Abstract:
A reflection device including a sheet made of a flexible material with at least one surface having reflective properties including a first edge and a second edge on opposite sides of a center line of the sheet where the first edge and the second edge of the sheet are shaped such that the flexible material is formed into a mostly parabolic shape in cross section when bent about a line perpendicular to the center line and the sheet is secured at two points along the center line.

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
     This application claims the benefit of the filing date of U.S. Provisional Application No. 61/299,476, entitled “A Parabolic Reflector,” filed on Jan. 29, 2010, which is incorporated herein by reference. 
    
    
     FIELD OF THE INVENTION 
     This invention deals with parabolic solar reflectors used to focus waves on a single focal point. 
     BACKGROUND OF THE INVENTION 
     A parabolic reflector is a device used to collect waves of energy including light waves, radio waves or sound waves. Waves of energy striking the surface of the parabolic reflector are reflected off of the surface of the reflector and focused on a focal point of the collector. The parabolic reflector can be used to concentrate waves to a single focal point which can be transmitted as a single beam transmitted parallel to the axis of the parabola. Parabolic collectors are currently used in solar collectors, radio wave collectors, lighting devices, radio telescopes and other applications where the collection of waves is of importance. 
     Currently, parabolic reflectors require a rigid support framework to hold the reflector in the shape of a parabola. Because of this rigid framework, parabolic reflectors are not very mobile and require extensive manpower and materials to install. Further, once installed, typical parabolic collectors are difficult to move. In addition, the support rigid framework increases the cost of using a typical parabolic collector due to the additional cost of material and manpower to assemble the reflector. 
     It would be beneficial to have an inexpensive mobile parabolic collector which will allow for a simplified assembly and better mobility. 
     SUMMARY OF THE INVENTION 
     Systems, methods, features, and advantages of the present invention will be or will become apparent to one with skill in the art upon examination of the following figures and detailed description. It is intended that all such additional systems, methods, features, and advantages be included within this description, be within the scope of the invention, and be protected by the accompanying claims. 
     One embodiment consistent with the present invention includes a reflection device comprising a sheet made of a flexible material with at least one surface having reflective properties including a first edge and a second edge on opposite sides of a center line of the sheet, where the first edge and the second edge of the sheet are shaped such that the flexible material is formed into a mostly parabolic shape in cross section when bent about a line perpendicular to the center line and the sheet is secured at two points along the center line. 
     In another embodiment consistent with the present invention, the first edge and second edge are offset from each other in a vertical direction such that the mostly parabolic shape is inclined by an angle θ. 
     In another embodiment consistent with the present invention, the length of the flexible material is defined by the equation 
     
       
         
           
             L 
             ⁢ 
             
               = 
               
                 X 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 1 
               
               
                 X 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 2 
               
             
             ⁢ 
             
                
               
                 
                   
                     
                       1 
                       4 
                     
                     ⁢ 
                     
                       
                         
                           4 
                           ⁢ 
                           
                             
                               ( 
                               
                                 
                                   X 
                                   ′ 
                                 
                                 A 
                               
                               ) 
                             
                             2 
                           
                         
                         + 
                         1 
                       
                     
                     * 
                     2 
                     ⁢ 
                     
                       ( 
                       
                         
                           X 
                           ′ 
                         
                         A 
                       
                       ) 
                     
                   
                   + 
                   ln 
                 
                 | 
                 
                   
                     
                       
                         4 
                         ⁢ 
                         
                           
                             ( 
                             
                               
                                 X 
                                 ′ 
                               
                               A 
                             
                             ) 
                           
                           2 
                         
                       
                       + 
                       1 
                     
                   
                   + 
                   
                     2 
                     ⁢ 
                     
                       ( 
                       
                         
                           X 
                           ′ 
                         
                         A 
                       
                       ) 
                     
                   
                 
               
                
             
           
         
       
     
     where A is a scaling factor defined by the equation 
     
       
         
           
             A 
             = 
             
               
                 1 
                 4 
               
               ⁢ 
               F 
             
           
         
       
     
     and F is the distance from the horizontal axis to the focal point of the parabola. 
     In another embodiment consistent with the present invention, the flexible material is made from a material that does not have memory capabilities, and 
     the shape of the first edge conforms to a line formed by a plurality of points defined by the equations 
     
       
         
           
             
               
                 
                   a 
                   = 
                   
                     
                       ∫ 
                       
                         X 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           1 
                           ′ 
                         
                       
                       
                         X 
                         ′ 
                       
                     
                     ⁢ 
                     
                       
                         
                           
                             1 
                             + 
                             
                               ( 
                               
                                 
                                   ⅆ 
                                   Y 
                                 
                                 
                                   ⅆ 
                                   X 
                                 
                               
                               ) 
                             
                           
                         
                         2 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                           
                         
                           = 
                           
                             
                                 
                               
                                 X 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   1 
                                   ′ 
                                 
                               
                               
                                 X 
                                 ′ 
                               
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               | 
                               
                                 
                                   1 
                                   4 
                                 
                                 ⁡ 
                                 
                                   [ 
                                   
                                     
                                       
                                         
                                           
                                             4 
                                             ⁢ 
                                             
                                               
                                                 ( 
                                                 
                                                   
                                                     X 
                                                     ′ 
                                                   
                                                   A 
                                                 
                                                 ) 
                                               
                                               2 
                                             
                                           
                                           + 
                                           1 
                                         
                                       
                                       * 
                                       2 
                                       ⁢ 
                                       
                                         
                                           X 
                                           ′ 
                                         
                                         A 
                                       
                                     
                                     + 
                                     
                                       ln 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       
                                          
                                         
                                           
                                             
                                               
                                                 4 
                                                 ⁢ 
                                                 
                                                   
                                                     ( 
                                                     
                                                       
                                                         X 
                                                         ′ 
                                                       
                                                       A 
                                                     
                                                     ) 
                                                   
                                                   2 
                                                 
                                               
                                               + 
                                               1 
                                             
                                           
                                           + 
                                           
                                             2 
                                             ⁢ 
                                             
                                               ( 
                                               
                                                 
                                                   X 
                                                   ′ 
                                                 
                                                 A 
                                               
                                               ) 
                                             
