Patent Publication Number: US-6665128-B1

Title: Corrective lens for copying books

Description:
FIELD OF THE INVENTION 
     The present invention relates to copying methods and apparatus and, more particularly, to a method and apparatus for correcting for the distortion that results when copying pages of a book pressed against the document support glass of a copying machine or a document scanner. 
     BACKGROUND OF THE INVENTION 
     Ever since the introduction of xerographic photocopiers many users have experienced difficulty in producing clear, non-distorted copies from opened books or volumes that have been pressed upon the flat document glass. This difficulty is mainly due to the fact that conventional photographic copiers are designed to produce copies from documents (generally, single sheets) that lay flat upon the document glass. Textbooks with hard bindings cannot be so oriented. 
     The crease of a typical textbook generally pyramids upwardly away from the glass, thus causing an upside-down “V-shaped” surface. This is especially so for thick books with hard covers. The crease falls outside of the focal length range for which the copier lens can properly project the image of the book onto the image receiving medium or pre-charged paper. In other words, the crease cannot be focused properly to provide an undistorted image. As a result, the copied crease is often blurred, distorted, and darkened. 
     In order to make the copies more legible, the operator of the copier is required to forcibly press on the backing of the book as it rests upon the glass. Sometimes this will work. However, many times the binding is too stiff and the crease cannot be projected into the fold. pressing with excessive force is also risky, as it may cause damage to the binding. In some cases, an immoderate amount of force may even break the glass. 
     DISCUSSION OF THE RELATED ART 
     In the last twenty-five years numerous apparatus have been invented or proposed to solve this problem. Most of them either do not work, or are impractical. A simple one-piece optical corrective or compensation lens was invented to cure these problems and was described in U.S. Pat. No. 6,313,954, granted to the present inventor on Nov. 6, 2001 for CORRECTIVE LENS FOR COPYING BOOKS. 
     The optical corrective lens of the &#39;954 patent is basically composed of three parts: a thick curvilinear optical lens in the center, and two very thin extension leaves or arms spread outward from each side of the center lens. The only optically functional part of the lens is the center portion. The main purpose of the extension leaves is to eliminate the glitches in the copies caused by light deflections along the edges of the center lens. The surface contours of the center lens are basically a composition of sections of circular surfaces or circular-like elliptical and parabolic surfaces. 
     Although the results were not perfect, the lens worked for making duplications from book pages using a commercial xerographic copier. However, it was later revealed that the lens was difficult to manufacture using any of the existing conventional and economical plastic fabrication processes like injection molding, blowing, or continuous extruding. This is due to the extremely thin cross-section of the large extension leaves, which had to be seamlessly fabricated as an integral part of a relatively thick center lens. To improve the lens&#39;s manufacturability, the thickness of the extension leaves must be substantially increased. However, the increase of the extension leaves will also increase the profile of the center lens; otherwise, the radius of each constituent section of the top surface of the center lens must be reduced accordingly. Either way, the optical performance of the lens will be substantially worsened. 
     Several such lenses were built to fine tune this design for thicker (0.050 inch) extension leaves. The quality of a typical xerographic copy of the pages of a thick reference book produced using a typical xerographic copier with any of these trial lenses is obviously much poorer than the similar copy demonstrated in Tai&#39;s patent application. There are two darkened strips proximate the centerline. Characters close to these darkened strips are over-magnified and darker than the rest of the characters in the copy. Characters in center areas a little bit farther away from the centerline are still somewhat insufficiently decompressed or under-magnified. Copies with such poor quality are unsightly but may still be considered acceptable for reading purposes. However, its application to text scanning and digitizing is questionable due to more stringent decompression requirements. 
     After a laborious attempt to improve the performance of the lens, it was clear that the types of surfaces proposed and suggested to construct the lens&#39;s most crucial bottom surface are inadequate. Thus, it was found that to make the basic design of this lens work properly, adequate curvature must be found for the constituent sections of the most crucial bottom surface. Without scientific theory or principle available to guide the work, some innovation or breakthrough was needed. 
