Method and apparatus for synthesizing and analyzing patterns utilizing novel "super-formula" operator

Abstract

Patterns (e.g, such as images, waveforms such as sounds, electromagnetic waves, or other signals, etc.) are synthesized, modulated and/or analyzed through the use of a computer programmed with a novel mathematical formula. The formula acts as a linear operator and can be used to create a variety of shapes, waveforms, and other representations. The formula greatly enhances ability in computer operations and provides a great savings in computer memory and a substantial increase in computing power.

1 . A method for display of widely variable naturalistic scenery, including simulated plants, simulated other organisms or simulated landscapes, on an amusement device with minimized data storage and processing requirements, the method comprising:
 creating with computer hardware a first shape that corresponds to at least a portion of a simulated natural form, including at least a portion of a simulated plant, a simulated other organism or a simulated landscape, based on a first set of parameter values input by said computer hardware into a representation of a formula that establishes shape position with respect to angle as:   shape position corresponds to a fraction in which the denominator of said fraction has a variable root parameter (n 1 ) set as the root of an expression including the absolute values of two terms, with the absolute value of a first term of said two terms being to an exponent based on a variable first exponent parameter (n 2 ) and the absolute value of a second term of said two terms being to an exponent based on a variable second exponent parameter (n 3 ), wherein said first term includes an axis sub term, having a first axis parameter (1/a or A), that is multiplied by a cosine sub term including the cosine of a value including a first rotational symmetry parameter (m 1 ) times an angle, and wherein said second term includes an axis sub term, having a second axis parameter (1/b or B), that is multiplied by a sine sub term including the sine of a value including a second rotational symmetry parameter (m 2 ) times an angle;
 wherein (i) said first and second axis parameters, said root parameter (n 1 ), said first and second exponent parameters (n 2 , n 3 ) and said first and second rotational symmetry parameters (m 1 , m 2 ) are finite real numbers, (ii) said root parameter (n 1 ) and said first and second axis parameters are not equal to zero, and (iii) said first and second axis parameters, said root parameter (n 1 ), said first and second exponent parameters (n 2 , n 3 ) and said first and second rotational symmetry parameters (m 1 , m 2 ) are not equal to 1; 
   displaying on a monitor of the amusement device a naturalistic image based on said first shape;   creating with said computer hardware additional shapes including said computer hardware inputting modified parameter values into said representation of said formula;   generating a positionwise display of a naturalistic scenery by displaying with said monitor additional naturalistic images based on said additional shapes, said additional naturalistic images being displayed on said monitor separated positionwise from said naturalistic image for said positionwise display of the naturalistic scenery;   whereby allowing display of widely variable naturalistic scenery, including simulated organisms, simulated plants or simulated landscapes, with minimized data storage and processing requirements due to minimized storage requirements for said parameter values and processing with said representation of said formula.   
     
     
         2 . A non-transitory computer readable medium tangibly storing a computer program product including instructions that, when executed, cause at least one computer processor of a computer system to perform a method for creating display of widely variable naturalistic scenery, including simulated plants, simulated other organisms or simulated landscapes, on an amusement device with minimized data storage and processing requirements, the method comprising:
 creating with computer hardware a first shape that corresponds to at least a portion of a simulated natural form, including at least a portion of a simulated plant, a simulated other organism or a simulated landscape, based on a first set of parameter values input by said computer hardware into a representation of a formula that establishes shape position with respect to angle as:   shape position corresponds to a fraction in which the denominator of said fraction has a variable root parameter (n 1 ) set as the root of an expression including the absolute values of two terms, with the absolute value of a first term of said two terms being to an exponent based on a variable first exponent parameter (n 2 ) and the absolute value of a second term of said two terms being to an exponent based on a variable second exponent parameter (n 3 ), wherein said first term includes an axis sub term, having a first axis parameter (1/a or A), that is multiplied by a cosine sub term including the cosine of a value including a first rotational symmetry parameter (m 1 ) times an angle, and wherein said second term includes an axis sub term, having a second axis parameter (1/b or B), that is multiplied by a sine sub term including the sine of a value including a second rotational symmetry parameter (m 2 ) times an angle;
 wherein (i) said first and second axis parameters, said root parameter (n 1 ), said first and second exponent parameters (n 2 , n 3 ) and said first and second rotational symmetry parameters (m 1 , m 2 ) are finite real numbers, (ii) said root parameter (n 1 ) and said first and second axis parameters are not equal to zero, and (iii) said first and second axis parameters, said root parameter (n 1 ), said first and second exponent parameters (n 2 , n 3 ) and said first and second rotational symmetry parameters (m 1 , m 2 ) are not equal to 1; 
   displaying on a monitor of the amusement device a naturalistic image based on said first shape;   creating with said computer hardware additional shapes including said computer hardware inputting modified parameter values into said representation of said formula;   generating a positionwise display of naturalistic scenery by displaying with said monitor additional naturalistic images based on said additional shapes, said additional naturalistic images being displayed on said monitor separated positionwise from said naturalistic image for said positionwise display of naturalistic scenery;   whereby allowing display of widely variable naturalistic scenery, including simulated organisms, simulated plants or simulated landscapes, with minimized data storage and processing requirements due to minimized storage requirements for said parameter values and processing with said representation of said formula.   
     
