Patent Publication Number: US-2021191331-A1

Title: Moon phase display mechanism

Description:
TECHNICAL FIELD OF THE INVENTION 
     The present invention relates to a moon phase display mechanism. More specifically, the moon phase display mechanism according to the invention is intended to equip a wearable object of small dimensions such as a timepiece, in particular a wristwatch. 
     Technological Background of the Invention 
     Timepieces, in particular wristwatches, equipped with a moon phase display mechanism have been known for a long time. These moon phase display mechanisms are, however, more decorative than they provide a piece of information allowing the owner of the watch to easily determine which quarter of the moon it is in. The simplest moon phase display mechanisms comprise a hand indicator that points to the various representations of the phases of the moon (first quarter, full moon, last quarter, new moon). Other known moon phase display mechanisms comprise a disc which carries two representations of the Moon, part of this disc being visible through an opening of adapted shape made in the dial of the watch and successively revealing a waxing moon, a full moon, a waning moon and a new moon. Such a presentation of the various phases of the Moon is very advantageous from an aesthetic point of view; however, the way the moon is represented has only a distant relation to the way the lunar star appears in the sky. Yet another moon phase display mechanism comprises a two-colour sphere that revolves completely on itself with each lunar cycle. Such a moon phase display mechanism allows the face of the moon to be represented realistically. However, because such a moon phase display mechanism uses a sphere to represent the different quarters of the moon, it is thick and occupies a large space, so that it is difficult to be integrated into a horological movement, in particular a wristwatch. 
     SUMMARY OF THE INVENTION 
     The present invention has the purpose of providing a moon phase display mechanism which provides a moon phase display which is in particular more faithful to reality and more easily understandable for the owner of the watch. 
     To this end, the present invention relates to a moon phase display mechanism driven by a horological movement, this moon phase display mechanism comprising a transparent support provided with an upper face and a lower face which extends at a distance from the upper face, a representation of the Moon being transferred, for example by printing or by engraving, to one of the upper or lower faces of this transparent support, a substrate being disposed under the transparent support, at a distance from the lower face of the latter, the moon phase display mechanism also comprising drive means moved by the horological movement and which are arranged to displace a shutter between the transparent support and the substrate, the shutter and the substrate having display contrasts which are inverted relative to each other, the shutter being displaced from an initial position to a final position for a duration of a lunar cycle, so as to reveal day after day the aspect of the Moon which changes from the new moon to the first quarter moon, then from the first quarter moon to the full moon, then to the last quarter moon and finally to the new moon, the shutter being returned from its final position to its initial position at the end of the lunar cycle. 
     According to a special embodiment of the invention, the drive means comprise a rectilinear rack which is driven by the horological movement and with which the shutter is fixedly coupled in translation. 
     Thanks to these features, the present invention provides a moon phase display mechanism allowing to display day after day the different aspects of the Moon in a manner which is original and easily understood by the user. In particular, the representation of the Moon which is provided by the moon phase display mechanism according to the invention is very close to the real aspect of the Moon in the sky, so that it is much simpler for the user to determine at which period of the lunar cycle the Moon is located. The moon phase display mechanism according to the invention is also thinner than those using a sphere rotating on itself, and therefore easier to integrate into a horological movement, in particular a wristwatch. In addition, regardless of the quarter wherein the Moon is located, its representation is always visible to the owner of the watch. It will also be noted that the moon phase display mechanism according to the invention allows to obtain a representation of the various phases of the realistic Moon, formed by two surfaces of different colours and separated by a terminator, that is to say the curve which separates the illuminated part from the dark part of the Moon, the profile of which is very realistic and very faithful to what the user can see when observing the Moon in the sky. This is in particular the case during the first and the last quarter moon, when the optical distortions are almost zero and when the terminator thus appears perfectly rectilinear. 
     According to another particular embodiment of the invention, the transparent support is in the form of a lens of plano-concave shape delimited upwardly, on the side of the observer, by a planar surface which receives the representation of the Moon, and delimited downwardly by a concave surface to which a preferably but not necessarily aspherical profile is given, this plano-concave lens being combined with a shutter to which is given a curved profile, preferably of the hyperbolic type. 
     Thanks to the combined use of a plano-concave, preferably aspherical lens, and of a shutter with a curved profile, preferably but not exhaustively of the hyperbolic type, the observer sees a terminator, that is to say the curve which separates the illuminated part from the dark part of the Moon, the profile of which is very realistic and very faithful to what the user can see when observing the Moon in the sky. Furthermore, the moon phase display mechanism is more compact than a moon phase display mechanism using a sphere and can thus be housed in a smaller volume such as that of a case of a wristwatch-type timepiece. By way of example, it is considered that for a representation of the Moon of the same diameter, the moon phase display mechanism according to the invention is half the thickness of a moon phase display mechanism using a sphere. Likewise, it is understood that, since the surface which receives the representation of the Moon is planar, the moon phase display mechanism according to the invention does not impede the movement of the displacement of the hands on the surface of the dial. 
    