     In U.S. Pat. No. 3,609,030, issued to Meyer L. Sugarman et al. on Sep. 28, 1971 for ELECTROSTATIC BOOK COPIER, a tabletop electrostatic book copier is illustrated, having an exposure station on its top. The book to be copied is placed on top of a vertical exposure station and the image of the book is projected directly to the pre-charged paper through a mirror and an optical lens. The image of the book surface is projected using an ordinary optical lens and a mirror found in conventional copiers. The image is projected directly onto a charged paper instead of onto an image-receiving medium. There is no teaching or suggestion of using a distorted lens to refocus the page of the document. 
     In U.S. Pat. No. 4,585,334, issued to Brian R. Malyon on Apr. 29, 1986 for DOCUMENT COPIERS, a document copier is shown that incorporates a scanner with a forty-five degree slanted scanning window glass mounted at the end of a rectangular housing. The ninety-degree corner, or the so-called wedge, between the window glass and one of the sidewalls points downwardly. The book to be copied is placed on a rack below the scanner window glass. The rack is constructed from two flat frames connected together at a ninety-degree angle, with its opening facing upward. The book to be copied is faced upwardly and rests on the rack with its inside pages opened at a ninety-degree angle. During the copying process, the rack first moves up, bringing the half opened book toward the scanner. It then stops at a predetermined position and keeps a surface of the book page away from the scanner window to prevent damaging the book. In this position, the opposite inside page of the book is under the scanner housing, facing the sidewall. It may or may not touch the slanted sidewall. The scanner then scans the surface of the inside page of the book under its window. 
     In U.S. Pat. No. 4,763,173, issued to Michael E. Harrigan et al. on Aug. 9, 1988 for IMAGING SYSTEM FOR COMPENSATING FOR IMAGE DISTORTION DUE TO WRINKLED OR CURLED DOCUMENTS, a document imaging system is shown including a positive, aspheric, Fresnel lens. The purpose of the system is to compensate for small, smooth wrinkles and curls of the original document to be copied. 
     In U.S. Pat. No. 5,071,252, issued to William Howseman, Jr. on Dec. 10, 1991 and Japanese patent No. JP 360,186,360A, issued to Kenich Watabiki, it is suggested that book size bundles of numerous short optical fibers shaped in a nearly flat triangular prism be placed between an opened book and the copier&#39;s document glass to optically bring the images of the book pages to the copier&#39;s document supporting glass. Neither of these inventors addressed the solutions of other related problems such as the illumination of the book page&#39;s surfaces, the solution to the decompression of the text recorded on the book pages, etc. The principles and mechanisms they employed are very different from that used in the invention proposed in this application. 
     In Japanese patent Number JP 63,254,437A issued to Tatsuya Shimoda on Oct. 21, 1988, the use of an optical apparatus composed of two slightly bent convex lenses connected together by a flexible hinge is suggested for compensating the pyramiding of the book pages adjacent to the book&#39;s binding. In the application, the flat ridge formed by the thin edges and the flexible hinge is proposed to be placed directly under the crease  7  formed by the facing inside book pages and the flat portions of the book pages  4  residing on the thicker center portions of the constituent lenses. Since convex lenses do not project the images of objects to a viewer closer than the lenses&#39; viewer side surfaces, the optical compensating effect is very limited for portions of book surfaces deep in the crease where the most compensation is needed. In addition, the thick center portions of the lenses under the flat portions of the book pages will adversely raise the entire profile of the book page&#39;s surfaces higher and, thus, the expected optical compensation effect is significantly further reduced. Although this invention also attempts to use convex lenses to compensate the added height of the portions of book pages adjacent the binding, it uses a very different mechanism and the spirit of his design is upon the flexible hinge that connects the edges of two convex lenses. That is very different from the single rigid lens design proposed in the invention described in this application. 
     In Japanese patent Number JP 55,052,073A issued to Hirosuke Okura on Apr. 16, 1980, a flat prism made of a transparent material is proposed to be used for compensating the added book page surface height in the crease formed adjacent to the binding of a book. prism is well known for its ability to deflect the rays of a light beam when the light beams approach and leave the prism with an angle. It neither has the capability to reposition the image of an object nor of deflecting a light beam when it strikes or leaves the prism perpendicular to any of its flat surfaces. Therefore, it is obvious that this device doesn&#39;t have the capability of repositioning the image of the book pages&#39; surfaces to help solve the blurring or image compression problem that the invention in this application attempts to solve. 
