows shifting of points inside a sphere in a direction defined by the normal vector of the surface of the dental restoration at the centre of the sphere; 
           [0040]      FIG. 2   b  shows shifting of points inside a sphere in a direction defined by the direction between the centre of the sphere and a selected point on the surface of the sphere; 
           [0041]      FIGS. 3   a  shows an exemplary modification of a part of the dental restoration, wherein the surface comprises contour lines; 
           [0042]      FIG. 3   b  shows another exemplary modification of the part of the dental restoration, wherein the surface comprises contour lines; 
           [0043]      FIG. 4   a  shows a side view of a half-sphere together with the shifted surface when shifting along the direction of the normal vector; 
           [0044]      FIG. 4   b  shows the shifted surface when a plurality of centres of spheres have been selected along a path; 
           [0045]      FIG. 4   c  shows a side view of a half-sphere together with the shifted surface when shifting along the direction between the centre of the sphere and a point on the surface of the sphere; and 
           [0046]      FIG. 4   d  shows the shifted surface when a plurality of centres of spheres have been selected along a path. 
       
    
    
     DETAILED DESCRIPTION 
       [0047]      FIG. 1  depicts a schematic representation of a digital model  1  of a dental restoration  2  according to one embodiment of the invention. In this case, a dental crown is depicted, however, the dental restoration  2  may also be a bridge, an abutment, an implant post, an inlay, an onlay, a veneer, a partial crown. 
         [0048]    As indicated by the cross  3 , a point on the surface of the dental restoration  2  has been selected and a sphere  4  has been automatically displayed, having its centre  3  at the selected point. A part of the surface of the dental restoration  2  is defined by the cross section  5  of the surface with the volume of the sphere  4 . It is an object of the present invention to modify this defined part of the surface (i.e. the cross section  5 ) either by adding of (digital) material to or removing (digital) material from the dental restoration  2 . In the shown case material has been added to the dental restoration  2  as indicated by the smooth surface inside the sphere  4  which defines the border of the added material. 
         [0049]    In an interface  6 , various buttons are provided by means of which a user may select an option or wherein he may insert number values. E.g. the user may select if he wants to “add”  7  material to or “remove”  8  material from the dental restoration  2 . The user may define the radius  9  of the sphere  4  as well as the strength  10  of the modification by defining number values by himself  13 ,  14 . He also has the possibility the select respective default values  11 ,  12 . In response to defining the radius  9  of the sphere  4  and the strength  10  of the adding material to or removing material from the dental restoration, the material amount  18  added or removed is automatically calculated and displayed in the interface  6 . 
         [0050]    The modification of the dental restoration  2  may be performed by shifting each point on the surface of the dental restoration  2  inside the sphere  4  as a function dependent on the distance of a point with respect to the centre  3  of the sphere. This way of shifting results in shifting values that are getting smaller from the centre  3  towards the circumference of the sphere  4 . 
         [0051]    The user may select if the shifting is to be performed in a direction defined by the normal vector  25  of the surface of the dental restoration at the centre  3  of the sphere  4  (default button  16 ) or if the shifting is to be performed in a direction  31  defined by the direction between the centre  3  of the sphere  4  and a selected point  28 ,  28 ′,  28 ″ on the surface of the sphere  4 . 
         [0052]    The direction of the shifting has to be associated to the fact if matter is to be added to or to be removed from the dental restoration  2 . In case material should be added the direction of shifting has to point from the surface of the dental restoration  2  to the region where no material is. In case material should be removed from the dental restoration  2 , the direction of shifting has to point from the surface of the dental restoration  2  to the region where material is present. 
         [0053]    In case the modification of the dental restoration  2  would result in a dental restoration  2  not well suited for use an alert  19  may be indicated in the interface  6 . As exemplary shown in the displayed interface  6  three alert buttons may exist, being related to undercut  20 , stability  21 , and preparation line  22  of the dental restoration  2 . One or several alert buttons may be highlighted and/or may be blinking in case the dental restoration  2  after performing the modification would comprise one or more undercuts, in case the preparation line of the dental restoration would be affected by the modification and/or in case the stability of the dental restoration  2  does not fulfil predefined criteria. In the displayed digital model  1  of the dental restoration  2  the respective areas may be indicated in a different colour and may be further highlighted by blinking. 
         [0054]    A performed modification step may be undone by the “back”  23  button, which when activated cancels the performed modification such that the dental restoration  2  looks, i.e. has the shape, as before performing the modification. It is also possible to undo a plurality of modification steps. 
         [0055]    Further, a button “return to initial image”  24  is provided to enable a user to return to the (initial) digital model from which he started his work. 