    
     
       BRIEF DESCRIPTION OF THE FIGURES 
       Other features and advantages of the present invention will emerge more clearly from the detailed description which follows of an exemplary embodiment of a moon phase display mechanism according to the invention, this example being given in a purely illustrative and non-limiting manner only in connection with the appended drawing on which: 
         FIG. 1  is a plan view of the moon phase display mechanism according to the invention; 
         FIG. 2  is a detail view on a larger scale of the oblong hole into which the pin carried by the finger protrudes; 
         FIG. 3A  is a detail view on a larger scale of the first lever in its intermediate position A 
         FIG. 3B  is a detail view on a larger scale of the first lever in its extreme position B wherein it bears against the top of the finger profile; 
         FIG. 4A  is a detail view on a larger scale of the first rack in its position C wherein its feeler beak is at the top of the cam profile; 
         FIG. 4B  is a detail view on a larger scale of the first rack in its position D wherein its feeler beak falls along the step of the cam; 
         FIG. 5  is a top view of the transparent support and the sheet metal from which the aspherical plano-concave lens and the shutter are obtained; 
         FIG. 6  is an elevational and sectional view of the optical assembly formed by the aspherical plano-concave lens, the shutter and the substrate; 
         FIG. 7  is a schematic top view which illustrates the aspect of the representation of the Moon as it can be perceived by the observer when the shutter begins to penetrate into the space which separates the aspherical plano-concave lens from the substrate; 
         FIG. 8A  is a schematic view of the moon phase display mechanism when it is in its extreme position E at the start of a lunar cycle; 
         FIG. 8B  is a view similar to that of  FIG. 8A  which illustrates the moon phase display mechanism according to the invention when it is in its extreme position F at the end of the lunar cycle, and 
         FIGS. 9A to 9L  illustrate the aspect of the terminator in several positions of the curved, preferably hyperbolic profile shutter. 
     
    
    