     In the aforementioned U.S. Pat. No. 6,313,954, issued to Chgarles C. Tai on Nov. 6, 2001 for CORRECTIVE LENS FOR COPYING BOOKS, a corrective lens for copying books is described that is capable of compensating the pyramiding of the book pages near the book&#39;s binding, although with some imperfections. These imperfections may be tolerable for making readable xerographic copies from books, but they cause severe problems in scanning the contents of books for digitizing purposes using less expensive desktop scanners. Furthermore, the imperfections caused by the use of the suggested cylindrical or cylindrical-like surfaces to construct the lens&#39;s bottom surface are amplified with the thickening of the lens&#39;s extension arms. The requirement of very thin extension leaves makes it difficult to manufacture using existing economical plastic fabricating processes without sacrificing its performance. Thus, the fundamental surface curvature must be redefined to substantially lower the manufacturing cost and benefit users by providing affordable book copying corrective lenses. 
     SUMMARY OF THE INVENTION 
     In accordance with the present invention, there is provided a substantially flexible, freely movable, and easily manufacturable corrective lens that yields undistorted copies of adjacent book pages presented for xerographic copying or scanning. The corrective lens is substantially large and rectangular in shape to cover various sizes of books. The lens is placed between the spread pages of the book to be copied and the document glass of the copier. The corrective lens has a mid-portion or center section extending into the crease of a typical opened book where it changes the focal point to accommodate the variable height of the text disposed at the inner edge of the pages. The lens extends outwardly from the center section to form substantially flat side portions, which structurally keep the pages of the open book in place during the copying procedure. 
     In order to assist the users of this lens in producing copies from books, the lens must be able to help them overcome the following four commonly faced problems: the compression of the image of the text, the blurring of the image of the text, the widened and darkened centerline, and the missing information recorded in the crease between facing book pages. To date, there is neither any scientific theory or principles available to do this nor any existing guidelines to point in the direction of the development of such a surface contour. Therefore, an innovative idea or breakthrough is needed. 
     A particular surface contour governed by a higher-order skewed hyperbolic-like equation was tried with astonishing results. The prototype lenses were built using surfaces with contours based on this equation and with substantially thicker extension leaves. They were tested and proven capable of simultaneously solving all of the aforementioned problems. Hereinafter, in order to simplify the name of this particularly shaped surface, the higher-order skewed hyperbolic-like curvilinear surface is called the curvilinear skewed hyperbolic surface or, simply, the skewed hyperbolic surface. Mathematically, the x- and y-coordinate relationship of a point on the contour of this particular type of curvilinear surface is governed, or can be defined, by the following equation: 
     
       
         ( X+Xo )×( Y+Yo ) n   =A+f ( X,Y )  (1) 
       
     
     where: 
     X is the x-directional Cartesian coordinate position of a point on the curve; 
     Y is the y-directional Cartesian coordinate position of that point on the curve; 
     Xo is an empirically determined X-positional constant for the curve; 
     Yo is an empirically determined y-positional constant for the curve; 
     n is an empirically determined skew power constant having a value between 0.9 and 2.5 for this particular application; 
     A is an empirically determined x-directional shape factor of the curve; and 
     f(X,Y) is an empirically determined small second-order correction factor that corrects the curvatures of the surfaces near the tip of the bottom surface as well as the connections between the curved center lens portion and the flat extension arms. 
     In the actual design, sections of these particularly shaped surfaces are used to construct the most optically crucial bottom surface of the lens. 
     From an image formation viewpoint, it is preferable to have the lens&#39;s top and bottom surfaces constructed using sections of differently curved skewed-hyperbolic surfaces. However, the lens&#39;s top surface also serves as a mechanism to stop and keep the surfaces of inside pages of the book to be copied at a predetermined position for better image projection. In addition, it is not as optically crucial as the bottom surface for obtaining a sharp and properly decompressed image of the book pages residing on the lens&#39;s top surface. Therefore, similar to the inventor&#39;s previous lens design, the lens&#39;s top surface is composed of surfaces with contours closely resembling the surface contour of the surfaces of inside pages of a typical thick textbook when it is opened and pressed upon a flat surface. The resemblance between the book&#39;s surface and the lens&#39;s top surface assures intimate contact when the book is opened and pressed onto the lens&#39;s top surface. 
