Patent Publication Number: US-8537173-B2

Title: Graphics rendering apparatus, graphics rendering method, Recording medium having recorded therein graphics rendering program, and integrated circuit for rendering a high-quality image using a scaling coefficient

Description:
BACKGROUND OF INVENTION 
     1. Technical Field 
     The present invention relates to a graphics rendering technique using polygon data and vector data. 
     2. Background Art 
     Graphical user interfaces (hereinafter, “GUIs”) originally developed for OS of personal computers have been now installed on digital TVs, portable telephones, and the like. There have been widespread GUIs. 
     Also, in recent years, there have become available many GUIs for displaying 3D images (hereinafter, “3D-GUIs”), as disclosed in Patent Literature 1. 
     As shown in  FIG. 34 , a 3D-GUI disclosed in the Patent Literature 1 displays, on a screen, a cubic CB and a menu screen (also referred to as “window” or “work space”) MENU, which is composed of an icon IC 1 , a character IC 2 , and so on, that is pasted onto each surface (polygon) PG of the cubic CB. 
     As shown in  FIG. 35 , there is an expectancy for a 3D-GUI that displays a plurality of cubes CB 1 , CB 2 , . . . , CB 7 , and zooms in the cube CB 1  that is selected by a user for display, and zooms out the other remaining cubes CB 2 , CB 3 , . . . , CB 7  around the cube CB 1  for display. 
     By the way, as this type of 3D-GUI, it is general to use a graphics rendering apparatus that stores a character font of a menu screen as vector data (outline data). 
     This type of graphics rendering apparatus, which has been conventionally used, performs processing roughly in accordance with a flow shown below. 
     Firstly, a character texture having an appropriate size is generated based on vector data (PG 1  shown in  FIG. 36 ). Next, the character texture is scaled in accordance with the size of a polygon PG (PG 2  or PG 3  shown in  FIG. 36 ). Then, the scaled character texture is mapped onto the polygon PG. 
     CITATION LIST 
     Patent Literature 
     
         
         [Patent Literature 1] U.S. Pat. No. 5,678,015 
       
    
     SUMMARY OF INVENTION 
     However, according to a conventional graphics rendering apparatus, when the cubes CB 1 , CB 2 , . . . , CB 7  are zoomed out for example, the polygon PG onto which the character texture has been mapped is accordingly zoomed out. This has sometimes crushed the character texture to be unreadable, such as the case of the PG 3  shown in  FIG. 36 . On the contrary when the cubes CB 1 , CB 2 , . . . , CB 7  are zoomed in, the polygon PG onto which the character texture has been mapped is accordingly zoomed in. This has sometimes caused aliasing on a contour of a character, such as the case of the PG 2  shown in  FIG. 36 . That is, zoom-in and zoom-out of a 3D image including a polygon PG has might decrease the quality of the 3D image. 
     The present invention was made in view of the above problem, and aims to provide a graphics rendering apparatus capable of displaying a high-quality 3D image. 
     In order to achieve the above aim, the graphics rendering apparatus relating to the present invention is a graphics rendering apparatus comprising: a scaling coefficient determination unit operable to determine, based on polygon data representing a polygon onto which a texture is to be mapped, a scaling coefficient that is a basis for scaling first vector data from which the texture is to be generated; a vector data conversion unit operable to generate second vector data by scaling the first vector data based on the scaling coefficient; a texture generation unit operable to generate a texture based on the second vector data; and a texture mapping unit operable to map the texture generated by the texture generation unit onto the polygon. 
     With this structure, first vector data is scaled based on the scaling coefficient. Then, a texture, which is generated using second vector data obtained by scaling the first vector data, is mapped onto a polygon. Accordingly, it is possible to prevent occurrence of crush of a character texture and aliasing caused by texture scaling. This enables display of a high-quality 3D image. 
     Also, the graphics rendering apparatus relating to the present invention may further comprise a virtual plate generation unit operable to generate, based on the polygon data and the first vector data, virtual plate data representing a virtual plate that includes a vector image represented by the first vector data, wherein the vector data conversion unit performs the scaling based on the virtual plate data. 
     With this structure, the vector data conversion unit performs scaling based on the virtual plate data in addition to the scaling coefficient, thereby realizing optimal scaling on the vector data. This enables display of a high-quality 3D image. 
     Also, in the graphics rendering apparatus relating to the present invention, the polygon may be rectangular, and may have a first side of one pair of opposite sides having a length Lx and a second side of the other pair of opposite sides having a length Ly, the scaling coefficient determination unit determines, as the scaling coefficient, a first scaling coefficient scx that corresponds to scaling in a direction along the first side and a second scaling coefficient scy that corresponds to scaling in a direction along the second side, such that expressions 1 and 2 are satisfied, respectively,
 