                                           
                                         
                                          
                                       
                                     
                                   
                                   ] 
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     equation 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     1 
                   
                   ) 
                 
               
             
             
               
                 
                   b 
                   = 
                   
                       
                   
                   ⁢ 
                   
                     
                       B 
                       * 
                       
                         
                           
                             
                               ⅆ 
                               2 
                             
                             ⁢ 
                             Y 
                           
                           
                             ⅆ 
                             
                               X 
                               2 
                             
                           
                         
                         
                           
                             
                               1 
                               + 
                               
                                 ( 
                                 
                                   
                                     ⅆ 
                                     Y 
                                   
                                   
                                     ⅆ 
                                     X 
                                   
                                 
                                 ) 
                               
                             
                           
                           2 
                         
                       
                     
                     = 
                     
                         
                       
                         B 
                         * 
                         
                           
                             
                               
                                 
                                   
                                     
                                       ( 
                                       
                                         
                                           cos 
                                           ⁢ 
                                           
                                               
                                           
                                           ⁢ 
                                           θ 
                                         
                                         + 
                                         
                                           2 
                                           ⁢ 
                                           
                                             ( 
                                             
                                               
                                                 X 
                                                 ′ 
                                               
                                               A 
                                             
                                             ) 
                                           
                                           ⁢ 
                                           sin 
                                           ⁢ 
                                           
                                               
                                           
                                           ⁢ 
                                           θ 
                                         
                                       
                                       ) 
                                     
                                     * 
                                     2 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     cos 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     θ 
                                   
                                   - 
                                 
                               
                             
                             
                               
                                 
                                   
                                     ( 
                                     
                                       
                                         
                                           - 
                                           sin 
                                         
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         θ 
                                       
                                       + 
                                       
                                         2 
                                         ⁢ 
                                         
                                           ( 
                                           
                                             
                                               X 
                                               ′ 
                                             
                                             A 
                                           
                                           ) 
                                         
                                         ⁢ 
                                         cos 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         θ 
                                       
                                     
                                     ) 
                                   
                                   * 
                                   2 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   sin 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   θ 
                                 
                               
                             
                           
                           
                             
                               
                                 ( 
                                 
                                   
                                     cos 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     θ 
                                   
                                   + 
                                   
                                     2 
                                     ⁢ 
                                     
                                       ( 
                                       
                                         
                                           X 
                                           ′ 
                                         
                                         A 
                                       
                                       ) 
                                     
                                     ⁢ 
                                     sin 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     θ 
                                   
                                 
                                 ) 
                               
                               3 
                             
                             * 
                             
                                 
                             
                             ⁢ 
                             
                               
                                 1 
                                 + 
                                 
                                   
                                     ( 
                                     
                                       
                                         
                                           
                                             - 
                                             sin 
                                           
                                           ⁢ 
                                           
                                               
                                           
                                           ⁢ 
                                           θ 
                                         
                                         + 
                                         
                                           2 
                                           ⁢ 
                                           
                                             ( 
                                             
                                               
                                                 X 
                                                 ′ 
                                               
                                               A 
                                             
                                             ) 
                                           
                                           ⁢ 
                                           cos 
                                           ⁢ 
                                           
                                               
                                           
                                           ⁢ 
                                           θ 
                                         
                                       
                                       
                                         
                                           cos 
                                           ⁢ 
                                           
                                               
                                           
                                           ⁢ 
                                           θ 
                                         
                                         + 
                                         
                                           2 
                                           ⁢ 
                                           
                                             ( 
                                             
                                               
                                                 X 
                                                 ′ 
                                               
                                               A 
                                             
                                             ) 
                                           
                                           ⁢ 
                                           sin 
                                           ⁢ 
                                           
                                               
                                           
                                           ⁢ 
                                           θ 
                                         
                                       
                                     
                                     ) 
                                   
                                   2 
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     equation 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     2 
                   
                   ) 
                 
               
             
           
         
       
     
     where a is a coordinate point in the vertical direction, b is a coordinate point in the horizontal direction, X1′ is a starting position in the horizontal position and X′ is a position along the horizontal direction, cutting the material along a line formed by the plurality of plotted points. 
     In another embodiment consistent with the present invention, at least one surface of the flexible material reflects light. 
     In another embodiment consistent with the present invention, at least one surface of the flexible material reflects energy waves. 
     In another embodiment consistent with the present invention, the reflection device includes a plurality of sheets which overlap one another to form a trough. 
     In another embodiment consistent with the present invention, the flexible material is made from a material that has memory capabilities, and 
     the shape of the first edge conforms to a line formed by a plurality of points defined by the equations 
     
       
         
           
             
               
                 
                   a 
                   = 
                   
                     
                       ∫ 
                       
                         X 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           1 
                           ′ 
                         
                       
                       
                         X 
                         ′ 
                       
                     
                     ⁢ 
                     
                       
                         
                           
                             1 
                             + 
                             
                               ( 
                               
                                 
                                   ⅆ 
                                   
                                     Y 
                                     ′ 
                                   
                                 
                                 
                                   ⅆ 
                                   
                                     X 
                                     ′ 
                                   
                                 
                               
                               ) 
                             
                           
                         
                         2 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                           
                         
                           = 
                           
                             
                                 
                               
                                 X 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   1 
                                   ′ 
                                 
                               
                               
                                 X 
                                 ′ 
                               
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               | 
                               
                                 
                                   1 
                                   4 
                                 
                                 ⁡ 
                                 
                                   [ 
                                   
                                     
                                       
                                         
                                           
                                             4 
                                             ⁢ 
                                             
                                               
                                                 ( 
                                                 
                                                   
                                                     X 
                                                     ′ 
                                                   
                                                   A 
                                                 
                                                 ) 
                                               
                                               2 
                                             
                                           
                                           + 
                                           1 
                                         
                                       
                                       * 
                                       2 
                                       ⁢ 
                                       
                                         ( 
                                         
                                           
                                             X 
                                             ′ 
                                           
                                           A 
                                         
                                         ) 
                                       
                                     
                                     + 
                                     
                                       ln 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       
                                          
                                         
                                           
                                             
                                               