     In order to further improve the performance of this newly developed corrective lens, the top surfaces of this lens are slightly modified. The two halves of the lens&#39;s top surface are slightly pushed further apart outwardly from the center to assure the intimate contact between the lens&#39;s top surface and the surfaces of the book&#39;s inside pages. In addition, the ridge of the lens&#39;s top surface is made slightly higher and sharper to make it capable of opening the crease further for better scanning of the text inside the crease. Furthermore, the portions of top surface near the bottom edges of the lens are slightly modified to smoothen the transitions of the lens&#39;s extension leaves from curved surfaces to flat top surfaces. 
     The extension arms or leaves of the lens are large, relatively thin, and rectangular in shape with almost constant cross-sections through their full length. They are thin but not substantially thin so as to prevent the lens from being injection molded. 
     The innovation of using higher-order curvilinear hyper-elliptical surfaces in this invention effectively improves the performance of the simple corrective lens, allowing the lens to help scan and digitize recorded text on inside pages of books. This also allows manufacturability using existing low-cost plastic fabricating processes without losing performance. Thus, the lens can be mass-produced at a substantially low manufacturing cost, benefiting anyone wishing to conduct book copying or scanning. The alternative design proposed in this invention extends its application to books two inches thick or more without substantially sacrificing the quality of the copies. 
     A lens with symmetrical left-hand and right-hand halves is most suitable for helping make clear copies of pages for most books. However, for making copies of the first and last few pages of a book, a non-symmetrical lens fits better to the non-symmetrical contour of the inside pages of a thick book. Therefore, the lens can be varied to accommodate different top surface contours. 
     Furthermore, the curvatures of the sloped portions of the inside pages of an opened, thick book are quite different and less curved than those of a thinner book. Therefore, as shown in FIG. 11, the profiles of the center lens portions of the top surfaces must be raised higher to accommodate the less curved surfaces; in other words, deepen the crease of the opened, thick book. As a result, a portion of these surfaces adjacent to the ridge would be raised higher than the maximum height that the lens can compensate. Although the portions of book pages residing on these portions of the lens may be within the binding edge margin and contain no text, the images would still appear blackened due to severe blurring. 
     In order to erase or white-out these unsightly darkened marks from the copies, one may coat these small portions of the bottom surfaces with opaque paint or cover them with films or thin covers. 
     In most cases, very thick books are also large in size and cannot fit two full pages within the boundary of the document glass of an ordinary copier. Therefore, a user of this corrective lens can only copy one page at a time. The partial image of the other page is unnecessary and, sometimes even undesirable. Thus, it is suggested in this invention to offset the top surfaces and the bottom surfaces. With one more degree of freedom allowed by this offsetting, the right-hand side of the lens in FIG. 11 can be designed to clearly capture a larger area of the book page or deepen the crease formed by the opposite book pages. On the other hand, this offset will double the width of the blackened area in the image of the left-hand page, normally unnecessary anyway. Similarly, the entire unnecessary side of a partial page image can be optically erased or whited out with paint, film, or a thin cover. 
     The lens improves the quality of the copies produced from the inside pages of a book simultaneously in five ways: 
     1) It works as an optical lens to project an image of the book surface closer to the copier&#39;s document support glass than the actual surfaces of the book pages, thus, helping the copier&#39;s lens to better focus and produce a crisp image of the book surface near its binding. 
     2) It optically stretches or decompresses the book image near its binding. As a result, the images of characters or pictures in this area are stretched back to their original shapes printed on the inside pages of the book. 
     3) The ridge helps to open up the book near the binding, enabling the images of the characters and pictures in this area to be more clearly scanned by the copier. 
     4) The top surfaces of the lens work as a guide to help the inside pages easily and smoothly slide outward without being squeezed toward each other. 
     5) It enables the use of thicker extension leaves to ease the difficulty in manufacturing without scarifying the lens&#39;s performance. 
     It is an object of this invention to provide improved means of perfecting the performance of the corrective lens of U.S. Pat. No. 6,313,954 and make it capable of helping users conduct text scanning and digitizing from inside pages of thick books or bound volumes. 
     It is another object of this invention to provide an improved means of making this lens capable of being inexpensively mass-produced using existing conventional plastic fabricating processes without losing valuable performance. 