 scx=C 1 *Lx (0 &lt;C 1)  [Expression 1]
 
 scy=C 2 *Ly (0 &lt;C 2)  [Expression 2]
 
the virtual plate is rectangular, and has a third side of one pair of opposite sides having a length Lplatex and a fourth side of the other pair of opposite sides having a length Lplatey, and
 
scale x=scx/L plate x   [Expression 3]
 
the vector data conversion unit performs the scaling, based on a first scaling rate scalex with respect to scaling in a direction along the third side that is determined such that an expression 3 is satisfied,
 
scale y=scx/L plate y   [Expression 4]
 
the vector data conversion unit performs the scaling, based on a second scaling rate scaley with respect to scaling in a direction along the fourth side that is determined such that an expression 4 is satisfied.
 
     With this structure, it is possible to scale the virtual plate so as to completely coincide in shape with the polygon. This enables display of a higher-quality 3D image. 
     Also, in the graphics rendering apparatus relating to the present invention, when a rectangular bounding box including the vector image has one pair of opposite sides each having a length VBx and the other pair of opposite sides each having a length VBy, the following conditions may be satisfied,
 
 L plate x=VBx (1+φ)(0&lt;φ&lt;1)  [Expression 5]
 
 L plate y=VBy (1+ψ)(0&lt;ψ&lt;1)  [Expression 6]
 
at least either of expressions 5 or 6 is satisfied, and
 
                     Lplatex   Lplatey     =     Lx   Ly             [     Expression   ⁢           ⁢   7     ]               
an expression 7 is also satisfied.
 
     With this structure, the virtual plate including the rectangular bounding box including the vector image resembles in shape the polygon. Accordingly, it is possible to perform optimal scaling by mapping onto the polygon. This enables display of a higher-quality 3D image. 
     Also, the graphics rendering apparatus relating to the present invention may further comprise a bounding box generation unit operable to generate a rectangular bounding box that includes the polygon based on the polygon data. 
     With this structure, even when the polygon is not rectangular, it is possible to perform scaling on the vector data, which is optimal to mapping onto the polygon. This enables display of a high-quality 3D image. 
     Also, in the graphics rendering apparatus relating to the present invention, when a rectangular bounding box including the bounding box has one pair of opposite sides each having a length PBx and the other pair of opposite sides each having a length PBy, the following conditions may be satisfied,
 
 L plate x=VBx (1+φ)(0&lt;φ&lt;1)  [Expression 8]
 
 L plate y=VBy (1+ψ)(0&lt;ψ&lt;1)  [Expression 9]
 
at least either of expressions 8 or 9 is satisfied, and
 
                     Lplatex   Lplatey     =     PBx   PBy             [     Expression   ⁢           ⁢   10     ]               
an expression 10 is also satisfied.
 