                                                 4 
                                                 ⁢ 
                                                 
                                                   
                                                     ( 
                                                     
                                                       
                                                         X 
                                                         ′ 
                                                       
                                                       A 
                                                     
                                                     ) 
                                                   
                                                   2 
                                                 
                                               
                                               + 
                                               1 
                                             
                                           
                                           + 
                                           
                                             2 
                                             ⁢ 
                                             
                                               ( 
                                               
                                                 
                                                   X 
                                                   ′ 
                                                 
                                                 A 
                                               
                                               ) 
                                             
                                           
                                         
                                          
                                       
                                     
                                   
                                   ] 
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     equation 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     1 
                   
                   ) 
                 
               
             
             
               
                 
                   b 
                   = 
                   
                     B 
                     * 
                     
                       ( 
                       
                         
                           
                             ( 
                             
                               
                                 X 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   2 
                                   ′ 
                                 
                               
                               A 
                             
                             ) 
                           
                           * 
                           
                             ( 
                             
                               
                                 X 
                                 ′ 
                               
                               A 
                             
                             ) 
                           
                         
                         + 
                         
                           
                             ( 
                             
                               
                                 X 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   1 
                                   ′ 
                                 
                               
                               A 
                             
                             ) 
                           
                           * 
                           
                             ( 
                             
                               
                                 X 
                                 ′ 
                               
                               A 
                             
                             ) 
                           
                         
                         - 
                         
                           
                             ( 
                             
                               
                                 X 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   1 
                                   ′ 
                                 
                               
                               A 
                             
                             ) 
                           
                           * 
                           
                             ( 
                             
                               
                                 X 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   2 
                                   ′ 
                                 
                               
                               A 
                             
                             ) 
                           
                         
                         - 
                         
                           
                             ( 
                             
                               
                                 X 
                                 ′ 
                               
                               A 
                             
                             ) 
                           
                           2 
                         
                       
                       ) 
                     
                     ⁢ 
                     
                       
                         ( 
                         
                           1 
                           + 
                           
                             4 
                             ⁢ 
                             
                               
                                 ( 
                                 
                                   
                                     X 
                                     ′ 
                                   
                                   A 
                                 
                                 ) 
                               
                               2 
                             
                           
                         
                         ) 
                       
                       
                         3 
                         / 
                         2 
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     equation 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     2 
                   
                   ) 
                 
               
             
           
         
       
     
     where a is a coordinate point in the vertical direction, b is a coordinate point in the horizontal direction, X1′ is a starting coordinate point in the horizontal position, X2′ is the ending coordinate point in the horizontal direction, X′ is a position between X1′ and X2′ in the horizontal direction. 
     In another embodiment consistent with the present invention, the flexible material is made of, Plexiglas, Lexan, fiberglass or carbon fiber. 
     In another embodiment consistent with the present invention, a plurality of flexible sheets are overlapped to form a trough structure. 
     Another embodiment consistent with the present invention include a method of producing a reflection device including the steps of shaping a first edge and a second edge on opposite sides of a center line of a sheet made of flexible material such that the flexible material is formed into a mostly parabolic shape in cross section when bent about a line perpendicular to a center line of the sheet and the sheet is secured at two points along the center line, where the sheet has at least one surface having reflective properties. 
     In another embodiment consistent with the present invention, the first edge and second edge are offset from each other in a vertical direction such that the mostly parabolic shape is inclined by an angle θ. 
     the length of the flexible material is defined by the equation 
     
       
         
           
             L 
             ⁢ 
             
               = 
               
                 X 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 1 
               
               
                 X 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 2 
               
             
             ⁢ 
             
                
               
                 
                   
                     
                       1 
                       4 
                     
                     ⁢ 
                     
                       
                         
                           4 
                           ⁢ 
                           
                             
                               ( 
                               
                                 
                                   X 
                                   ′ 
                                 
                                 A 
                               
                               ) 
                             
                             2 
                           
                         
                         + 
                         1 
                       
                     
                     * 
                     2 
                     ⁢ 
                     
                       ( 
                       
                         
                           X 
                           ′ 
                         
                         A 
                       
                       ) 
                     
                   
                   + 
                   ln 
                 
                 | 
                 
                   
                     
                       
                         4 
                         ⁢ 
                         
                           
                             ( 
                             
                               
                                 X 
                                 ′ 
                               
                               A 
                             
                             ) 
                           
                           2 
                         
                       
                       + 
                       1 
                     
                   
                   + 
                   
                     2 
                     ⁢ 
                     
                       ( 
                       
                         
                           X 
                           ′ 
                         
                         A 
                       
                       ) 
                     
                   
                 
               
                
             
           
         
       
     
     where A is a scaling factor defined by the equation 
     
       
         
           
             A 
             = 
             
               
                 1 
                 4 
               
               ⁢ 
               F 
             
           
         
       
     
     where F is the distance from the horizontal axis to the focal point of the parabola. 
     In another embodiment consistent with the present invention, the flexible material is made from a material that does not have memory capabilities, and the shape of the first edge conforms to a line formed by a plurality of points defined by the equations 
     
       
         
           
             
               
                 
                   a 
                   = 
                   
                     
                       ∫ 
                       
                         X 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           1 
                           ′ 
                         
                       
                       
                         X 
                         ′ 
                       
                     
                     ⁢ 
                     
                       
                         
                           
                             1 
                             + 
                             
                               ( 
                               
                                 
                                   ⅆ 
                                   Y 
                                 
                                 
                                   ⅆ 
                                   X 
                                 
                               
                               ) 
                             
                           
                         
                         2 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                           
                         
                           = 
                           
                             
                                 
                               
                                 X 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   1 
                                   ′ 
                                 
                               
                               
                                 X 
                                 ′ 
                               
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               | 
                               
                                 
                                   1 
                                   4 
                                 
                                 ⁡ 
                                 
                                   [ 
                                   
                                     
                                       
                                         
                                           
                                             4 
                                             ⁢ 
                                             
                                               
                                                 ( 
                                                 
                                                   
                                                     X 
                                                     ′ 
                                                   
                                                   A 
                                                 
                                                 ) 
                                               
                                               2 
                                             
                                           
                                           + 
                                           1 
                                         
                                       
                                       * 
                                       2 
                                       ⁢ 
                                       