     DETAILED DESCRIPTION OF AN EMBODIMENT OF THE INVENTION 
     The present invention proceeds from the general inventive idea which consists in transferring a representation of the Moon on either one of the two upper and lower faces of a transparent support which is disposed above and at a distance from a substrate, with a shutter interposed between the transparent support and the substrate. The face of the Moon can be represented in a colour similar to that of the substrate, while the shutter and the substrate have inverted contrasts: if the substrate is bright, then the shutter will be dark and, conversely, if the substrate is dark, the shutter will be bright. Assuming, only as an illustrative example, that the representation of the Moon and the substrate are dark and that the shutter is bright, it is understood that when the shutter is not in the space between the transparent support and the dark substrate, the representation of the Moon which is above the dark substrate is not perceptible to the observer. Then, as the bright shutter penetrates into the space which separates the transparent support and the dark substrate, the representation of the Moon gradually becomes visible to the user. The present invention thus provides a mechanism which is more compact than the moon phase display mechanisms which comprise a sphere and which allows the moon phases to be displayed in an original and much more realistic manner than most prior art moon phase display devices allow. Consequently, it is much easier for the observer to understand what period of the lunar cycle he is in. Furthermore, the realism is further increased if, in accordance with a special embodiment of the invention, the transparent support is given a plano-concave profile, preferably but not necessarily an aspherical profile, and such a transparent support is combined with a curved shutter, preferably hyperbolic in profile. Indeed, such a combination allows to obtain a terminator whose profile is very faithful to the one observed in reality as the Moon waxes, becomes full, then wanes and the lunar cycle resumes. 
     Housed for example in a timepiece such as a wristwatch, the moon phase display mechanism  1  according to the invention is driven by a horological movement, that is to say a mechanism whose operation depends on the division of time. More specifically, the horological movement comprises a motion-work mobile, one pinion of which (not visible in the figures) drives a twenty-four-hour wheel  2  which, as its name suggests, is arranged so as to perform one complete revolution by day. 
     The twenty-four-hour wheel  2  carries a finger  4  on an axis  6  of which this finger  4  is mounted free in rotation. In order to be able to pivot relative to the twenty-four-hour wheel  2 , the finger  4  is mounted on the axis  6  with a slight axial play thanks to a ring  8  engaged on this axis  6 . Moreover, the finger  4  is provided with a pin  10  which protrudes into an oblong hole  12  formed in the thickness of the twenty-four-hour wheel  2  and which limits the freedom of pivoting of the finger  4  relative to the twenty-four-hour wheel  2  (see  FIG. 2 ). It is therefore understood that when the pin  10  abuts against an inner wall  14  of the oblong hole  12 , it is driven in rotation by the twenty-four-hour wheel  2  and in turn drives the finger  4  which, as well, performs a complete revolution in twenty-four-hours. 
     The moon phase display mechanism  1  according to the invention also comprises a first lever  16  which is pivotally mounted about a pivot axis  18  and which is elastically applied against a first part  20   a  of a finger  4  profile  20  by an upper spring  22 . The presence of a starwheel  24  whose position is indexed by a jumper  26  which is held elastically against a toothing  28  of this starwheel  24  by a lower spring  30  is also noted in the drawing. 
     The twenty-four-hour wheel  2  rotates in the clockwise direction, bringing with it the finger  4 . The first lever  16  thus slides along the first part  20   a  of the finger  4  profile  20  and, after passing through an intermediate position A ( FIG. 3A ), is in an extreme position B ( FIG. 3B ) wherein it is supported by a foot  32  against a top  34  of the finger  4  profile  20 . Moreover, the first lever  16  is engaged by a beak  36  with the toothing  28  of the starwheel  24 . When, for example around midnight, the finger  4  advances further, the first lever  16  exceeds the extreme position B wherein it is supported against the top  34  of the finger  4  profile  20 , and drives the starwheel  24  by one pitch in the counter-clockwise direction. This movement is allowed by the fact that when the first lever  16  exceeds the top  34  of the finger  4  profile  20 , a lever effect occurs on the finger  4  which causes the pivoting of this finger  4  and the concomitant displacement of the pin  10  which, in abutment against one end of the oblong hole  12  formed in the thickness of the twenty-four-hour wheel  2 , will displace to abut against the opposite end of this oblong hole  12 . Then, the first lever  16  begins to slide again along a second part  20   b  of the finger  4  profile  20  which is located after the top  34  of this profile  20 . It will be noted that at the very moment when the first lever  16  causes the advance of the starwheel  24  by one pitch, the jumper  26  switches, against the return force of the lower spring  30 , from a groove between two consecutive teeth of the toothing  28  of the starwheel  24  to the immediately following groove of this toothing  28 . By falling into the following groove, the jumper  26  allows the starwheel  24  to complete its one-pitch advance and once again ensures the precise positioning of this starwheel  24 . 