     
         3 . A computer-implemented method for creating positionwise display of widely variable naturalistic scenery, including simulated plants, simulated other organisms or simulated landscapes, on an amusement device, the method comprising:
 generating one or more first images of shapes used in nature, said shapes including at least a portion of a simulated plant, simulated other organism or simulated landscape, with formula parameters input into a representation of a formula that establishes position r with respect to angle Φ as:   
       
         
           
             
               r 
               = 
               
                 1 
                 
                   
                     
                       
                          
                         
                           
                             
                               1 
                               a 
                             
                             · 
                             cos 
                           
                            
                           
                               
                           
                            
                           
                             
                               
                                 m 
                                 1 
                               
                               · 
                               φ 
                             
                             4 
                           
                         
                          
                       
                       
                         n 
                         2 
                       
                     
                     + 
                     
                       
                          
                         
                           
                             
                               1 
                               b 
                             
                             · 
                             sin 
                           
                            
                           
                               
                           
                            
                           
                             
                               
                                 m 
                                 2 
                               
                               · 
                               φ 
                             
                             4 
                           
                         
                          
                       
                       
                         n 
                         3 
                       
                     
                   
                   
                     n 
                     1 
                   
                 
               
             
           
         
         
           wherein (i) the parameters a, b, n 1 , n 2 , n 3 , m 1 , and m 2  are finite real numbers, (ii) the parameters a, b, and n 1 ≠0, and (iii) the parameters n 1 , n 2 , n 3 , m 1 , and m 2 ≠1; 
         
         creating additional images of shapes used in nature with said representation of said formula by moderation of the formula parameters used to create the one or more first images to create new images with variation in said shapes; and 
         generating widely variable naturalistic scenery, including simulated plants, simulated other organisms or simulated landscapes, by displaying said one or more first images together on the amusement device and creating a positionwise arrangement for display of said additional images on the amusement device. 
       
     
     
         4 . The method of  claim 1 , wherein said display is at least part of a video. 
     
     
         5 . The method of  claim 3 , wherein said arrangement is at least part of a video. 
     
     
         6 . The method of  claim 1 , wherein said display is at least part of a computer animation. 
     
     
         7 . The method of  claim 3 , wherein said arrangement is at least part of a computer animation. 
     
     
         8 . The method of  claim 1 , wherein said creating with said computer hardware additional shapes including said computer hardware inputting modified values into said representation of said formula includes incrementing parameter values such that shapes appear to continually change in real time. 
     
     
         9 . The method of  claim 3 , wherein said moderation of said formula parameters includes incrementing parameter values such that shapes appear to continually change in real time. 
     
     
         10 . The method of  claim 1 , further including modifying said first shape by performing at least one operation on said first shape. 
     
     
         11 . The method of  claim 10 , wherein said at least one operation includes transforming said first shape to at least one non-orthogonal lattice. 
     
     
         12 . The method of  claim 10 , wherein said at least one operation includes flattening, skewing, elongating, enlarging, rotating, moving and/or translating said first shape. 
     
     
         13 . The method of  claim 1 , wherein said naturalistic scenery includes at least part of the simulated living organism. 
     
     
         14 . The method of  claim 1 , wherein said naturalistic scenery includes at least part of the simulated plant. 
     
     
         15 . The method of  claim 1 , wherein said naturalistic scenery includes at least part of the simulated landscape. 
     