                                         
                                           X 
                                           ′ 
                                         
                                         A 
                                       
                                     
                                     + 
                                     
                                       ln 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       
                                          
                                         
                                           
                                             
                                               
                                                 4 
                                                 ⁢ 
                                                 
                                                   
                                                     ( 
                                                     
                                                       
                                                         X 
                                                         ′ 
                                                       
                                                       A 
                                                     
                                                     ) 
                                                   
                                                   2 
                                                 
                                               
                                               + 
                                               1 
                                             
                                           
                                           + 
                                           
                                             2 
                                             ⁢ 
                                             
                                               ( 
                                               
                                                 
                                                   X 
                                                   ′ 
                                                 
                                                 A 
                                               
                                               ) 
                                             
                                           
                                         
                                          
                                       
                                     
                                   
                                   ] 
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     equation 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     1 
                   
                   ) 
                 
               
             
             
               
                 
                   b 
                   = 
                   
                       
                   
                   ⁢ 
                   
                     
                       B 
                       * 
                       
                         
                           
                             
                               ⅆ 
                               2 
                             
                             ⁢ 
                             Y 
                           
                           
                             ⅆ 
                             
                               X 
                               2 
                             
                           
                         
                         
                           
                             
                               1 
                               + 
                               
                                 ( 
                                 
                                   
                                     ⅆ 
                                     Y 
                                   
                                   
                                     ⅆ 
                                     X 
                                   
                                 
                                 ) 
                               
                             
                           
                           2 
                         
                       
                     
                     = 
                     
                         
                       
                         B 
                         * 
                         
                           
                             
                               
                                 
                                   
                                     
                                       ( 
                                       
                                         
                                           cos 
                                           ⁢ 
                                           
                                               
                                           
                                           ⁢ 
                                           θ 
                                         
                                         + 
                                         
                                           2 
                                           ⁢ 
                                           
                                             ( 
                                             
                                               
                                                 X 
                                                 ′ 
                                               
                                               A 
                                             
                                             ) 
                                           
                                           ⁢ 
                                           sin 
                                           ⁢ 
                                           
                                               
                                           
                                           ⁢ 
                                           θ 
                                         
                                       
                                       ) 
                                     
                                     * 
                                     2 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     cos 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     θ 
                                   
                                   - 
                                 
                               
                             
                             
                               
                                 
                                   
                                     ( 
                                     
                                       
                                         
                                           - 
                                           sin 
                                         
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         θ 
                                       
                                       + 
                                       
                                         2 
                                         ⁢ 
                                         
                                           ( 
                                           
                                             
                                               X 
                                               ′ 
                                             
                                             A 
                                           
                                           ) 
                                         
                                         ⁢ 
                                         cos 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         θ 
                                       
                                     
                                     ) 
                                   
                                   * 
                                   2 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   sin 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   θ 
                                 
                               
                             
                           
                           
                             
                               
                                 ( 
                                 
                                   
                                     cos 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     θ 
                                   
                                   + 
                                   
                                     2 
                                     ⁢ 
                                     
                                       ( 
                                       
                                         
                                           X 
                                           ′ 
                                         
                                         A 
                                       
                                       ) 
                                     
                                     ⁢ 
                                     sin 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     θ 
                                   
                                 
                                 ) 
                               
                               3 
                             
                             * 
                             
                                 
                             
                             ⁢ 
                             
                               
                                 1 
                                 + 
                                 
                                   
                                     ( 
                                     
                                       
                                         
                                           
                                             - 
                                             sin 
                                           
                                           ⁢ 
                                           
                                               
                                           
                                           ⁢ 
                                           θ 
                                         
                                         + 
                                         
                                           2 
                                           ⁢ 
                                           
                                             ( 
                                             
                                               
                                                 X 
                                                 ′ 
                                               
                                               A 
                                             
                                             ) 
                                           
                                           ⁢ 
                                           cos 
                                           ⁢ 
                                           
                                               
                                           
                                           ⁢ 
                                           θ 
                                         
                                       
                                       
                                         
                                           cos 
                                           ⁢ 
                                           
                                               
                                           
                                           ⁢ 
                                           θ 
                                         
                                         + 
                                         
                                           2 
                                           ⁢ 
                                           
                                             ( 
                                             
                                               
                                                 X 
                                                 ′ 
                                               
                                               A 
                                             
                                             ) 
                                           
                                           ⁢ 
                                           sin 
                                           ⁢ 
                                           
                                               
                                           
                                           ⁢ 
                                           θ 
                                         
                                       
                                     
                                     ) 
                                   
                                   2 
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     equation 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     2 
                   
                   ) 
                 
               
             
           
         
       
     
     where a is a coordinate point in the vertical direction, b is a coordinate point in the horizontal direction, X1′ is a starting position in the horizontal position and X′ is a position along the horizontal direction, cutting the material along a line formed by the plurality of plotted points. 
     In another embodiment consistent with the present invention, at least one surface of the flexible material reflects light. 
     In another embodiment consistent with the present invention, at least one surface of the flexible material reflects light. 
     In another embodiment consistent with the present invention, at least one surface of the flexible material reflects energy waves. 
     In another embodiment consistent with the present invention, where a plurality of sheets overlap one another to form a trough. 
     In another embodiment consistent with the present invention, the flexible material is made from a material that has memory capabilities, and the shape of the first edge conforms to a line formed by a plurality of points defined by the equations 
     
       
         
           
             
               
                 
                   a 
                   = 
                   
                     
                       ∫ 
                       
                         X 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           1 
                           ′ 
                         
                       
                       
                         X 
                         ′ 
                       
                     
                     ⁢ 
                     
                       
                         
                           
                             1 
                             + 
                             
                               ( 
                               
                                 
                                   ⅆ 
                                   
                                     Y 
                                     ′ 
                                   
                                 
                                 
                                   ⅆ 
                                   
                                     X 
                                     ′ 
                                   
                                 
                               
                               ) 
                             
                           
                         
                         2 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                           
                         
                           = 
                           
                             
                                 
                               
                                 X 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   1 
                                   ′ 
                                 
                               
                               
