Abstract:
A relief print master is created by a printhead that moves in a slow scan direction. The nozzles of the printhead jet droplets of a polymerisable liquid on a rotating drum. The different nozzles jet droplets simultaneously on different layers that have different diameters. As a result, the droplets jetted by different nozzles travel over different distances before landing. The effect of this is that the droplets undergo different position lag as they land on the different layers. By rotating the printhead in a plane that is orthogonal to the jetting direction this effect can be compensated for.

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
       [0001]    This application is a 371 National Stage Application of PCT/EP2011/063625, filed Aug. 8, 2011. This application claims the benefit of U.S. Provisional Application No. 61/375,251, filed Aug. 20, 2010, which is incorporated by reference herein in its entirety. In addition, this application claims the benefit of European Application No. 10173538.9, filed Aug. 20, 2010, which is also incorporated by reference herein in its entirety. 
     
    
     BACKGROUND OF THE INVENTION 
       [0002]    1. Field of the Invention 
         [0003]    The invention deals with the field of three dimensional printing, more specifically the printing of relief features on a rotating cylindrical support using a fluid depositing apparatus such as an inkjet printhead. Even more specifically, the invention deals with the field of creating a flexographic print master on a rotating drum by a depositing printhead that moves in a slow scan direction and deposits curable liquid such as a UV-curable liquid. 
         [0004]    2. Description of the Related Art 
         [0005]    In flexographic printing or flexography a flexible cylindrical relief print master is used for transferring a fast drying ink from an anilox roller to a printable substrate. The print master can be a flexible plate that is mounted on a cylinder, or it can be a cylindrical sleeve. 
         [0006]    The raised portions of the relief print master define the image features that are to be printed. 
         [0007]    Because the flexographic print master has elastic properties, the process is particularly suitable for printing on a wide range of printable substrates including, for example, corrugated fiberboard, plastic films, or even metal sheets. 
         [0008]    A traditional method for creating a print master uses a light sensitive polymerisable sheet that is exposed by a UV radiation source through a negative film or a negative mask layer (“LAMS”-system) that defines the image features. Under the influence of the UV radiation, the sheet will polymerize underneath the transparent portions of the film. The remaining portions are removed, and what remains is a positive relief print plate. 
         [0009]    In the unpublished applications EP08172281.1 and EP08172280.3, both assigned to Agfa Graphics NV and having a priority date of 2008 Dec. 19, a digital solution is presented for creating a relief print master using a fluid droplet depositing printhead. 
         [0010]    The application EP08172280.3 teaches that a relief print master can be digitally represented by a stack of two-dimensional layers and discloses a method for calculating these two-dimensional layers. 
         [0011]    The application EP08172281.1 teaches a method for spatially diffusing nozzle related artifacts in the three dimensions of the stack of two-dimensional layers. 
         [0012]    Both applications also teach a composition of a fluid that can be used for printing a relief print master, and a method and apparatus for printing such a relief print master. 
         [0013]      FIG. 1  shows an embodiment of such an apparatus  100 .  140  is a rotating drum that is driven by a motor  110 . A printhead  160  moves in a slow scan direction Y parallel with the axis of the drum at a linear velocity that is locked with the rotational speed X of the drum. The printhead jets droplets of a polymerisable fluid onto a removable sleeve  130  that is mounted on the drum  140 . These droplets are gradually cured by a curing source  150  that moves along with the printhead and provides local curing. When the relief print master  130  has been printed, the curing source  170  provides an optional and final curing step that determines the final physical characteristics of the relief print master  120 . 