         [0056]    A plurality of subsequent points, i.e. points lying on a line, on the surface of a dental restoration  2  may be selected by the user, e.g. by selecting one point and then dragging a line along a desired path. Such a selection may be performed by a clicking a mouse button and the following dragging of the path may be performed by keeping the mouse click active. An end of the dragging may be achieved by releasing the mouse button. The distance (measured in space or along the surface of the digital model) between subsequent points on the desired path, i.e. centres  3  of the spheres  4 , may be defined by the user or may have a default value. 
         [0057]    The modifications of the dental restoration  2  along the desired path may become permanent after releasing the mouse button, wherein permanent means that the modification is displayed in a fixed manner, but wherein it is still possible to undo such a modification, e.g. by selecting the back button  23  of the interface  6 . 
         [0058]    In case, a further point  3  and/or a further desired path would be selected and would overlap with an already existing modification, the further modification would be performed to the already existing modification as if the existing modification would be a common surface of the dental restoration  2 . This means when in this further modification material is added to the dental restoration or if material is removed from the dental restoration so far unmodified areas on the dental restoration as well as areas already modified in a former modification step would be treated the same way. 
         [0059]    However, in case while, i.e. keeping the mouse click active, selecting a plurality of subsequent points on the surface of the dental restoration  2  and an overlap in one or several positions along the desired path exists, a modification in the one or more respective positions will be performed (after releasing the mouse button) only once. This means that in the one or more respective positions no addition of shifting values is performed. 
         [0060]    In  FIGS. 2   a  and  2   b  depict two variants of shifting points of the surface of the dental restoration  2  inside the sphere  4 . 
         [0061]      FIG. 2   a  shows how some points  26   1 ,  26   2 ,  26   3 ,  26   4 ,  26   5 ,  26   6  (which are located at a diameter curve  27 ) on the surface of the dental restoration  2  are shifted when the shifting is performed in a direction given by the normal vector  25  of the surface of the dental restoration  2  at the centre  3  of the sphere  4 . The size of the shifting is larger for points  26   1 - 26   6  nearer to the centre  3  than for points  26   1 - 26   6  further away from the centre  3 . This ensures a smooth change between the region where the surface of the dental restoration  2  is modified and the region where no modification takes place. The size distribution of the shifting may be given by the following function s(a)=f * exp(−(b−a) 2 )/(|r|/2) 2 ), wherein b is the centre  3  of the sphere  4 , a is a point on the surface of the dental restoration  2  and r is the radius  9  of the sphere  4  (“*” stands for the mathematical operation of a multiplication). 
         [0062]      FIG. 2   b  shows how some points  26   1 - 26   6  on the surface of the dental restoration  2  are shifted when the shifting is performed in a direction  31  given by the direction between the centre  3  of the sphere  4  and a selected point  28  on the surface of the sphere  4 . 
         [0063]    It is also possible that after a shifting of the points  26   1 - 26   6  has been performed, the user selects a shifted point  26   1 ′- 26   6 ′,  26   1 ″- 26   6 ″ and drags it in some other position. Then, automatically all the other points are shifted additionally in this new direction. 
         [0064]    In case subsequent centres  3  of spheres  4  have been selected on the surface of the dental restoration  2 , for those points that are part of two or more spheres  4  the shifting being associated to the two or more spheres  4  may be added up (in three dimensions). 
         [0065]      FIG. 3   a  shows an enlarged view of the sphere  4  inside which the modification of the dental restoration  2  takes place. Indicated is the modified surface of the dental restoration  2 , wherein the shape of the surface is clarified by showing five contour lines  30   1 ,  30   2 ,  30   3 ,  30   4 ,  30   5 . It is possible to use more or less contour lines for representing the shape of the shifted surface. The direction  31  of shifting was determined by a selected point  28 ′ on the surface of the sphere  4 . 
         [0066]    The user has the possibility to vary the shape of the modified surface by dragging a line along a desired path  29 , wherein a plurality of points on the surface of the sphere may be traversed. The change of the shape of the surface is shown to the user in real time such that he can immediately see what changes in the shape would result by a selection of another point on the surface of the sphere  4 , i.e. a selection of another direction of shifting. 
         [0067]    In  FIG. 3   b , the shape of the modified surface is shown in case another point  28 ″ on the surface of the sphere  4  is selected and thus the points  26   1 - 26   6  on the surface of the dental restoration  2  are shifted in along a direction defined by the direction between the centre  3  of the sphere  4  and said point  28 ″ on the surface of the sphere  4 . The shape of the shifted surface of the dental restoration  2  is indicated by five contour lines  30   1 ′,  30   2 ′,  30   3 ′,  30   4 ′,  30   5 ′. The shifted surface of the dental restoration  2  varies therefore with respect to the shifted surface shown in  FIG. 3   a.    