                                 X 
                                 ′ 
                               
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               | 
                               
                                 
                                   1 
                                   4 
                                 
                                 ⁡ 
                                 
                                   [ 
                                   
                                     
                                       
                                         
                                           
                                             4 
                                             ⁢ 
                                             
                                               
                                                 ( 
                                                 
                                                   
                                                     X 
                                                     ′ 
                                                   
                                                   A 
                                                 
                                                 ) 
                                               
                                               2 
                                             
                                           
                                           + 
                                           1 
                                         
                                       
                                       * 
                                       2 
                                       ⁢ 
                                       
                                         ( 
                                         
                                           
                                             X 
                                             ′ 
                                           
                                           A 
                                         
                                         ) 
                                       
                                     
                                     + 
                                     
                                       ln 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       
                                          
                                         
                                           
                                             
                                               
                                                 4 
                                                 ⁢ 
                                                 
                                                   
                                                     ( 
                                                     
                                                       
                                                         X 
                                                         ′ 
                                                       
                                                       A 
                                                     
                                                     ) 
                                                   
                                                   2 
                                                 
                                               
                                               + 
                                               1 
                                             
                                           
                                           + 
                                           
                                             2 
                                             ⁢ 
                                             
                                               ( 
                                               
                                                 
                                                   X 
                                                   ′ 
                                                 
                                                 A 
                                               
                                               ) 
                                             
                                           
                                         
                                          
                                       
                                     
                                   
                                   ] 
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     equation 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     1 
                   
                   ) 
                 
               
             
             
               
                 
                   b 
                   = 
                   
                     B 
                     * 
                     
                       ( 
                       
                         
                           
                             ( 
                             
                               
                                 X 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   2 
                                   ′ 
                                 
                               
                               A 
                             
                             ) 
                           
                           * 
                           
                             ( 
                             
                               
                                 X 
                                 ′ 
                               
                               A 
                             
                             ) 
                           
                         
                         + 
                         
                           
                             ( 
                             
                               
                                 X 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   1 
                                   ′ 
                                 
                               
                               A 
                             
                             ) 
                           
                           * 
                           
                             ( 
                             
                               
                                 X 
                                 ′ 
                               
                               A 
                             
                             ) 
                           
                         
                         - 
                         
                           
                             ( 
                             
                               
                                 X 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   1 
                                   ′ 
                                 
                               
                               A 
                             
                             ) 
                           
                           * 
                           
                             ( 
                             
                               
                                 X 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   2 
                                   ′ 
                                 
                               
                               A 
                             
                             ) 
                           
                         
                         - 
                         
                           
                             ( 
                             
                               
                                 X 
                                 ′ 
                               
                               A 
                             
                             ) 
                           
                           2 
                         
                       
                       ) 
                     
                     ⁢ 
                     
                       
                         ( 
                         
                           1 
                           + 
                           
                             4 
                             ⁢ 
                             
                               
                                 ( 
                                 
                                   
                                     X 
                                     ′ 
                                   
                                   A 
                                 
                                 ) 
                               
                               2 
                             
                           
                         
                         ) 
                       
                       
                         3 
                         / 
                         2 
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     equation 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     2 
                   
                   ) 
                 
               
             
           
         
       
     
     where a is a coordinate point in the vertical direction, b is a coordinate point in the horizontal direction, X1′ is a starting coordinate point in the horizontal position, X2′ is the ending coordinate point in the horizontal direction, X′ is a position between X1′ and X2′ in the horizontal direction. 
     In another embodiment consistent with the present invention, the flexible material is made of, Plexiglas, Lexan, fiberglass or carbon fiber. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       The accompanying drawings, which are incorporated in and constitute a part of this specification, illustrate an implementation of the present invention and, together with the description, serve to explain the advantages and principles of the invention. In the drawings: 
         FIG. 1  depicts a side view of a parabolic reflector consistent with the present invention. 
         FIG. 2  depicts a plan view of a parabolic reflector consistent with the present invention. 
         FIG. 3A  depicts a plurality of parabolic reflectors consistent with the present invention which are mounted together and restrained by two rods. 
         FIG. 3B  depicts a parabolic reflector connecting to a support unit consistent with the present invention. 
         FIG. 4  depicts a side view of a parabolic reflector consistent with the present invention. 
         FIG. 5  depicts a plan view of the sheet before bending which is consistent with the present invention. 
         FIG. 6  depicts a parabolic reflector consistent with the present invention which is secured by a securing mechanism. 
         FIG. 7  is a functional block diagram of a method of producing a gravity formed parabolic reflector consistent with the present invention. 
     
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
     While various embodiments of the present invention are described herein, it will be apparent to those of skill in the art that many more embodiments and implementations are possible that are within the scope of this invention. Accordingly, the present invention is not to be restricted except in light of the attached claims and their equivalents. 
     As disclosed in further detail herein, is a parabolic reflector consisting of a sheet of material secured at two edges. The width of the material is configured such the material self-forming into a geometrically correct parabola when secured along two edges of the material. Further, the present invention also entails a method of forming the sheet of material such that the material forms a geometrically correct parabola opposed to a catenary. By producing a geometrically correct parabola, energy waves are tightly focused on a focal point which improves the performance of the reflector without requiring a rigid support structure to form the parabola. 
       FIG. 1  depicts a side view of a parabolic reflector  100  consistent with the present invention. The reflector includes a sheet of material  102 , which is secured at two edges  104  and  106 . The shape of the parabola is effective to direct waves of energy or light transmitted parallel to the axis of the parabola to the focal point of the parabola  108 . The sides of the material when laid flat are shaped such that the surface of the material  110  hangs down to form the parabolic shape without the use of a rigid support structure. In one embodiment, the sheet is made from a flexible material including, but not limited to reflective plastic, cloth, silk, mylar, or any other flexible material that does not resist bending. 
     As  FIG. 1  depicts, the X and Y axis are introduced parallel to the horizontal and vertical respectively. The X′ and Y′ axis are introduced proper to the parabola with the Y′ axis coinciding with the axis of the parabola and inclined by an angle theta Θ from the vertical. The angle theta Θ represents the angle incident rays and the axis of the parabola are offset from the vertical direction. 
     For a parabola supported at Cartesian positions (−D1, H1) and (D2, H2) with a span S parallel to the X′ axis, a parabola is defined on the same Cartesian coordinate system between the positions X1′  104  and X2′  106  using a scaling factor A calculated by the equation: 
     