         [0014]    An example of a printhead is shown in  FIG. 3 . The printhead  300  has nozzles  310  that are arranged on a single axis  320  and that have a periodic nozzle pitch  330 . The orifices of the nozzles are located in a plane that corresponds with the nozzle plate. 
         [0015]      FIG. 2  demonstrates that, as the printhead moves from left to right in the direction Y, droplets  250  are jetted onto the sleeve  240 , whereby the “leading” portion  211  of the printhead  210  prints droplets that belong to a layer  220  having a relatively smaller diameter, whereas the “trailing” portion  212  of the printhead  210  prints droplets on a layer  230  having a relatively larger diameter. 
         [0016]    Because in the apparatus in  FIGS. 1 and 2  the linear velocity of the printhead in the direction Y is locked with the rotational speed X of the cylindrical sleeve  130 ,  240 , each nozzle of the printhead jets fluid along a spiral path on the rotating drum. This is illustrated in  FIG. 4 , where it is shown that fluid droplets ejected by nozzle  1  describe a spiral path  420  that has a pitch  410 . 
         [0017]    In  FIG. 4 , the pitch  410  of the spiral path  420  was selected to be exactly equal to the length of the nozzle pitch  430  of the printhead  440 . In a more general preferred embodiment the pitch of the spiral path is an integer multiple “N” of the nozzle pitch. In such a case there are N interlaced spiral paths. 
         [0018]    A prior art system such as the one depicted in  FIG. 2  and  FIG. 4  suffers from an unexpected problem. 
         [0019]    The droplets that are ejected by the nozzles of the printhead  210 ,  440  have a finite velocity while they travel to their landing position. As a result it takes some time for them to reach their landing position on the rotating drum. The effect can be described as “landing position lag”. This landing position lag—by itself—poses no problem. However, in the prior art system shown in  FIG. 2 , the nozzles near the leading edge of the printhead eject droplets that land on a layer of the print master having a relatively smaller diameter, whereas the nozzles near the trailing edge of the printhead eject droplets that land on a layer having a relatively larger diameter. The effect of this is that the droplets ejected by nozzles near the leading edge of the printhead be subject to more landing position lag compared with droplets ejected by nozzles near the trailing edge of the printhead. This results in a distortion of the three-dimensional grid that makes up the relief print master, since droplets that are intended to be stacked on top of each other in the different layers will be shifted relatively to each other. This weakens the matrix of droplets that make up the relief print master. 
       SUMMARY OF THE INVENTION 
       [0020]    In order to overcome the problems described above, preferred embodiments of the invention reduce the geometrical distortion of the matrix of cured droplets that make up the relief print master and that results from the effects of landing position lag in a prior art system as the one shown in  FIG. 2 . 
         [0021]    Preferred embodiments of the invention are realized by a system described below wherein the printhead is rotated in the plane that corresponds with its nozzle plate by an amount that reduces the effects of landing position lag. 
         [0022]    Various specific preferred embodiments are also described below. 
         [0023]    The above and other elements, features, steps, characteristics and advantages of the present invention will become more apparent from the following detailed description of the preferred embodiments with reference to the attached drawings. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0024]      FIG. 1  shows an embodiment of an apparatus for printing a relief print master on a sleeve. 
           [0025]      FIG. 2  shows a different view of an embodiment of an apparatus for printing a relief print master on a sleeve. 
           [0026]      FIG. 3  shows a printhead with a single row of nozzles. 
           [0027]      FIG. 4  shows a spiral path on which the fluid droplets ejected by the nozzles of a printhead as in  FIG. 3  land. 
           [0028]      FIG. 5  shows projections in the Y-Z, X-Y and X-Z planes that demonstrate the effect of landing position lag in a prior art system. 
           [0029]      FIG. 6  shows projections in the Y-Z, X-Y and X-Z planes that demonstrate how the effect of the landing position lag is reduced by rotating the printhead in a XY-plane that is parallel to a plane that is tangential to the cylindrical sleeve. 
       