     A further object of this invention is to provide improved means for compensating for the distance between the crease of a book and regions proximate thereto, during a photocopying process that does not require complex structural modifications to conventional copying machines. 
     It is a further object of this invention to provide improved means for the lens to act as a wedge to spread the pages of a book closest to the binding thereof. 
     It is a further object of this invention to provide an improved freely movable lens for placement between an open book to be copied and the document glass of a copier, wherein the lens changes its local focal point in accordance with the distance between the book page&#39;s surface and the copier&#39;s document glass across the crease. 
     A further object of this invention is to provide means to alter the lens and make it capable of producing clear, crisp, and distortion-free copies or scanned images from extra thick books. 
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS 
     A complete understanding of the present invention may be obtained by reference to the accompanying drawings, when considered in conjunction with the subsequent detailed description, in which: 
     FIG. 1 illustrates a perspective view of a book that is disposed upon the document glass of a typical xerographic copying machine; 
     FIG. 2 shows a plan view of typically distorted adjacent pages of the book copied by the copying machine shown in FIG. 1; 
     FIG. 3 depicts another plan view of typically distorted adjacent pages of the book copied by pressing down the book harder onto the copying machine shown in FIG. 1; 
     FIG. 4 illustrates still another plan view of typically distorted adjacent pages of the book copied by pressing down the book very hard onto the copying machine shown in FIG. 1; 
     FIG. 5 depicts a perspective view of an unimproved corrective lens having two very thin extended flat leaves and a thicker center portion with cylindrical bottom surfaces as described in U.S. Pat. No. 6,313,954; 
     FIG. 6 shows a sectional side view of the corrective lens depicted in FIG. 5; 
     FIG. 7 shows a plan view of a xerographic copy of adjacent pages of a book, using the unimproved corrective lens as shown in FIGS. 5 and 6 but with a pair of slightly thicker extension leaves; 
     FIG. 8 depicts a perspective view of the corrective lens in accordance with this invention; 
     FIG. 9 shows a sectional side view of the improved corrective lens depicted in FIG. 8; 
     FIG. 10 illustrates a plan view of the adjacent pages of a typical thick reference book copied utilizing the lens suggested in FIGS. 8 and 9; and 
     FIG. 11 depicts a perspective view of an alternate embodiment of this invention. 
    
    
     DESCRIPTION OF THE PREFERRED EMBODIMENT 
     Generally speaking, the invention features a-corrective lens for copying pages of a book pressed flat on the document glass of a xerographic copy machine or an electronic document scanner. It is made of a rigid or semi-rigid optically transparent material like glass, acrylic, polycarbonate, nylon, or other plastic material. The surfaces of the lens may be covered with a protective coating to increase their resistance to scratching and scoring. 
     The corrective lens is specially shaped to enable a snug fit in the space formed between the inside pages of the book to be copied and the document supporting glass of the copier without substantially lifting the book. The corrective lens has a substantially pyramidal center extending into the crease of a typical book. The lens fans out from the pyramidal center to extend to substantially flat side portions. 
     Now referring to FIG. 1, a perspective view is illustrated of a book  1  that is pressed against the document supporting glass  2  of a typical xerographic copying machine  3 . It will be observed that the binding  6  of the book  1  is relatively stiff, and the crease  5  between book pages or leaves  4  forms a pyramidal void  7  between the book  1  and the glass  2 . 
     The crease  5  of typical textbooks generally pyramids upwardly away from the glass  2 , thus causing an upside-down, V-shaped void  7 , as shown. The lens of a typical copying machine  3  cannot capture the text or information disposed in the crease  5 , resulting in distorted copies, as shown in FIGS. 2 through 4. 
     FIG. 2 illustrates a photocopied sheet  9  produced by a conventional copier such as shown in FIG. 1, illustrating that the image of the crease  10  can be distorted with text  12  that runs into a wide black centerline  11 . 
     FIG. 3 depicts a typical copy of the book pages where the binding is pressed toward the glass, as shown in FIG.  1 . It will be observed that the text  16  disappears into a black centerline  15 , appearing much narrower than that of the centerline shown in FIG.  2 . 
     FIG. 4 depicts a copy of the book after a substantially greater force is applied to the binding than was applied to either of the copies depicted in FIGS. 2 and 3. It will be observed that although the centerline  17  is substantially diminished in width and blackness, text  18  still runs into the centerline  17  and is illegible. 