       
         
           
             A 
             = 
             
               
                 
                   S 
                   2 
                 
                 
                   ( 
                   
                     
                       
                         H 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         1 
                       
                     
                     + 
                     
                       
                         H 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         2 
                       
                     
                   
                   ) 
                 
               
               = 
               
                 
                   
                     D 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       1 
                       2 
                     
                   
                   
                     H 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     1 
                   
                 
                 = 
                 
                   
                     
                       D 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         2 
                         2 
                       
                     
                     
                       H 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       2 
                     
                   
                   = 
                   
                     
                       1 
                       4 
                     
                     ⁢ 
                     F 
                   
                 
               
             
           
         
       
     
     Where F is the distance from the X′ axis to the focal point of the parabola  108 . 
     The length of the parabolic reflector is defined by the following equation: 
             L   ⁢     =     X   ⁢           ⁢     1   ′         X   ⁢           ⁢     2   ′         ⁢              A   *     1   4     ⁢         4   ⁢       X   ′     A       +   1       *   2   ⁢       X   ′     A       +   ln     |           4   ⁢       (       X   ′     A     )     2       +   1       +     2   ⁢       X   ′     A                      
An operator may select a piece of material longer than the length calculated using the above mentioned equation.
 
     Next, an equation for cutting two sides of a piece of material that will form the parabolic shape when the material is suspended from two edges is determined.  FIG. 2  depicts a plan view of a parabolic reflector  100  consistent with the present invention. In the plan view, the parabolic reflector includes a first end  202 , a second end  204  and two sides  206 . 
     The width, W, of the parabolic reflector  100  varies along the horizontal X axis such that the distribution of weight across the reflector is proportional to the equation 
                   ⅆ   2     ⁢   Y       ⅆ     X   2               1   +     (       ⅆ   Y       ⅆ   X       )         2           
which results in the slack of the material  102  forming a parabolic shape with a focal point offset by the angle theta Θ when viewed from the side. To produce this effect, the sides of a rectangular sheet  206  of material are manipulated such that the shape of the material distributes the weight in such a manner as to form the material into a parabola with a focal point offset by the angle theta Θ.
 
     The curve defining the sides  206  of the sheet of material  102  is created by sequentially plotting positions along the a and b axis for the interval X1′ to X2′, as shown in  FIG. 2 . For each X′ over the interval X1′ to X2′ along the length of the parabola L, the horizontal position on the line is calculated using the following equation: 
             a   =         ∫     X   ⁢           ⁢     1   ′         X   ′       ⁢         1   +     (       ⅆ   Y       ⅆ   X       )         2       ⁢           =                 ⁢             X   ⁢           ⁢     1   ′         X   ′       ⁢           ⁢     |       1   4     ⁡     [             4   ⁢       (       X   ′     A     )     2       +   1       *   2   ⁢       X   ′     A       +     ln   ⁢           ⁢                4   ⁢       (       X   ′     A     )     2       +   1       +     2   ⁢     (       X   ′     A     )                  ]                     
and the position b is determined by the equation:
 
             b   =       B   *           ⅆ   2     ⁢   Y       ⅆ     X   2               1   +     (       ⅆ   Y       ⅆ   X       )         2         =     B   *           (       cos   ⁢           ⁢   θ     +     2   ⁢     (       X   ′     A     )     ⁢   sin   ⁢           ⁢   θ       )     *   2   ⁢   cos   ⁢           ⁢   θ     -       (         -   sin     ⁢           ⁢   θ     +     2   ⁢     (       X   ′     A     )     ⁢   cos   ⁢           ⁢   θ       )     *   2   ⁢   sin   ⁢           ⁢   θ             (       cos   ⁢           ⁢   θ     +     2   ⁢     (       X   ′     A     )     ⁢   sin   ⁢           ⁢   θ       )     3     *       1   +       (           -   sin     ⁢           ⁢   θ     +     2   ⁢     (       X   ′     A     )     ⁢   cos   ⁢           ⁢   θ           cos   ⁢           ⁢   θ     +     2   ⁢     (       X   ′     A     )     ⁢   sin   ⁢           ⁢   θ         )     2                       
Where B is scale factor calculated by the equation:
 
     
       
         
           
             B 
             = 
             
               
                 1 
                 2 
               
               ⁢ 
               
                 W 
                 max 
               
               ⁢ 
               
                   
               
               ⁢ 
               over 
               ⁢ 
               
                   
               
               ⁢ 
               
                   
                 
                   
                     [ 
                     
                       
                         X 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           1 
                           ′ 
                         
                       
                       , 
                       
                         X 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           2 
                           ′ 
                         
                       
                     
                     ] 
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   of 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                       
                     
                       ( 
                       
                         
                           
                             
                               ( 
                               
                                 
                                   cos 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   θ 
                                 
                                 + 
                                 
                                   2 
                                   ⁢ 
                                   
                                     ( 
                                     
                                       
                                         X 
                                         ′ 
                                       
                                       A 
                                     
                                     ) 
                                   
                                   ⁢ 
                                   sin 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   θ 
                                 
                               
                               ) 
                             
                             * 
                             2 
                             ⁢ 
                             cos 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             θ 
                           
                           - 
                           
                             
                               ( 
                               
                                 
                                   
                                     - 
                                     sin 
                                   
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   θ 
                                 
                                 + 
                                 
                                   2 
                                   ⁢ 
                                   
                                     ( 
                                     
                                       
                                         X 
                                         ′ 
                                       
                                       A 
                                     
                                     ) 
                                   
                                   ⁢ 
                                   cos 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   θ 
                                 
                               
                               ) 
                             
                             * 
                             2 
                             ⁢ 
                             sin 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             θ 
                           
                         
                         
                           
                             
                               ( 
                               
                                 
                                   cos 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   θ 
                                 
                                 + 
                                 
                                   2 
                                   ⁢ 
                                   
                                     ( 
                                     
                                       
                                         X 
                                         ′ 
                                       
                                       A 
                                     
                                     ) 
                                   