    
    
     DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS 
       [0030]      FIG. 4  shows a prior art system that is suitable for creating a relief print master and that can serve as the basis for an improved system according to preferred embodiments of the current invention. 
         [0031]      FIG. 5  shows projections on three different orthogonal planes of the relevant portions of the prior art system in  FIG. 4 . 
         [0032]    In  FIG. 4  the cylindrical support  400  rotates at a frequency of NumberofRevolutionsperSecond along a central axis  470 . 
         [0033]    In  FIGS. 4 and 5 , a printhead unit  440 ,  520  has nozzles that are arranged on a nozzle row  530 . The distance between the nozzle row  530  and the central axis of the rotating cylinder is referred to by the variable NozzlePlateDistance. In the prior art system shown in  FIG. 4  and  FIG. 5  the nozzle row  530  is parallel with the central axis  470  of the rotating cylindrical support  400 . 
         [0034]    Every nozzle of the printhead has an index number j that in  FIG. 4  and  FIG. 5  ranges from 1 to 5. The distance between two adjacent nozzles is the nozzle pitch, represented by the variable NozzlePitch and is indicated with the reference signs  430  (in  FIG. 4) and 540  (in  FIG. 5 ). In the remaining part of this document a nozzle having an index number j will be referred to as “nozzle[j]”. 
         [0035]    The Y dimension in  FIG. 4  and  FIG. 5  is parallel with the central axis  470  of the drum  400  (in  FIG. 4 ). The Y direction corresponds with the movement of the printhead in the Y dimension and is indicated by the arrow in the drawings. The speed of the movement of the printhead  440 ,  520  in the Y-dimension is locked with the frequency of the rotating cylindrical drum support. 
         [0036]    The X dimension in  FIG. 4  and  FIG. 5  indicates the direction in which a point on the surface of the rotating drum moves. Because in the context of the preferred embodiments of the current invention the diameter of the cylindrical support is significantly larger than the displacement of a point on the surface during the relevant time frame of the rotation of the cylindrical support, the X-dimension can be locally approximated by a straight line that is tangential to the surface of the drum and that is orthogonal with regard to the central axis of the cylindrical drum. A rendering of the X-dimension is indicated in the X-Z projection (on the right-hand side) of  FIG. 5 , where the layers  511 ,  512 ,  513 ,  514  and  515  have a nearly flat curvature. In the calculations that follow, the X-dimension is locally approximated by a straight line. 
         [0037]    The Z direction is orthogonal to both the X and Y dimensions and indicates the height with regard to a reference surface in an X-Y plane. In  FIG. 5  the nozzle plane, i.e. a fictitious plane in which the orifices of the nozzles are located serves as a reference plane. 
         [0038]    In a more general preferred embodiment, a printhead unit according to the current invention can have any number of nozzles on a nozzle row higher than one. Also, in a more general preferred embodiment a printhead unit can optionally have multiple parallel nozzle rows that can be staggered, for example for increasing the resolution of the printhead unit compared with the resolution of the individual printheads. In that case, the multiple parallel rows are located in a plane that is parallel with a tangent plane of the rotating cylindrical support. 
         [0039]    The nozzles  1 ,  2 ,  3 ,  4  and  5  of the printhead unit  520  eject droplets that land on the different layers  511 ,  512 ,  513 ,  514  and  515 . The landing positions are indicated with the reference numbers  1 ′,  2 ′,  3 ′,  4 ′ and  5 ′. 
         [0040]    The positions  1 ′,  2 ′,  3 ′,  4 ′ and  5 ′ of the landed droplets can be connected by a curve  550 . 
         [0041]    The printhead  440 ,  520  has a leading edge portion that contains a nozzle that jets onto a layer having a relatively smaller diameter and a trailing edge portion that comprises a nozzle that jets onto a layer having a relatively (with regard to the layer on which the nozzle belonging to the leading edge jets) larger diameter. For example, in  FIG. 5  a nozzle[4] which jets onto layer  514  could belong to a leading edge portion of the printhead, whereas nozzle[2] jetting onto layer  512  would belong to a trailing edge portion of the printhead. 
       Part 1 of the Mathematical Analysis 
       [0042]    In  FIG. 5  any given layer  511 ,  512 ,  513 ,  514  and  515  on the drum has a diameter represented by the variable Diameter[i] in which i is an index number that refers to the layer. In the remaining part of the text, a layer having an index number i will be referred to as “layer[i]”. 
         [0043]    The circumference of such a layer i is represented by the variable Circumference[i] and has a value equal to: 
         [0000]      Circumference[ i]=PI* Diameter[ i]   
         [0044]    The sleeve rotates in an X-direction at a frequency that is represented by the variable NumberofRevolutionsperSecond. The circumferential speed of a given layer i of the sleeve is represented by the variable CircumferentialSpeed[i] and expresses the displacement Δx[i] of a surface point on the layer in the X dimension per time unit. 
         [0000]      CircumferentialSpeed[ i]=Δx[i]/Δt    
         [0000]    The value of CircumferentialSpeed[i] is equal to: 
         [0000]    
       