     With this structure, the virtual plate resembles in shape the bounding box including the polygon. Accordingly, it is possible to perform optimal scaling by mapping onto the polygon. This enables display of a higher-quality 3D image. 
     Also, in the graphics rendering apparatus relating to the present invention, the virtual plate may include a plurality of rectangular bounding boxes each including a vector image. 
     With this structure, it is possible to collectively perform scaling processing on a plurality of vector data pieces, thereby improving the processing efficiency. 
     Also, the graphics rendering apparatus relating to the present invention may further comprise: a 3D image processing unit operable to process the input polygon data and including the scaling coefficient determination unit; a 2D image processing unit operable to process the input first vector data and including the vector data conversion unit; a data number counting unit operable to count the number of polygon data pieces input by the 3D image processing unit and the number of first vector data pieces input by the 2D image processing unit; and a processing method setup unit operable to cause the 3D image processing unit to perform processing that is to be performed by the 2D image processing unit, based on the number of polygon data pieces and the number of first vector data pieces counted by the data number counting unit. 
     With this structure, depending on the processing load on the 3D image processing apparatus and the processing load on the 2D image processing apparatus, the processing method setup unit causes the 3D image processing apparatus and the 2D image processing apparatus to share processing that is to be performed by the vector data conversion unit. This can improve the processing capacity. 
     Also, the graphics rendering method relating to the present invention is a graphics rendering method to be executed by a computer, the graphics rendering method comprising: a scaling coefficient determining step of determining, based on polygon data representing a polygon onto which a texture is to be mapped, a scaling coefficient that is a basis for scaling first vector data from which the texture is to be generated; a vector data converting step of generating second vector data by scaling the first vector data based on the scaling coefficient; a texture generating step of generating a texture based on the second vector data; and a texture mapping step of mapping the texture generated in the texture generating step onto the polygon. 
     Also, the graphics rendering program relating to the present invention is a graphics rendering program for causing a computer to execute graphics rendering processing, the graphics rendering processing comprising: a scaling coefficient determining step of determining, based on polygon data representing a polygon onto which a texture is to be mapped, a scaling coefficient that is a basis for scaling first vector data from which the texture is to be generated; a vector data converting step of generating second vector data by scaling the first vector data based on the scaling coefficient; a texture generating step of generating a texture based on the second vector data; and a texture mapping step of mapping the texture generated in the texture generating step onto the polygon. 
     Also, the recording medium relating to the present invention is a recording medium having recorded therein a graphic rendering program for causing a computer to execute graphics rendering processing, the graphics rendering processing comprising: a scaling coefficient determining step of determining, based on polygon data representing a polygon onto which a texture is to be mapped, a scaling coefficient that is a basis for scaling first vector data from which the texture is to be generated; a vector data converting step of generating second vector data by scaling the first vector data based on the scaling coefficient; a texture generating step of generating a texture based on the second vector data; and a texture mapping step of mapping the texture generated in the texture generating step onto the polygon. 
     Also, the integrated circuit relating to the present invention is an integrated circuit for graphics rendering comprising: a scaling coefficient determination unit operable to determine, based on polygon data representing a polygon onto which a texture is to be mapped, a scaling coefficient that is a basis for scaling first vector data from which the texture is to be generated; a vector data conversion unit operable to generate second vector data by scaling the first vector data based on the scaling coefficient; a texture generation unit operable to generate a texture based on the second vector data; and a texture mapping unit operable to map the texture generated by the texture generation unit onto the polygon. 
     With this structure, it is possible to reduce the size of the graphics rendering apparatus. 
    
    
     
       BRIEF DESCRIPTION OF DRAWINGS 
         FIG. 1  is a plane view showing a polygon relating to an Embodiment 1. 
         FIG. 2  shows polygon data relating to the Embodiment 1. 
         FIGS. 3A ,  3 B, and  3 C show vector data relating to the Embodiment 1. 
         FIG. 4  shows vector data relating to the Embodiment 1. 
         FIG. 5  shows the structure of a graphics rendering apparatus relating to the Embodiment 1. 
         FIG. 6  shows model view conversion relating to the Embodiment 1. 
         FIG. 7  shows projection conversion relating to the Embodiment 1. 
         FIG. 8  shows viewport conversion relating to the Embodiment 1. 
         FIG. 9  shows a conception of a scaling value table relating to the Embodiment 1. 
         FIG. 10  shows mapping of a texture onto a polygon relating to the Embodiment 1. 
         FIG. 11  is a flow chart showing operations of the graphics rendering apparatus relating to the Embodiment 1. 
         FIG. 12  is a flow chart showing scaling coefficient calculation processing relating to the Embodiment 1. 
         FIG. 13  is a flow chart showing operations of the graphics rendering apparatus relating to the Embodiment 1. 
         FIG. 14  is a flow chart showing scaling coefficient determination processing relating to the Embodiment 1. 
         FIG. 15  is a flow chart showing scaling coefficient determination processing relating to the Embodiment 1. 
         FIG. 16  shows the structure of a graphics rendering apparatus relating to an Embodiment 2. 
         FIGS. 17A ,  17 B,  17 C, and  17 D explain virtual plate data relating to the Embodiment 2. 
         FIG. 18  is a flow chart showing operations of the graphics rendering apparatus relating to the Embodiment 2. 
         FIG. 19  is a flow chart showing virtual plate generation processing relating to the Embodiment 2. 
         FIG. 20  shows the virtual plate generation processing relating to the Embodiment 2. 
         FIGS. 21A and 21B  show polygon data relating to an Embodiment 3. 
         FIG. 22  shows bounding box data relating to the Embodiment 3. 
         FIG. 23  shows the structure of a graphics rendering apparatus relating to the Embodiment 3. 
         FIG. 24  shows mapping of a texture relating to the Embodiment 3. 
         FIG. 25  is a flow chart showing operations of the graphics rendering apparatus relating to the Embodiment 3. 
         FIG. 26  is a flow chart showing bounding box calculation relating to the Embodiment 3. 
         FIG. 27  shows the structure of a graphics rendering apparatus relating to an Embodiment 4. 
         FIG. 28  is a flow chart showing operations of the graphics rendering apparatus relating to the Embodiment 4. 
         FIG. 29  shows vector data relating to the Embodiment 4. 
         FIG. 30  shows virtual plate generation processing relating to the Embodiment 4. 
         FIG. 31  is a flow chart showing operations of the graphics rendering apparatus relating to the Embodiment 4. 
         FIG. 32  shows operations of a graphics rendering apparatus relating to a modification example. 
         FIGS. 33A and 33B  show operations of a graphics rendering apparatus relating to a modification example. 
         FIG. 34  shows the outline of a conventional example. 
         FIG. 35  shows the outline of a conventional example. 
         FIG. 36  shows the outline of a conventional example. 
     