         [0068]      FIGS. 4   a - 4   d  show in a simplified case the resulting shifted surface of a dental restoration  2  in a sectional side view when a plurality of points on the surface of the dental restoration  2  (i.e. centres  3  of spheres  4 ) along a path have been selected. 
         [0069]    In the figures the (initial) surface of the dental restoration  2  to be modified is assumed to be flat and thus the path  32  along which the selected points lie is a two-dimensional line. However, in the general case the surface of the dental restoration  2  to be modified would not be flat but would have a curved shape, and thus, the path would be a three-dimensional line. 
         [0070]      FIG. 4   a  shows a sectional side view of a half-sphere  4  together with a side view of the shifted surface  33  when the points on the surface of the dental restoration are shifted along the direction of the normal vector of the surface of the dental restoration at the centre  3  of the sphere. The shifted surface  33  has a rotational symmetry. 
         [0071]    In  FIG. 4   a , the path  32  is only shown as an indication. The path  32 , in some sense, is generated when one point of the surface of the dental restoration is selected as a centre  3  of a sphere  4 , e.g. by a mouse click, and the selection is kept active by keeping the mouse click active and moving the mouse along a desired path  32  on the surface of the dental restoration  2 . At a certain distance from the first centre  3  of the first sphere  4 , a second centre  3  of a sphere  4  would be established and this would repeat until the moving of the mouse along the desired path  32  stops. 
         [0072]    to In  FIG. 4   b  the first and the last half-spheres  4  and their corresponding centres  3  are shown and the movement (of the mouse) along a desired path  32  is indicated by the arrow. 
         [0073]    As soon as a point on the surface of the dental restoration is selected, a sphere  4  having its centre  3  at this point is displayed automatically and, as shown in the depicted example, (digital) material is added to the dental restoration  2 . This addition is indicated by the shifted surface of the dental restoration  2 . In the displayed example, the shifting of points of the surface of the dental restoration  2  which results from selecting a plurality of points on the surface of the dental restoration  2  as centres  3  for spheres  4 , is added up for each single point in case two, three, four or more spheres overlap. In case of such an overlap a point of the surface of the dental restoration  2  will be respectively shifted two, three, four or more times, and the values of each of these two, three, four or more shifting processes are added up. This can also be seen in the sectional side view of the final surface  34  of the modified dental restoration in  FIG. 4   b . The slopes of the sectional view of the final surface  34  are different than the ones of surface  33  in case only one point on the surface of the dental restoration  2  has been selected. Also the maximum height of the final surface  34  is larger than the maximum height of the surface  33 . This is a result of adding up of the single shifting values. 
         [0074]    If two, three, four or more centres  3  of spheres  4  are selected, in general, it is possible that for each sphere  4  the points on the surface of the dental restoration  2  are shifted in accordance to a defined function and for each sphere  4  of the plurality of spheres  4  the performed shifting starts from the initial, i.e. unmodified, surface of the dental restoration  2 . Thus, for each sphere  4  the surface of the dental restoration  2  may be shifted as shown in  FIG. 4   a  (and  FIG. 4   c ; see description below). The total shifting of the points inside the plurality of spheres  4  may be performed by generating an envelop to the shifted surfaces of the dental restoration  2  resulting from the shifts in each of the plurality of spheres  4 . This means that a point is shifted only by the amount which corresponds to the largest shifting given by one of multiple spheres that include a point. If a point that lies three times inside a sphere would be shifted e.g. due to lying in the first sphere by 0.1 mm, due to lying in the second sphere by 0.2 mm and by lying in the third sphere by 0.3 mm it would be shifted by 0.3 mm, which corresponds to the largest of the three values. As described beforehand the shifting may also amount to 0.6 mm which is the sum of the individual shifting values. 
         [0075]      FIGS. 4   c  and  4   d  show the scenario as in  FIGS. 4   a  and  4   b , but in the case when shifting is performed along the direction between the centre  3  of the sphere  4  and a selected point on the surface of the sphere  4 . The shifted surface  35  has no rotational symmetry. In the sectional side view of the shifted surface  35  this gets obvious as the peak of the shifted surface in  FIG. 4   c  is displaced to the left and thus the slopes of the sectional side view of the shifted surface  35  are asymmetric. As a consequence also the slopes of the sectional view of the final surface  36  as shown in  FIG. 4   d  are asymmetric.