         
           
             
               
                 
                   
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       Part 2 of the Mathematical Analysis 
       [0045]    A nozzle[j] ejects a droplet at a time point t 1  with a speed equal to DropletVelocity in the Z-dimension. The value of the speed DropletVelocity is a characteristic of the printhead unit and is expressed by: 
         [0000]      DropletVelocity= dz/dt    
         [0000]    Δz[i][j] is the distance between a nozzle[j] and the surface of a layer[i] on which the droplets ejected by nozzle[j] land. For example, in  FIG. 5  Δz[3][3] (indicated by the reference sign  560 ) is the distance between nozzle[3] and the layer[3] on which the droplets ejected by nozzle[3] land. 
         [0046]    If it is assumed that the droplet velocity is constant over the trajectory Δz[i][j], the time Δt[i][j] it takes for the droplet to travel over the distance Δz[i][j] is expressed by: 
         [0000]      Δ t[i][j]=Δz[i][j] /DropletVelocity
 
         [0047]    The droplet ejected by a nozzle[j] arrives at the surface of the layer[i] at a time t 2  which is equal to: 
         [0000]        t 2= t 1+Δ z[i][j] /DropletVelocity
 
       Part 3 of the Mathematical Analysis 
       [0048]    Referring to  FIG. 5 , the x-coordinate of the position of nozzle[j] can be referred to as x[j][0]. 
         [0049]    Similarly, x[j][i] refers to the x-coordinate of a droplet that was ejected by nozzle[j] and that has landed on layer[i]. 
         [0050]    The difference between the x-coordinate x[j][0] of the nozzle[j] and the x-coordinate x[j][i] is referred to as Δx[j][i] and is defined as: 
         [0000]      Δ x[j][i]=x[j][i]−x[j][ 0]
 
         [0051]    While a droplet ejected by a nozzle[j] travels from the orifice of the nozzle to the surface of a layer[i] of the drum, this surface has moved during a period Δt[i][j] over a distance Δx[i][j] in the x dimension that is equal to: 
         [0000]      Δ x[i][j]= CircumferentialSpeed[ i]*Δt[i][j] 
 
         [0052]    Substituting in the above expression the variables CircumferentialSpeed[i] and Δt[i][j] leads to: 
         [0000]      Δ x[i][j ]=CircumferentialSpeed[ i]*Δz[i][i] /DropletVelocity
 
         [0000]      Δ x[i][j]=PI *Diameter[ i ]*NumberofRevolutionsperSecond*Δ z[i][j] /DropletVelocity
 
         [0053]    If the nozzle plate of a printhead is located at a distance having a value NozzlePlateDistance from the axis of the drum, and a layer[i] on the drum has a diameter equal to Diameter[i], then the distance Δz[i][j] between a nozzle[j] and a layer[i] can be expressed as: 
         [0000]      Δ z[i][j] =NozzlePlateDistance−Diameter[ i]/ 2
 