                                   ⁢ 
                                   sin 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   θ 
                                 
                               
                               ) 
                             
                             3 
                           
                           * 
                           
                             
                               1 
                               + 
                               
                                 
                                   ( 
                                   
                                     
                                       
                                         
                                           - 
                                           sin 
                                         
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         θ 
                                       
                                       + 
                                       
                                         2 
                                         ⁢ 
                                         
                                           ( 
                                           
                                             
                                               X 
                                               ′ 
                                             
                                             A 
                                           
                                           ) 
                                         
                                         ⁢ 
                                         cos 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         θ 
                                       
                                     
                                     
                                       
                                         cos 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         θ 
                                       
                                       + 
                                       
                                         2 
                                         ⁢ 
                                         
                                           ( 
                                           
                                             
                                               X 
                                               ′ 
                                             
                                             A 
                                           
                                           ) 
                                         
                                         ⁢ 
                                         sin 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         θ 
                                       
                                     
                                   
                                   ) 
                                 
                                 2 
                               
                             
                           
                         
                       
                       ) 
                     
                   
                 
               
             
           
         
       
     
     When the two lines defining the sides of the parabola in plan view are determined using the above equations, the material  102  may be suspended, without tension, by two parallel rods and the material  102  will form a parabola which is effective to reflect light in such a manner that the light is focused on the focal point of the parabola  108 . 
       FIG. 3A  depicts a plurality of parabolic reflectors  100  consistent with the present invention that are secured to a first support unit  302  and a second support unit  304 . The length of each of the sheets of material  102  and the shape of the sides of sheet of material  102  shown in  FIG. 3A  are calculated using the above equations. 
     Each of the first ends  202  of the parabolic reflectors  100  are secured to a first support unit  302  and each of the second ends  204  of the parabolic reflectors  100  are secured to a second support unit  304 . The parabolic reflectors  100  are each secured to the support units using securing methods including, but not limited to, nailing each of the ends  202  and  204  to the support units  302  and  304 , gluing each of the ends  202  and  204  to the support units  302  and  304 , stapling each of the ends  202  and  204  to the support units  302  and  304 , or any other acceptable method of securing the ends  202  and  204  to the support units. 
       FIG. 3B  depicts one embodiment of a parabolic reflector  100  connecting to a support unit. As depicted in  FIG. 3B , one of the ends  202  or  204  is folded over and secured to form a sleeve  306  through which a support unit  302  or  304  is inserted. The sleeve  306  is formed by wrapping excess material at the ends  202  or  204  around one of the support units  302  or  304  and securing the end  202  or  204  to material  102 . The end  202  or  204  is secured to the material  102  using securing methods including, but not limited to, sewing the end  202  or  204  to the material  102 , gluing  202  or  204  to the material  102  or any other method of securing the ends  202  or  204  to the material to form a sleeve  306 . 
     In one embodiment consistent with the present invention, the length of the material  102  is increased by an amount equal to the circumference of the support unit  302  or  304 . 
     In another embodiment consistent with the present invention, a plurality of parabolic reflectors  100  are arranged such that each of the parabolic reflectors  100  overlaps the adjacent parabolic reflector creating a trough  308 , as shown in  FIG. 3A . With this arrangement, it is possible to concentrate light to multiple focal points over a large area. In addition, since the parabolic reflectors only require two edges to secure them in place, installation and manufacturing costs are greatly reduced. 
     The present embodiment has numerous applications including, but not limited to, a solar trough for use in high temperature solar collection, a daylight reflector for daylight harvesting of light, as a reflector for artificial lighting system, or as a mobile solar or radio wave collector. Because the present embodiment does not require a support structure, it can be assembled and disassembled quickly making it ideal for portable or temporary applications. Further, since the present embodiment does not require a support structure, the cost of manufacturing the parabolic reflector is reduced. 
       FIG. 4  depicts a side view of a parabolic reflector  400  consistent with the present invention. An elastic bendable flat sheet  402  of uniform thickness and variable width is secured at two points  404  and  406  using a securing mechanism  410  including, but not limited to, a wire, a string, a rod or any other mechanism, which can secure the ends of the sheet  402 . The width of the sheet  402  is adjusted along its length manipulating the sheet&#39;s resistance to bending such that the sheet  402  forms a parabolic shape. By forming the sheet into a parabolic shape, light entering the parabola is reflected in such a manner that the light is focused on the focal point of the parabola  408 . 
     The parabolic sheet is cut from an elastic bendable flat sheet  402  having a W as selected by an operator. The length of the sheet L is a function of the profile of parabola as shown in  FIG. 4  and is defined by the following equation: 
             L   =             X   ⁢           ⁢   1       X   ⁢           ⁢   2       ⁢     ❘         1   4     ⁢         4   ⁢       (       X   ′     A     )     2       +   1       *   2   ⁢     (       X   ′     A     )       +     ln   ⁢                4   ⁢       (       X   ′     A     )     2       +   1       +     2   ⁢     (       X   ′     A     )                            
Where A is a scaling factor calculated by the equation:
 
             A   =         S   2       (         H   ⁢           ⁢   1       +       H   ⁢           ⁢   2         )       =         D   ⁢           ⁢     1   2         H   ⁢           ⁢   1       =         D   ⁢           ⁢     2   2         H   ⁢           ⁢   2       =       1   4     ⁢   F                 
as discussed above.
 