     According to a preferred but non-limiting embodiment of the moon phase display mechanism according to the invention, the latter also comprises a manual device for correcting the moon phase display. Referred to as a whole by the general reference numeral  38 , this manual correction device comprises a second lever  40  pivoted about an axis  42  and which comprises an actuating means  44  such as a pin at an end opposite to the pivot axis  42 . This second lever  40  comprises for example a folded area  46  against which rests a corrector (not visible in the drawing) when the latter is actuated against the elastic return force of a spring by the owner of the wristwatch from outside the volume of the watch case. Under the effect of actuation of the corrector, the second lever  40  pivots about its axis  42  and in turn controls the pivoting of the first lever  16  so as to cause the starwheel  24  to advance by one pitch. This advance of the starwheel  24  takes place under the same conditions as those described above when the first lever exceeds the top  34  of the finger  4  profile  20 . 
     According to a preferred embodiment given for only a purely illustrative and non-limiting purpose, a complete revolution of the starwheel  24  corresponds to two successive lunar cycles, a lunar cycle corresponding to the time which elapses between two new successive moons and which is also called lunar month. To this end, the moon phase display mechanism according to the invention is completed by a first pinion  50  mounted coaxially and fixed in rotation on the starwheel  24 , by a setting-wheel  56  as well as by a cam  52  on the axis of rotation of which a second pinion  54  is fixedly mounted. The first pinion  50  drives the second pinion  54  via the setting-wheel  56 , the toothing ratios of this kinematic chain being calculated so that the cam  52  performs one complete revolution per lunar cycle. 
     The cam  52  has a snail-like profile  58  provided with a substantially rectilinear step  60 . A first rack  62  provided with a toothed sector  64  is also provided with a feeler beak  66  by which it permanently follows the cam  52  profile  58 . Shortly before the start of a new lunar cycle, for example at around midnight, the feeler beak  66  of the first rack  62  is at the top of the cam  52  profile  58  (position C— FIG. 4A ), then falls along the step  60  of the cam  52  (position D— FIG. 4B ). During this movement, the first rack  62  which, by its toothed sector  64 , is in permanent engagement with a third pinion  68 , rotates this third pinion  68  in the clockwise direction by an amount corresponding to the drop of the feeler beak  66  along the step  60 . 
     By rotating, the third pinion  68  rotates a wheel  70  with which it forms a mobile  69 . In other words, the third pinion  68  is mounted on the wheel  70  in a coaxial manner and fixed in rotation relative to this wheel  70 . Consequently, the wheel  70  transmits its rotational movement to drive means  72  of the moon phase display mechanism  1  which comprise a lower wheel  74  and an upper wheel  78  mounted free in rotation on an axis of rotation  76 . The lower wheel  74  meshes with a rectilinear rack  80  which in turn meshes with the upper wheel  78 . 
     According to the invention, the lower wheel  74  is responsible for controlling the moon phase display mechanism  1 . To this end, by pivoting, the lower wheel  74  drives the rectilinear rack  80  in translation and pushes it back into a first extreme position E illustrated in  FIG. 8A  which corresponds to the start of a new lunar cycle. Subsequently, when, after having fallen along the step  60  of the cam  52  at the start of the lunar cycle, the feeler beak  66  begins again to follow the cam  52  profile  58 , the feeler beak  66  is gradually pushed back in the clockwise direction to a second extreme position F (see  FIG. 8B ), so that the third pinion  68 , and therefore the wheel  70 , rotate in the counter-clockwise direction. Consequently, the lower wheel  74  rotates in the clockwise direction and drives the rectilinear rack  80  in translation from the right to the left of the drawing from its first extreme position E which corresponds to the start of a new lunar cycle to its second extreme position F illustrated in  FIG. 8B  which corresponds to the end of the lunar cycle. Once the feeler beak  66  has travelled the entire length of the cam  52  profile  58 , it will again find itself at the top of the step  60  of the cam  52  and, at the start of a new lunar cycle, the feeler beak  66  will fall along the step  60 , which will cause the return of the rectilinear rack  80  in its initial position. 
     The moon phase display mechanism according to the invention is supplemented by a device that allows to take-up clearances and return this moon phase display mechanism to its extreme position E at the end of a lunar cycle. This device consists of the upper wheel  78  engaged, on the one hand, with the toothing of the rectilinear rack  80 , and on the other hand, with an intermediate wheel  82  of an intermediate mobile  84  which also comprises an intermediate pinion  86 . This intermediate pinion  86  meshes with a toothed sector  88  of a second rack  90  which is elastically constrained by the return force of a fourth spring  92 . Thanks to this arrangement, all the plays of the kinematic chain which extends between the first rack  62  and the second rack  90 , are taken up so that the positioning of the rectilinear rack  80  is always precise. 