         [0054]    By substituting this expression for Δz[i][j] into the expression for Δx[i][j], the following new expression is obtained for Δx[i][j]: 
         [0000]      Δ x[i][j]=PI *Diameter[ i ]*NumberofRevolutionsperSecond*(NozzlePlateDistance−Diameter[ i]/ 2)/DropletVelocity
 
         [0055]    The above expression provides the value for the x-coordinate of the landing position: 
         [0000]        x[j][i]=x[j][ 0 ]+Δx[j][i]   
         [0000]        x[j][i]=x[j][ 0 ]+PI* Diameter[ i ]*NumberofRevolutionsperSecond*(NozzlePlateDistance−Diameter[ i]/ 2)/DropletVelocity
 
         [0056]    Defining a constant K having a value equal to: 
         [0000]        K=PI *NumberofRevolutionsperSecond/DropletVelocity 
         [0000]    optionally simplifies the expression for Δx[i][j] to: 
         [0000]      Δ x[i][j]=K* Diameter[ i ]*(NozzlePlateDistance−Diameter[ i]/ 2)
 
       Part 4: Interpretation of the Mathematical Analysis 
       [0057]    For a given nozzle[j], the expression for Δx[i][j] is a quadratic function of the Diameter[i] of the layer[i] on which its ejected droplets land. 
         [0058]    K is a constant of which the sign depends on the sign of variable NumberofRevolutionsperSecond. In what follows it is assumed that both the variables NumberofRevolutionsperSecond and hence K have a positive sign. 
         [0059]    The structural relation between the drum and the printhead dictates that for an arbitrary layer the following constraint must be met: 
         [0000]      Diameter[ i]/ 2&lt;=NozzlePlateDistance 
         [0000]    The value of Δx[i][j] becomes 0 in the special case that: 
         [0000]      Diameter[ i]/ 2=NozzlePlateDistance 
         [0060]    As the value Diameter[i] of the diameter of a layer linearly decreases, the absolute value of Δx[i][j] quadratically increases. 
         [0061]    In a practical situation the variations of the variable Diameter[i] are small compared with the value of NozzlePlateDistance. 
         [0062]    In that case the quadratic function can be locally approximated by a straight line. The slope of this straight line is expressed by the first derivative of the quadratic function: 
         [0000]      δ(Δ x[i][j ])/δ(Diameter[ i ])= K *(NozzlePlateDistance−Diameter[ i ])
 
         [0063]    For layers nearby the nozzle plate, the variable Diameter[i] has a value that is approximately equal to 2*NozzlePlateDistance, the value of the first derivative is equal to: 
         [0000]      δ(Δ x[i][j ])/δ(Diameter[ i ])=− K *NozzlePlateDistance
 
         [0064]    In that case the local expression for Δx[i][j] becomes: 
         [0000]      Δ x[i][j]≈K *NozzlePlateDistance*(2*NozzlePlateDistance−Diameter[ i ])
 
       Part 5: Correction 
       [0065]    Referring to  FIG. 5 , the landing position of a droplet ejected by the nozzle[ 1 ] on the layer[ 1 ] is shifted over a distance Δx[ 1 ][ 1 ] (reference sign  562 ) in the opposite X direction, whereas the landing position of a droplet ejected by the nozzle[ 5 ] on the layer[ 5 ] is shifted over a distance Δx[ 5 ][ 5 ] (reference sign  563 ) in the opposite X direction. The distances Δx[ 1 ][ 1 ] and Δx[ 5 ][ 5 ] can be expressed using the previous equations: 
         [0000]      Δ x[ 1][1 ]=K* Diameter[1](NozzlePlateDistance−Diameter[1]/2)
 
         [0000]      Δ x[ 5][5 ]=K* Diameter[5](NozzlePlateDistance−Diameter[5]/2)
 