     FIG. 5 shows the optical copying corrective or compensation lens in U.S. Pat. No. 6,313,954 that is closely similar to the present invention. It is basically composed of three parts: a thick curvilinear optical lens  27  in the center, and two extension leaves or arms  22  and  23  spread outward from each side of the center lens  27 . The only optically functional part of the lens  27  is the center portion. The main purpose of the extension leaves  22  and  23  is to eliminate the glitches in the copies caused by light deflections along the edges of the center lens  27 . The surface contours of the center lens  27  are basically a composition of sections of circular surfaces or circular-like elliptical surfaces. 
     FIG. 6 is a more detailed sectional side view of the corrective lens shown in FIG.  5 . The extension leaves  32  and  33  are substantially thin. The contours of the left- and right-hand halves of the upper surfaces  34  and  35  as well as the lower surfaces  36  and  37  of the center lens portion  30  of the apparatus are basically circularly shaped. 
     FIG. 7 illustrates a photocopy of the inside page of a thick book produced by a conventional xerographic copier using the corrective lens shown in FIG. 5 but with thicker (0.050 inch) extension leaves. It is obvious that the text of the entire page is quite legible; however, the image of the crease adjacent the binding of the opened book is obviously unsightly and slightly distorted. The centerline  40  is widened. There are also two strips of darkened marks  41  and  42  next to and along the centerline  40 . The last few columns of characters  43  and  44  next to the binder margins are over-magnified and appear larger and darker than normal. Characters  45  and  46  a little bit further away from the centerline  40  are slightly compressed or under-magnified. Although these characters  43  to  46  are legible, they may not be electronically recognizable by the software of less expensive electronic scanners. 
     Referring to FIGS. 8 and 9, an improved optical compensation lens  51  of this invention is shown. The lens  51  is designed to fit inside the tight space between the book and the document supporting glass of the copier shown in FIG.  1 . The lens  51  comprises an optically transparent material like glass, Lucite®, Plexiglas®, polycarbonate, nylon, acrylic, or other types of transparent materials. 
     In FIG. 8, the upper surface  55   a  and  56   a  of lens  51 , as well as the lower surface  55   b  and  56   b  may be coated with a thin, protective, non-abrasive material like Teflon or silicon-based coating to enhance its resistance to scratching and scoring. The leaves  52  and  53  are thin but not substantially thin enough to prevent them from being injection molded. They have a nearly constant cross-sectional shape along their full length until the center portion  57  is reached. The center portion  57  is an optical lens severely distorted to a nearly triangular shape to enable it to fit snugly inside the crease of the open book as the book lies flat and is pressed on the document supporting glass, as illustrated in FIG.  1 . 
     The lens  51  has a variable focal length to compensate the variation of distance between the flat document supporting glass and the surfaces of the book pages residing on the lens&#39;s top surfaces  55   a  and  56   a . As the lens approaches the crease  58  from its leaves  52  and  53 , the lens portion  57  thickens towards the center  54 , which peaks in a pyramidal fashion. 
     The contour of respective top surfaces  55   a  and  56   a  resembles the surface contour of the inside pages  4  in FIG.  1 . Therefore, when an opened book is placed on top of and is pressed on lens  51 , the inside page surface  4  to be copied can rest upon, and make intimate contact with, the top surfaces  55   a  and  56   a  of the lens  51 . 
     In operation, the wedge shaped ridge  54  is pressed into the pyramidal void between the adjacent pages, the force of such procedure allowing the wedge shaped ridge  54  to spread the pages of the book so that the contour of the book&#39;s inside pages  4  thereof matches the curvature of the ridge  54 . Thus, this area  7  in FIG. 1 is opened for the copier to properly scan the text of both pages  4 , without obstruction. In order to enhance the performance, the ridge  54  is made sharper and taller. In addition, the two halves  55   a  and  56   a  of the top surface are made farther apart from the center of the lens portion  57  to assure intimate contact between the book pages and the lens&#39;s top surface. 
     The slope of the bottom surface  55   b  and  56   b  of lens  51  may be (but is not necessarily) similar to that of the corresponding top surface  55   a  and  56   a . Although the shapes of the top  55   a  and  56   a  and bottom surfaces  55   b  and  56   b  are approximately similar, their detailed curvatures are different so that the lens  51  can properly compensate the image of the book surface. 
