Method and apparatus for transforming and rendering graphical curves

Abstract

The present invention provides a method and apparatus for rendering images on a computer screen. Under the invention, a portion of a base image is described using a path. The path is then transformed using a non-affine transform to produce a transformed path. The transformed path is then rendered onto the computer screen. Another aspect of the present invention is a method and apparatus for rendering curves of any order and any dimension. In particular, the present invention provides a means for converting a function of any order and any dimension that describes one segment of a curve into a function that describes a different sized segment or an adjoining segment.

1 . A method of displaying an image on a computer screen, the method comprising: 
 describing at least a portion of a base image as a path comprising a function of at least one variable, the path representing multiple pixels;    performing a non-affine transform on the path instead of the multiple pixels represented by the path to produce a transformed path by performing the non-affine transform on the function including the variable; and    rendering the transformed path onto the computer screen.    
   
   
       2 . The method of  claim 1  wherein performing a non-affine transform comprises performing a bilinear transform.  
   
   
       3 . The method of  claim 2  wherein describing the portion of the base image as a path comprises describing the portion using a function of order n.  
   
   
       4 . The method of  claim 3  wherein performing a bilinear transform produces a transformed function of order 2n.  
   
   
       5 . The method of  claim 3  wherein describing the portion of the base image as a path comprises describing the portion as a function of order one.  
   
   
       6 . The method of  claim 3  wherein describing the portion of the base image as a path comprises describing the portion as a function of order three.  
   
   
       7 . The method of  claim 1  wherein performing a non-affine transform comprises performing a perspective transform.  
   
   
       8 . The method of  claim 7  wherein performing a perspective transform produces a rational function of order n.  
   
   
       9 . The method of  claim 1  wherein rendering the transformed path comprises approximating the transformed path as a series of lines and rendering each line in the series of lines.  
   
   
       10 . A computer-readable medium having computer-executable components for performing steps comprising: 
 generating a function of a variable to describe multiple pixels of an image for a computer screen;    transforming the function instead of the multiple pixels using a non-affine transform applied to the entire function including the variable to produce a transformed function; and    converting the transformed function into an image on the computer screen.    
   
   
       11 . The computer-readable medium of  claim 10  wherein transforming the function comprises transforming a function representing a smooth curve.  
   
   
       12 . The computer-readable medium of  claim 10  wherein transforming the function comprises using a bilinear transform.  
   
   
       13 . The computer-readable medium of  claim 12  wherein generating a function to describe an image comprises generating a function of order n and wherein transforming the function produces a transformed function of order 2n.  
   
   
       14 . The computer-readable medium of  claim 10  wherein transforming the function comprises using a perspective transform.  
   
   
       15 . The computer-readable medium of  claim 10  wherein converting the transformed function into an image comprises converting the transformed function into a series of lines and converting each line into an image.