         [0066]    The difference (Δx[ 5 ][ 5 ]−Δx[ 1 ][ 1 ]) in the x dimension between the landing positions of droplets ejected by nozzle[ 1 ] and nozzle[ 5 ] is expressed by: 
         [0000]      (Δ x[ 5][5]−Δ x[ 1][1])= K *NozzlePlateDistance*(Diameter[5]−Diameter[1])− K *(Diameter[5] 2 −Diameter[1] 2 )/2
 
         [0067]    All the values in the above expression are design parameters of the system so that the value of (Δx[ 5 ][ 5 ]−Δx[ 1 ][ 1 ] can be easily evaluated. 
         [0068]    In  FIG. 6 , the printhead  520  is rotated in the x-y plane by an angle α around a rotation center that corresponds with the position of the nozzle[ 1 ]. 
         [0069]    As a result of this rotation, the landing position of a droplet that is ejected by the nozzle[ 5 ] is moved over a distance having a value ΔxRotatedHead[ 5 ] in the x direction. 
         [0070]    The displacement of ΔxRotatedHead[ 5 ] is expressed by 
         [0000]      Δ x RotatedHead[5]=sin(α)*(5−1)*NozzlePitch
 
         [0071]    By selecting an appropriate value for α, it is possible to obtain that the difference (Δx[ 5 ][ 5 ]−Δx[ 1 ][ 1 ]) between the landing positions of droplets ejected by nozzle[ 1 ] and nozzle[ 5 ] in the X-dimension is exactly compensated by the displacement ΔxRotatedHead[n] that results from rotating the printhead with an angle α. 
         [0072]    Mathematically, this translates into the following requirement: 
         [0000]      (Δ x[ 5][5 ]−Δx[ 1][1])=Δ x RotatedHead[5]
 
         [0073]    The value for α that should be selected to meet this condition is: 
         [0000]      α= a  sin {(Δ x[ 5][5]−Δ[1][1])/((5−1)*NozzlePitch)}
 
         [0074]    As  FIG. 6  shows, the rotation of the printhead with the angle α considerably flattens out the curve  650  that connects the landing positions of the ejected droplets. 
         [0075]      FIG. 6  shows a specific case in which the printhead  520  is rotated so that the droplets ejected by nozzles  1  and  5  fall on the same line that is parallel with the Y axis.  FIG. 6  is also a specific case in which there are as many layers on which droplets land as there are nozzles, whereby every nozzle ejects droplets on a different layer. 
         [0076]    In a more general case a printhead has N nozzles having index numbers i (i=1, 2, 3, . . . N) and ejects droplets on M layers having index numbers j (j=1, 2, 3, . . . M). 
         [0077]    The generalized formula for obtaining the angle α for rotating the printhead so that the droplets of two different nozzles, having index numbers j1 and j2 (1&lt;=j1&lt;j2&lt;=N) and that jet on layers having index numbers i1 and i2 (1&lt;=i1&lt;=i2&lt;=M), fall on a line parallel with the Y dimension is: 
         [0000]      α= a  sin {(Δ x[i 2][ j 2]−Δ x[i 1][ j 1])/(( j 2− j 1)*NozzlePitch)}
 
         [0000]    In which:
       Δx[i2][j2] refers to the distance measured in the z-direction between the nozzle having an index number j2 and the layer having an index number i2 on which the droplets of said nozzle land;   Δx[i1][j1] refers to the distance measured in the z-direction between the nozzle having an index number j1 and the layer having an index number i1 on which the droplets of said nozzle land.       
 