     According to the invention, the moon phase display mechanism  1  comprises the rectilinear rack  80  with which a shutter  94  is fixedly coupled in translation. The moon phase display mechanism  1  also comprises, on the side of an observer  96 , a transparent support  98  provided with an upper face  100  which extends parallel to and at a distance from a lower face  102 . A representation  104  of the Moon, for example in the form of a decal, is transferred to the upper face  100  of the transparent support  98 . This representation  104  of the Moon could also be transferred to the lower face  102  of the transparent support  98 . A substrate  106  is, relative to the observer  96 , disposed under the transparent support  98 , at a distance from the latter. The shutter  94  is mounted on the rectilinear rack  80  so as to be able to gradually penetrate into the space which separates the transparent support  98  from the substrate  106  when the rectilinear rack  80  is driven by the lower wheel  74 . The shutter  94  and the substrate  106  have inverted contrasts: either the shutter  94  is bright and the substrate  106  as well as the representation  104  of the Moon are dark, or the shutter  94  is dark and the substrate  106  as well as the representation  104  of the Moon are bright. Assuming, only by way of example, that the representation  104  of the Moon and the substrate  106  are dark and that the shutter  94  is bright and reflective, it is understood that when the shutter  94  is not in the space located between the transparent support  98  and the dark substrate  106 , the representation  104  of the Moon is located above the dark substrate  106  and is therefore not perceptible by the observer  96 . Then, as the bright and reflective shutter  94  penetrates into the space which separates the transparent support  98  from the dark substrate  106 , the representation  104  of the Moon gradually becomes perceptible by the user. More specifically, as the shutter  94  begins to penetrate into the space between the transparent support  98  and the dark substrate  106 , the observer  96  gradually sees the first quarter moon appear. Then, when the reflective shutter  94  is completely between the transparent support  98  and the dark substrate  106 , the observer  96  sees the complete representation  104  of the Moon: it is the full moon. Then, the shutter  94  continues its rectilinear movement in the same direction and begins to leave the space between the transparent support  98  and the dark substrate  106 , so that the observer  96  gradually sees the last quarter of the moon appear, a situation which corresponds to the moment when the shutter  94  leaves the same free surface as hidden surface. Finally, when the shutter  94  is completely out of the space between the transparent support  98  and the dark substrate  106 , the observer  96  no longer sees the representation  104  of the Moon again (on the assumption that the substrate  106  has the same colour as the representation  104  of the Moon) and therefore knows that the lunar cycle has ended and that a new lunar cycle will begin. Thus, thanks to the invention, the observer  96  has an easily understandable representation of the various phases of the moon: new moon, first quarter moon, full moon, last quarter moon and then again new moon. 
     According to a particular embodiment of the invention, the transparent support  98  is in the form of a lens  108  of plano-concave shape delimited upwardly, on the side of the observer  96 , by a planar surface  110  which receives the representation  104  of the Moon, and delimited downwardly by a concave surface  112  to which a preferably aspherical profile is given. This aspherical plano-concave lens  108  is combined with a shutter  94  folded in its centre to give it a curved profile, preferably but not necessarily a hyperbolic profile. An image of the Moon whose terminator, that is to say the curve which separates the dark part from the illuminated part of the Moon, best approximates the real aspect of the Moon in the sky is thus obtained. 
     To determine the geometric dimensions of the aspherical plano-concave lens  108  and of the hyperbolic profile shutter  94 , use is made of computer-aided optical system design software such as that marketed under the brand LightTools, whose version 8 which has been published in 2019 was used for the purposes of the present invention. 
     Once the dimensions of the representation  104  of the Moon which it is desired to be able to display by means of the moon phase display mechanism according to the invention have been defined, the main parameters on which it is possible to act in order to obtain a realistic representation of the phases of the Moon are:
         the material from which the aspherical plano-concave lens  108  will be made and therefore the refractive index of the latter;   the profile of the aspherical concave surface  112  of the aspherical plano-concave lens  108  and therefore its conic constant;   the dimensions of the shutter  94 ;   the distance which separates the top of the arch formed by the aspherical concave surface  112  and the shutter  94 ;   the curved, preferably hyperbolic, profile of the shutter  94  and therefore its conic constant.       