    
    
     DETAILED DESCRIPTION OF INVENTION 
     Embodiment 1 
     &lt;1&gt; Data 
     &lt;1-1&gt; Polygon Data 
     According to the present embodiment, as shown in  FIG. 1 , polygon data PD 1  representing a shape of an object (hereinafter, “polygon”) PG onto which a texture is mapped is used (see  FIG. 2 ). 
     The polygon data PD 1  is composed of coordinate data Pi (i=0, 1, 2, 3) representing four points on the 2D coordinate system (model coordinate system), as shown in  FIG. 1 . The coordinate data Pi (i=0, 1, 2, 3) is composed of an X component and a Y component, and represents four points p (P 0 (x0,y0)), p (P 1 (x1,y0)), p (P 2 (x1,y1)), and p (P 4 (x0,y1)), as shown in  FIG. 1 . In this case, the polygon data PD 1  is a rectangular polygon PG as indicated by a hatched part in  FIG. 1 . 
     Also, the polygon data PD 1  includes attribute information BD 1  relating to a color value for determining a color of the polygon PG (such as a color value represented in hexadecimal notation) and attribute information BD 2  relating to a planar normal vector N (see  FIG. 1 ) including a polygon PG necessary for rendering. 
     Note that polygon data PD 2 , PD 3 , PD 4 , PD 5 , and PD 6  shown in  FIG. 2  are described later. 
     &lt;1-2&gt; Vector Data 
     Vector data VD 1  (first vector data) is data that defines a shape of a character rendered on a 2D surface as shown in  FIG. 3A . As shown in  FIG. 3B , the vector data VD 1  is composed of coordinate data (hereinafter, “vertice data”) Vi (xi,yi) (i=0, 1, . . . , 44) representing a plurality of vertices p(Vi) (45 vertices in the example shown in  FIG. 3B ) on a contour of the character and coordinate data (hereinafter, “control point data”) Si(xi,yi) (i=0, 1, . . . , 42) representing a control point p(Si) that defines a curved line drawn between adjacent vertices p(Vi) and p(Vi+1) along the contour, as shown in  FIG. 4 . 
     The curved line drawn between adjacent vertices p(Vi) and p(Vi+1) along the contour of the character represented by the vector data VD 1  is a Bezier curve. For example, as shown in  FIG. 3C , a curved line drawn between vertices p(V 13 ) and p(V 14 ) that is inscribed in two sides of a rectangle having vertices p(V 13 ) and p(V 14 ) and a control point p(S 13 ) as three vertices. 
     Also, the vector data VD 1  includes attribute information BD 21  relating to a color value designating a color for filling the inside of a character texture generated based on the vector data VD 1 , as shown in  FIG. 4 . 
     Note that vector data VD 2  and VD 3  shown in  FIG. 4  are described later. 
     &lt;2&gt; Structure 
     A graphics rendering apparatus  10  relating to the present embodiment includes, as shown in  FIG. 5 , a 3D image processing apparatus  10   a  for processing polygon data PD 1  input by a polygon data input unit  210 , a 2D image processing apparatus  10   b  for processing vector data VD 1  (first vector data) input by a vector data input unit  220 , a frame buffer  22 , a texture buffer  21 , a projection conversion matrix setup unit  14   a  for setting up a parameter relating to a projection conversion matrix P for use by the 3D image processing apparatus  10   a.    
     &lt;2-1&gt; 3D Image Processing Apparatus 
     The 3D image processing apparatus  10   a  includes a processor (not shown) and a memory (not shown). The processor appropriately reads and executes a program, thereby realizing a polygon data input reception unit  11 , a 3D coordinate conversion unit  12 , a scaling coefficient determination unit  13 , a projection conversion unit  14 , a polygon rasterization unit  15 , and an image display unit  10   c.    
     &lt;2-1-1&gt; Polygon Data Input Reception Unit 
     The polygon data input reception unit  11  receives the polygon data PD 1  input by a user via the polygon data input unit  210 . 
     Here, the polygon data input reception unit  11  re-arranges received data such that coordinate data having a smallest vertice on the X coordinate and the Y coordinate is P 0  and vertices represented by coordinate data P 1 , P 2 , and P 3  are arranged in a counterclockwise direction of the contour of the polygon PG. Note that  FIG. 2  shows the polygon data PD 1  after re-arrangement. 
     &lt;2-1-2&gt; 3D Image Conversion Unit 
     When the polygon data PD 1  is input by the polygon data input reception unit  11 , the 3D coordinate conversion unit  12  performs an operation for adding 0 as a Z component and 1 as a W component to the coordinate data Pi (i=0, 1, 2, 3) constituting the polygon data PD 1 . Then, the 3D coordinate conversion unit  12  converts the polygon data PD 1  into four pieces of coordinate data represented by the 4D coordinate system, namely, P 0   m  (x0,y0,0,1), P 1   m  (x1,y0,0,1), P 2   m  (x1,y1,0,1), and P 3   m  (x0,y1,0,1) (polygon data PD 2 ) (see  FIG. 2 ). 
     In this way, the polygon data PD 1  is converted into the polygon data PD 2  represented by the 4D coordinate system. As a result, by performing multiplication of the polygon data PD 2  by the matrix, it is possible to represent all of coordinate conversion for translating the polygon PG, coordinate conversion for scaling the polygon PG, and coordinate conversion for rotating the polygon PG. 
     The 3D coordinate conversion unit  12  performs an operation on coordinate data Pim (i=0, 1, 2, 3) constituting the polygon data PD 2  based on an Expression (11) to appropriately perform translation conversion, scaling conversion, or rotation conversion (hereinafter, any combination of these conversions is referred to as “model view conversion”). This results in generation of coordinate data Pie (i=0, 1, 2, 3) representing four vertices of the polygon PG arranged on a 3D space having a view point as an original point (hereinafter, “view coordinate system”) as shown in  FIG. 6 , namely, polygon data PD 3  (see  FIG. 2 ).
 