       FIG. 5  depicts a plan view of the sheet  402  before bending, which is consistent with the present invention. The length of the sheet  402  L is calculated using the equation discussed above. As  FIG. 5  demonstrates, the width of the sheet W varies in the horizontal direction. When the sheet is secured with a securing mechanism, it bends into the shape of a parabola without the need for any additional support other than the securing mechanism. 
     The curve defining each side of the Sheet is created by sequentially plotting positions along the a and b axis for the interval X1′ to X2′, as shown in  FIG. 5 . For each X′ over the interval X1′ to X2′, a position is calculated using the following equation: 
                   a   =       ∫     X   ⁢           ⁢     1   ′         X   ′       ⁢         1   +     (       ⅆ     Y   ′         ⅆ     X   ′         )         2                   =             X   ⁢           ⁢     1   ′         X   ′       ⁢     ❘       1   4     [             4   ⁢       (       X   ′     A     )     2       +   1       *   2   ⁢     (       X   ′     A     )       +     ln   ⁢                4   ⁢       (       X   ′     A     )     3       +   1       +     2   ⁢     (       X   ′     A     )                  ]                     
and the position b is determined by the equation:
 
             b   =     B   *     (         (       X   ⁢           ⁢     2   ′       A     )     *     (       X   ′     A     )       +       (       X   ⁢           ⁢     1   ′       A     )     *     (       X   ′     A     )       -       (       X   ⁢           ⁢     1   ′       A     )     *     (       X   ⁢           ⁢     2   ′       A     )       -       (       X   ′     A     )     2       )     ⁢       (     1   +     4   ⁢       (       X   ′     A     )     2         )       3   /   2               
Where B is a scale factor given by the equation:
 
     
       
         
           
             B 
             = 
             
               
                 1 
                 2 
               
               ⁢ 
               
                 W 
                 max 
               
               ⁢ 
               
                   
               
               ⁢ 
               over 
               ⁢ 
               
                   
               
               ⁢ 
               
                   
                 
                   
                     [ 
                     
                       
                         X 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         1 
                       
                       , 
                       
                         X 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         2 
                       
                     
                     ] 
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   of 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                       
                     
                       
                         ( 
                         
                           
                             
                               ( 
                               
                                 
                                   X 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     2 
                                     ′ 
                                   
                                 
                                 A 
                               
                               ) 
                             
                             * 
                             
                               ( 
                               
                                 
                                   X 
                                   ′ 
                                 
                                 A 
                               
                               ) 
                             
                           
                           + 
                           
                             
                               ( 
                               
                                 
                                   X 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     1 
                                     ′ 
                                   
                                 
                                 A 
                               
                               ) 
                             
                             * 
                             
                               ( 
                               
                                 
                                   X 
                                   ′ 
                                 
                                 A 
                               
                               ) 
                             
                           
                           - 
                           
                             
                               ( 
                               
                                 
                                   X 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     1 
                                     ′ 
                                   
                                 
                                 A 
                               
                               ) 
                             
                             * 
                             
                               ( 
                               
                                 
                                   X 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     2 
                                     ′ 
                                   
                                 
                                 A 
                               
                               ) 
                             
                           
                           - 
                           
                             
                               ( 
                               
                                 
                                   X 
                                   ′ 
                                 
                                 A 
                               
                               ) 
                             
                             2 
                           
                         
                         ) 
                       
                       ⁢ 
                       
                         
                           ( 
                           
                             1 
                             + 
                             
                               4 
                               ⁢ 
                               
                                 
                                   ( 
                                   
                                     
                                       X 
                                       ′ 
                                     
                                     A 
                                   
                                   ) 
                                 
                                 2 
                               
                             
                           
                           ) 
                         
                         
                           3 
                           / 
                           2 
                         
                       
                     
                   
                 
               
             
           
         
       
     
     The length of the cable or string required to connect the ends of the parabola,  404  and  406 , is defined by the following equation:
 
√{square root over ((H2−H1) 2 +S 2 )}
 
       FIG. 6  depicts a parabolic reflector  400  consistent with the present invention that is secured by a securing mechanism  410 . The length of the securing mechanism  410  is determined using the equation to determine the length of the cable of string disclosed above. When the securing mechanism  410  is connected to both securing points  404  and  408 , the sides of the material  402  are pulled together to form the parabolic shape calculated using the method described above. 
     The securing mechanism  410  is coupled to the parabolic reflectors at securing points  404  and  408  using a coupling method including, but not limited to, gluing the securing mechanism  410  to a securing point  404  or  408 , gluing the securing mechanism to a securing point  404  or  408 , welding the securing mechanism  410  to a securing point  404  or  408 , or by any other suitable method of coupling the securing mechanism  410  to a securing point  404  or  408 . 
     In one embodiment consistent with the present invention, a collector  602  is situated at the focal point of the parabola to collect rays reflected from the parabola. 
       FIG. 7  is a functional block diagram of a method of producing a gravity formed parabolic reflector will be described performed by a cutting system which includes a memory, a processor and a user interface. First, the securing points and the span between the securing points for the parabolic reflector are identified on a Cartesian coordinate system and are inputted into the cutting system by the user interface (Step  702 ). Next, the cutting system calculates the length and width of the reflector at the points between the two securing points. (Step  704 ). The cutting system then selects a suitable piece of material which is capable of being cut into the desired shape based on the length and width calculations. 
     Once the length and width of the reflector are calculated between both of the securing points, the selected piece of material is loaded into the cutting system (Step  706 ). The material is automatically fed into the cutting system via a material feeding unit. The cutting system then identifies the position on the material where the securing units will be positioned and cuts the material to the desired length (Step  708 ). The cutting system then moves a cutting unit in the horizontal direction and shapes the sides of the material based on the horizontal position of the cutting unit based on the equations listed above (Step  710 ). 
     Once the material is cut, the parabolic reflector is secured in position using a securing unit, including, but not limited to a baton, roller or any other suitable securing mechanism. Once secured, the reflector will assume a parabolic shape due to the weight distribution of the material. 
     In another embodiment consistent with the present invention, an elastic bendable parabolic reflector is produced using the same method depicted in  FIG. 6 . The elastic bendable parabolic reflector is made of a stiff material with memory capabilities including, but not limited to, plastic, Plexiglas, Lexan, fiberglass or carbon fiber sheet, or any other bendable material with memory capabilities. In addition, the elastic bendable sheet is secured by two securing units which include, but are not limited to a cable, wire or any other securing means. 
     The present embodiment has numerous applications including, but not limited to, a solar trough for use in high temperature solar collection, a daylight reflector for daylight harvesting of light, as a reflector for artificial lighting system, as a mobile solar collector or radio wave collector or as a lunar high temperature solar collector which can be used in a zero gravity environment. Because the present embodiment does not require a support structure, it can be assembled and disassembled quickly making it ideal for portable or temporary applications. Further, since the present embodiment does not require a support structure, the cost of manufacturing the parabolic reflector is reduced.