     Only by way of a preferred example, the aspherical plano-concave lens  108  is made of a transparent material whose refractive index is preferably comprised between 1.60 and 1.85, with an optimum value in the vicinity of 1.78. This value was selected after numerous tests which allowed to observe that, the higher the value of the refractive index of the material from which the lens is made, the closer the lens had to be brought to the shutter  94  so that it the latter is not visible to the observer through this lens. It is easily understood that this is favourable from the point of view of space requirement in the case where it is desired to integrate a moon phase display device in accordance with the present invention into a timepiece of the wristwatch type. On the other hand, the higher the refractive index, the more expensive and difficult the corresponding material is to be machined. Furthermore, it has been realised that when the lens gets too close to the shutter  94 , one ends up seeing the image of the peripheral edge of the lens which forms an opaline to lactescent crown around the representation  104  of the Moon, which is not acceptable. Likewise, it was found that by selecting too low refractive index values, the image of the shutter  94  with a curved and preferably hyperbolic profile which gradually covered the representation of the Moon was not very aesthetic, nor really realistic compared to the true representation of the Moon. This is why a value of the order of 1.60 to 1.85 and preferably equal to or substantially equal to 1.78 for the optical refractive index of the material from which the aspherical plano-concave lens  108  is made appeared to be an optimum allowing to provide the best compromise between the optical refractive index of the material from which the aspherical plano-concave lens  108  is made and the distance separating the aspherical concave surface  112  of the aspherical plano-concave lens  108  and the shutter  94 , and thus obtain a moon phase display mechanism whose space requirement is compatible with the dimensions of the timepiece wherein this mechanism is intended to be housed while providing a terminator whose profile is suitable. An example of a material which is well suited for the purposes of the present invention is the glass produced and marketed by Schott under the reference N-SF 11. 
     The dimensions of a block of transparent or at the very least translucent material such as a glass cylinder or polymer cylinder such as polycarbonate from which the aspherical plano-concave lens  108  is obtained are then introduced into the computer-aided design software. In the present case, the aspherical plano-concave lens  108  is obtained by machining a cylindrical glass block whose diameter D is comprised between 6 mm and 7 mm and whose height H is comprised between 0.9 mm and 1.1 mm (see  FIG. 5 ). 
     As regards the hyperbolic profile shutter  94 , this is obtained from a rectangular sheet metal whose thickness e is preferably but not exclusively comprised between 0.08 mm and 0.2 mm, and whose length l of the side which extends parallel to the direction of displacement of the shutter  94  is selected to be comprised between 7 mm and 8 mm, while the width L of the side which extends perpendicular to the direction of displacement of this shutter  94  is selected to be comprised between 9 mm and 10 mm. This sheet metal is provided at its centre with a fold  114  which extends in a direction parallel to the direction of displacement of the shutter  94  and preferably has flat edges  116  parallel to the fold  114 . It will indeed be noted that it is not necessary for the shutter  94  to maintain its hyperbolic profile to its ends because, in these areas, the optical distortion effect is produced essentially by the aspherical plano-concave lens  108 . These flat edges  116  therefore only have the function of completely obstructing the field of vision provided by the aspherical plano-concave lens  108  and, due to their flatness, these edges  116  allow to reduce the space requirement of the moon phase display mechanism. 
     The profile of the aspherical concave surface  112  of the aspherical plano-concave lens  108  is determined by the values of the distances r and z(r). If the central axis of symmetry of the aspherical plano-concave lens  108  is called S, the distance r corresponds to the distance which separates each point of the central axis of symmetry S from the point of the aspherical concave surface  112  which is located opposite thereto (see  FIG. 6 ). Likewise, the hyperbolic profile of the shutter  94  is determined by the distance r′ which separates each point of the plane of symmetry S′ of this shutter  94  from the surface of the latter. These distances r, r′ are determined by means of the same relation below: 
     