     The bottom surface  55   b  and  56   b  of this lens  51  is composed of sections of surfaces with curvatures defined by the curvilinear skewed-hyperbolic equation given previously. 
     The sections of surfaces near the connections between the center lens  57  and extending leaves  52  and  53  are slightly modified to smoothen the abrupt curvature change from curved surfaces to flat surfaces of the extension leaves  52  and  53 . The extension leaves  52  and  53  are substantially thin but still thick enough for letting this lens  51  be manufactured using existing less expensive plastic fabrication processes. They are also large and rectangular shaped, with a nearly constant cross-section along their span. The extension arm  52  on the left-hand side of the lens  51  is longer than the length of the right-hand extension arm  53 . The purpose of the differential length of the extension arms  52  and  53  is to provide the user of this lens  51  the freedom to duplicate the text in either a single page of a large book or, more economically, duplicate the text in both adjacent pages of a smaller book onto a single piece of copying paper. 
     The sectional view with an enlarged center lens portion shown in FIG. 9 illustrates the lens&#39;s  51  improvements. The extension leaves  62  and  63  are thin but obviously thicker than the extension leaves  32  and  33  shown in FIG.  6 . The ridge  61  of the center lens portion is higher and sharper for making it easier to insert into the crease of the book to open up and spread the book pages, as shown in FIG.  1 . The main bodies of the lens portion of the top surfaces  64  and  65  are circular with corrections near the ridge  61  and connections between the curved portions and flat portions of these surfaces. The curved lens portion of the bottom surfaces  66  and  67  are portions of higher-power hyperbolic curves with a slight correction near its apex  68  and connection between these curved portions and flat portions of the bottom surfaces  66  and  67 . 
     FIG. 10 shows the text in the two opposite book pages  71  clearly and crisply duplicated without visible distortion or missing information recorded on these pages. The separation line  70  between the images of these adjacent book pages is thin, clear, and crisp. 
     Referring to FIG. 11, an alternate embodiment  51  of the lens  80  is shown. The lens  80  is made with less curved top surfaces  85   a  and  86   a  to accommodate the less curved surface contour of a book two or more inches thick. Thus, the book pages residing on the curved portions of the lens  80  will be pushed higher than that of a book residing on the top surface of the lens  51 . As a result, the part of the book pages on the lens&#39;s surface near the ridge  84  will be blurred and leave an unsightly darkened mark in the copy. Therefore, the bottom surfaces  85   b  and  86   b  are shifted toward one side of the lens  80 ; in this case, it is the right-hand side. This offset arrangement of the top  85   a  and  86   a  and bottom  85   b  and  86   b  surfaces improves the appearance of the copies by not capturing the blurred image of a binding area in which there is no text in most cases. 
     A thin cover or thin opaque film  89  is attached on the bottom side  85  of the left side of the lens  80 . This cover or film  89  is used to block the light passing through the portion of the lens  80  on which it is attached so that the copier (not shown) will not capture the image of the portion of the book page residing on the left-side top surface  85   a  of the portion of the lens  80  that is covered by the cover or film  89 . Thus, it optically removes or eliminates the unsightly blackened marks and the undesirable partial page image from the copies produced using lens  80 . Since the left half of the lens  80  is nonfunctional, its extension leave  82  is designed to be substantially shorter for easy handling and storage. 
     This cover or film  89  can be made either as an integral part of the lens  80  or a separate part, easily attachable to and removable from the bottom surface of the lens  80 . Although this cover  89  doesn&#39;t enhance the functions of the lens  80 , it adds some desirable features as well as value to the lens  80 , benefiting users at a minimal cost. Of course, the application of this cover or film  89  is not limited to this particular lens  80 . It can also be made for and applied to the general purpose lens shown as  51  in FIG. 8 as an inexpensive add-on feature for the convenience of users. 
     Since other modifications and changes varied to fit particular operating requirements and environments will be apparent to those skilled in the art, the invention is not considered limited to the example chosen for purposes of disclosure, and covers all changes and modifications which do not constitute departures from the true spirit and scope of this invention. 
     Having thus described the invention, what is desired to be protected by Letters patent is presented in the subsequently appended claims.