         [0080]    Using the above formula leads to a compensation that—under the given assumptions—will bring the landing positions of droplets ejected by the nozzle[j1] and nozzle[j2] on a line that is parallel with the Y dimension. 
         [0081]    In the example shown in  FIG. 6  the angle α was optimized so that the droplets ejected by the first nozzle and the last nozzle of the nozzle row would land on the same x coordinate. In a more general case nozzle[j1] belongs to a trailing portion of the printhead, whereas nozzle[j2] belongs to a leading portion of the printhead. The optimal selection of j1 and j2 can depend on what criterion is used for the “flattening” of the curve  650 . Examples of such criterions are: minimizing the maximum deviation in the X-dimension between the landing positions of the droplets ejected by the nozzles of a printhead, or minimizing the root mean square value of the deviations. In general satisfactory results are obtained with a selection of j1=1 and j2=N. 
         [0082]    In  FIG. 6  the rotation of the printhead has a rotational center that corresponds with the X-Y position of nozzle[1]. In a more general preferred embodiment, preferred embodiments of the invention can be brought into practice using a different rotational center, such as for example the X-Y position of nozzle[3] or—even more in general, any other location within the nozzle plate of the printhead. 
         [0083]    In the above mathematical analysis it was explicitly assumed that the speed of the droplets between the time they leave the nozzle plate and the time they land on a layer remains constant. This is only approximately true. In a real situation, the speed of a droplet ejected by a nozzle diminishes while it travels through space from the orifice towards its landing position. The effect of this is that the difference of the landing position along the X-dimension of droplets landing on layers with different diameters increases even more than what is predicted by the expression for Δx[j][i]. In that case compensation is necessary by rotating the printhead by an amount that is larger than the value of α that is predicted in the above formula for this angle. Consequently, a preferred embodiment of the current invention specifies the value for a using the following inequality: 
         [0000]      α= r*a  sin {(Δ x[i 2][ j 2]−Δ x[i 1][ j 1])/(( j 2− j 1)*NozzlePitch)}
 
         [0000]    in which: 1.0&lt;=r 
         [0084]    In another preferred embodiment a meets the following constraint: 
         [0000]      α= r*a  sin {(Δ x[i 2][ j 2]−Δ x[i 1][ j 1])/(( j 2− j 1)*NozzlePitch)}
 
         [0000]    In which: 1.0&lt;=r&lt;=2.0 
         [0085]    In yet another preferred embodiment a meets the following constraint: 
         [0000]      α= r*a  sin {(Δ x[i 2][ j 2]−Δ x[i 1][ j 1])/(( j 2− j 1)*NozzlePitch)}
 
         [0000]    In which: 1.0&lt;=r&lt;=1.1 
         [0086]    There may be instances that it is not necessary or even desirable to rotate the printhead by an amount that achieves maximum compensation for the x coordinate of the landing positions of droplets ejected by nozzles on different layers. 
         [0087]    In one preferred embodiment the rotation by the angle α meets the following constraint: 
         [0000]      α= r*a  sin {(Δ x[i 2][ j 2]−Δ x[i 1][ j 1])/(( j 2− j 1)*NozzlePitch)}
 
         [0000]    In which: 0.1&lt;=r&lt;=1.0 
         [0088]    In another preferred embodiment the rotation by the angle α meets the following constraint: 
         [0000]      α= r*a  sin {(Δ x[i 2][ j 2]−Δ x[i 1][ j 1])/(( j 2− j 1)*NozzlePitch)}
 
         [0000]    In which: 0.5&lt;=r&lt;=1.0 
         [0089]    In yet another preferred embodiment the rotation by the angle α meets the following constraint: 
         [0000]      α= r*a  sin {(Δ x[i 2][ j 2]−Δ x[i 1][ j 1])/(( j 2− j 1)*NozzlePitch)}
 
         [0000]    In which: 0.9&lt;=r&lt;=1.0 
         [0090]    Having explained the preferred embodiments of the invention in the context of preparing a relief print master, it should be clear to the person skilled in the art that the same inventive concepts can be used for creating other three-dimensional objects on a cylindrical drum than a relief print master for flexography. In general, any relief object that fits on a cylindrical drum and that can be printed using curable liquid can benefit from using the invention. 
         [0091]    While preferred embodiments of the present invention have been described above, it is to be understood that variations and modifications will be apparent to those skilled in the art without departing from the scope and spirit of the present invention. The scope of the present invention, therefore, is to be determined solely by the following claims.