       
         
           
             
               z 
                
               
                 ( 
                 r 
                 ) 
               
             
             = 
             
               
                 
                   
                     r 
                     2 
                   
                   R 
                 
                 
                   1 
                   + 
                   
                     
                       1 
                       - 
                       
                         
                           ( 
                           
                             1 
                             + 
                             k 
                           
                           ) 
                         
                          
                         
                           
                             r 
                             2 
                           
                           
                             R 
                             2 
                           
                         
                       
                     
                   
                 
               
               + 
               
                 
                   ∑ 
                   
                     n 
                     = 
                     1 
                   
                   N 
                 
                  
                 
                   
                     A 
                     n 
                   
                   · 
                   
                     r 
                     n 
                   
                 
               
             
           
         
       
     
     where  k=−e   2    
     As visible in  FIG. 6 , the origin of the function z(r) corresponds to the point O which is located at the top of the arch formed by the aspherical concave surface  112 . The value of the function z(r) corresponds, in each point of the arch formed by the aspherical concave surface  112 , to the height of this point considered from the base of the aspherical plano-concave lens  108 . 
     The values of the constants R and k which characterise the aspherical plano-concave lens  108 , as well as those of the constants R′ and k′ which characterise the shutter  94  will be determined by successive iterations in the manner described below. As for the coefficients A n , they are coefficients of a polynomial sum the values of which will also be determined by iterations. 
     As for the aspherical plano-concave lens  108 , the constant R corresponds to the radius of curvature of the aspherical concave surface  112  at the point O which is located at the top of the arch formed by this aspherical concave surface  112 . So that the terminator T which is the dividing line between the dark part and the illuminated part of the Moon appears rectilinear in the middle of the lunar cycle, it is necessary that in the vicinity of the point O the aspherical concave surface  112  is practically planar. To this end, a very large radius of curvature R value, of the order of several thousands of millimetres, is initially introduced into the computer-aided design software. As for the constant “k” which is called “conic constant”, it is a quantity which describes the conical sections. Conical section means a plane curve defined by the intersection of a cone of revolution with a plane. When the section plane does not pass through the top of the cone, its intersection with this cone corresponds to one of the following plane curves: ellipse, parabola or hyperbola. 
     Note that k=−e 2  with e corresponding to the eccentricity of the conical section. The eccentricity of a conical section is a positive real number which characterises only the shape of this conical section; the eccentricity of a conical section can be interpreted as a measure of the amount by which a conical section deviates from a circle. Thus, the eccentricity of a circle is zero. The eccentricity of an ellipse that is not a circle is strictly comprised between zero and one. The eccentricity of a parabola is equal to 1 and the eccentricity of a hyperbola is greater than 1. 
     The conical constant k is involved in the equation 
         y   2 −2 Rx +( k+ 1) x   2 =0
 