     
         16 . The method of  claim 1 , wherein said representation of said formula includes a representation of: 
       
         
           
             
               r 
               = 
               
                 1 
                 
                   
                     
                       
                          
                         
                           
                             
                               1 
                               a 
                             
                             · 
                             cos 
                           
                            
                           
                               
                           
                            
                           
                             
                               
                                 m 
                                 1 
                               
                               · 
                               φ 
                             
                             4 
                           
                         
                          
                       
                       
                         n 
                         2 
                       
                     
                     + 
                     
                       
                          
                         
                           
                             
                               1 
                               b 
                             
                             · 
                             sin 
                           
                            
                           
                               
                           
                            
                           
                             
                               
                                 m 
                                 2 
                               
                               · 
                               φ 
                             
                             4 
                           
                         
                          
                       
                       
                         n 
                         3 
                       
                     
                   
                   
                     n 
                     1 
                   
                 
               
             
           
         
         where parameters a, b, n i  and m i  are finite real numbers, n 1 ≠0, and a and b≠0, and 
         wherein said shape position is a radius r, and said angle or orientation is an angle Φ. 
       
     
     
         17 . The method of  claim 1 , wherein said representation of said formula includes a representation of: 
       
         
           
             
               r 
               = 
               
                 1 
                 
                   
                     
                       
                          
                         
                           
                             
                               1 
                               a 
                             
                             · 
                             cos 
                           
                            
                           
                               
                           
                            
                           
                             
                               
                                 m 
                                 1 
                               
                               · 
                               φ 
                             
                             4 
                           
                         
                          
                       
                       
                         n 
                         2 
                       
                     
                     ± 
                     
                       
                          
                         
                           
                             
                               1 
                               b 
                             
                             · 
                             sin 
                           
                            
                           
                               
                           
                            
                           
                             
                               
                                 m 
                                 2 
                               
                               · 
                               φ 
                             
                             4 
                           
                         
                          
                       
                       
                         n 
                         3 
                       
                     
                   
                   
                     n 
                     1 
                   
                 
               
             
           
         
         where parameters a, b, n i  and m i  are finite real numbers, n 1 ≠0, and a and b≠0, and 
         wherein said shape position is a radius r, and said angle or orientation is an angle Φ, and 
         where 
       
       
         
           
             
               
                 
                   
                      
                     
                       
                         
                           1 
                           a 
                         
                         · 
                         cos 
                       
                        
                       
                           
                       
                        
                       
                         
                           
                             m 
                             1 
                           
                           · 
                           φ 
                         
                         4 
                       
                     
                      
                   
                   
                     n 
                     2 
                   
                 
                 ± 
                 
                   
                      
                     
                       
                         
                           1 
                           b 
                         
                         · 
                         sin 
                       
                        
                       
                           
                       
                        
                       
                         
                           
                             m 
                             2 
                           
                           · 
                           φ 
                         
                         4 
                       
                     
                      
                   
                   
                     n 
                     3 
                   
                 
               
               > 
               0. 
             
           
         
       
     
     
         18 . The method of  claim 1 , wherein said representation of said formula includes a representation of: 
       
         
           
             
               r 
               = 
               
                 1 
                 
                   
                     
                       
                          
                         
                           
                             A 
                             · 
                             cos 
                           
                            
                           
                               
                           
                            
                           
                             
                               
                                 m 
                                 1 
                               
                               · 
                               φ 
                             
                             4 
                           
                         
                          
                       
                       
                         n 
                         2 
                       
                     
                     + 
                     
                       
                          
                         
                           
                             B 
                             · 
                             sin 
                           
                            
                           
                               
                           
                            
                           
                             
                               
                                 m 
                                 2 
                               
                               · 
                               φ 
                             
                             4 
                           
                         
                          
                       
                       
                         n 
                         3 
                       
                     
                   
                   
                     n 
                     1 
                   
                 
               
             
           
         
         wherein said shape position is a radius r, and said angle or orientation is an angle Φ. 
       
     
     
         19 . The method of  claim 1 , wherein said sets of parameter values includes data stored remotely on at least one remote data storage device, and further including said computer hardware receiving data of parameter sets from said at least one data storage device remotely on-line or over the Internet. 
     
     
         20 . The method of  claim 1 , wherein said at least a portion of the simulated natural form includes a combination of said first shape and at least one other shape. 
     
     
         21 - 28 . (canceled)