     which describes a conical section whose apex is at the origin and whose tangent extends along the “y” axis and where R is the radius of curvature for x=0. This formula is used in geometric optics to describe the optical surface of a lens. In this case, it was initially indicated to the computer-aided design software that the conic constant was zero (k=0), in other words, one was dealing with a circle. 
     Consequently, as regards the aspherical plano-concave lens  108 , the simulation is started with a zero value of the conical constant k and a very large value of the radius of curvature R. 
     The same is true as regards the shutter  94  for which the simulation is started with a zero value of the conical constant k′ and a very large value of the radius of curvature R′. It is important to note that the shutter  94  can be considered as the object whose image is perceived through the aspherical plano-concave lens  108  and, as such, its geometric features can be determined by a computer-aided optical system design software such as LightTools. 
     Finally, it is considered that the aspherical plano-concave lens  108  is of even order, so that one starts by arbitrarily selecting values for the coefficients A 4 , A 6  and A 8 . In the initial choice of the values of the coefficients A 4 , A 6  and A 8 , the person skilled in the art is guided by the fact that he knows that the values of these coefficients are very low and that they keep decreasing as the index n increases. However, decision is made to stop at the coefficient A 8  because the contribution of the higher order coefficients on the improvement of the aspect resulting from the terminator T is negligible. Regarding the coefficient A 2 , this is ignored because the first term of the expression z(r) already contains the square of the variable r. 
     Using the computer-aided design software, a representation  118  of the Moon and its terminator T is simulated for several shutter positions  94  (see  FIGS. 7 and 9A to 9L ). In  FIG. 9A , it is the start of a lunar cycle. In  FIG. 9C , the Moon is in its first quarter. In  FIG. 9F , it is the middle of the lunar cycle and the Moon is full.  FIG. 91  corresponds to the last quarter of the Moon and in  FIG. 9L , it is the new Moon. To carry out the simulations, one begins, for example, to vary the values of the parameters A n  as well as of the conical constant k and of the radius of curvature R which characterise the aspherical plano-concave lens  108 , while keeping the values of the parameters A′, as well as of the conical constant k′ and of the radius of curvature R′ which characterise the shutter  94  unchanged, and observes on the computer screen the aspect resulting from the terminator T. The experiment is repeated, this time keeping constant the values of the parameters A n , k and R which characterise the aspherical plano-concave lens  108 , and varying the values of the parameters A n  as well as of k′ and R′ which characterise the shutter  94 , and the aspect resulting from the terminator T is observed on the computer screen by means of the “Photorealistic Rendering” function of the LighTools software. This function allows to view the entire device formed by the aspherical plano-concave lens, the hyperbolic shutter and the substrate as if this device was photographed at the desired angles and distances. Thanks to the “Photorealistic Rendering” function, it is thus possible to verify that the desired optical effect is suitable. Thus, one proceeds step by step until obtaining a profile of the terminator T that is considered faithful to its real aspect and which is satisfactory. Of course, this is a purely subjective criterion which is left to the discretion of each individual. 
     It will be noted that for the dimensional features of the aspherical plano-concave lens  108  and of the shutter  94  mentioned above, the most satisfactory results as regards the visual aspect of the terminator T were obtained for the values k=−1 and R=20840 mm and A 4 =3.769·10 −3 , A 6 =2.9534·10 −5  and A 8 =−1.407·10 −7  as regards the aspherical plano-concave lens  108 , and for the values k′=−4.922 and R′=2.556 mm and A 4 =1.654·10 −5 , A 6 =−1.511·10 −6  and A 8 =4.686·10 −8  as regards the shutter  94 . It will be observed that as regards the value of the conic constant k, the value retained for the aspherical plano-concave lens  108  is equal to −1, which corresponds to a parabolic profile. As for the value of the conic constant k′ which characterises the profile of the shutter  98 , it is less than −1, which corresponds to a hyperbolic profile. 
     Thus, the point O which is located at the top of the arch formed by the aspherical concave surface  112  is located at a distance P equal to 0.78 mm relative to the base of the cylindrical glass block. Consequently, it is deduced that at this point O, the thickness of the aspherical plano-concave lens  108  is 0.22 mm. This is the minimum thickness of the aspherical plano-concave lens  108 . 
     It goes without saying that the present invention is not limited to the embodiment which has just been described and that various simple modifications and variants can be considered by the person skilled in the art without departing from the scope of the invention as defined by the appended claims. It should be noted in particular that, in the case where the shutter is bright, it can be covered with a layer of phosphorescent material such as that marketed under the registered trademark Super-LumiNova®. It should also be noted that in order to avoid light reflection phenomena, the surface of the shutter can advantageously have roughness. Always with the same concern to limit light reflections as much as possible, the plano-concave lens can be the subject of an anti-reflective treatment and its edges can be metallised. According to a particular embodiment of the invention not shown in the drawing, provision may be made to provide the cam  52  with two steps  60 . Given that the star wheel  24  makes a complete revolution in two lunar cycles, it is then possible to directly engage the star wheel  24  with the cam  52 , and thus to save the pinions  50  and  54  and the setting-wheel  56 . 
     NOMENCLATURE 
     
         
           1 . Moon phase display mechanism 
           2 . Twenty-four-hour wheel 
           4 . Finger 
           6 . Axis 
           8 . Ring 
           10 . Pin 
           12 . Oblong hole 
           14 . Inner wall 
           16 . First lever 
           18 . Pivot axis 
           20 . Profile 
           22 . Upper spring 
           24 . Starwheel 
           26 . Jumper 
           28 . Toothing 
           30 . Lower spring 
           32 . Foot 
           34 . Top 
           36 . Beak 
           38 . Manual correction device 
           40 . Second lever 
           42 . Pivot axis 
           44 . Actuating means 
           46 . Folded area 
           50 . First pinion 
           52 . Cam 
           54 . Second pinion 
           56 . Setting-wheel 
           58 . Profile 
           60 . Step 
           62 . First rack 
           64 . Toothed sector 
           66 . Feeler beak 
           68 . Third pinion 
           69 . Mobile 
           70 . Wheel 
           72 . Drive means 
           74 . Lower wheel 
           76 . Axis of rotation 
           78 . Upper wheel 
           80 . Rectilinear rack 
           82 . Intermediate wheel 
           84 . Intermediate mobile 
           86 . Intermediate pinion 
           88 . Toothed sector 
           90 . Second rack 
           92 . Fourth spring 
           94 . Shutter 
           96 . Observer 
           98 . Transparent support 
           100 . Upper face 
           102 . Lower face 
           104 . Representation of the Moon 
           106 . Substrate 
           108 . Aspherical plano-concave lens 
           110 . Planar surface 
           112 . Aspherical concave surface 
           114 . Fold 
           116 . Flat edges 
           118 . Representation of the Moon