 Pie=M*Pim  ( i= 0,1,2,3)  [Expression 11]
 
     Here, the value “Pim” (i=0, 1, 2, 3) represents coordinate data constituting the polygon data PD 2 . The value “Pie” (i=0, 1, 2, 3) represents coordinate data constituting the polygon data PD 3 . The sign “M” represents a matrix (model view matrix) on which one of translation conversion, scaling conversion, and rotation conversion, or any combination of these conversions is to be performed. The sign “*” represents multiplication of a matrix by a vector. 
     Then, the 3D coordinate conversion unit  12  inputs the generated polygon data PD 3  into the scaling coefficient determination unit  13 . 
     &lt;2-1-3&gt; Scaling Coefficient Determination Unit 
     The scaling coefficient determination unit  13  calculates a first scaling coefficient scx and a second scaling coefficient scy that are necessary for calculating a scaling rate (scaling value) of the vector data VD 1 . 
     Here, based on information relating to a projection conversion matrix P (described later) input by the projection conversion matrix setup unit  14   a , the scaling coefficient determination unit  13  judges whether the projection conversion matrix P is for perspective projection conversion or for parallel projection conversion. Based on a result of the judgment, the scaling coefficient determination unit  13  changes a calculation method of the first scaling coefficient scx and the second scaling coefficient scy. Note that the scaling coefficient determination unit  13  judges a type of the projection conversion matrix P based on whether the fourth row of the projection conversion matrix P is (0001) or not (see Expressions (16) and (17) shown later). 
     &lt;2-1-3-1&gt; Case where Projection Conversion Matrix P is for Perspective Projection Conversion 
     The scaling coefficient determination unit  13  calculates, with respect to the polygon PG, a length Lx in the X direction (length of a first side) and a length Ly in the Y direction (length of a second side). 
     Here, the polygon data PD 3  input by the scaling coefficient determination unit  13  is generated by the polygon input reception unit  11 , based on the polygon data PD 1  after data re-arrangement as shown in  FIG. 1 . Accordingly, the length Lx of the polygon PG in the X direction is a distance between a vertice p(P 0   e ) and a vertice p(P 1   e ) (or a distance between a vertice p(P 2   e ) and a vertice p(P 3   e )). The length Ly of the polygon PG in the Y direction is a distance between the vertice p(P 1   e ) and the vertice p(P 2   e ) (or a distance between the vertice p(P 0   e ) and the vertice p(P 3   e )). 
     Accordingly, the scaling coefficient determination unit  13  calculates, with respect to the polygon PG, the length Lx in the X direction and the length Ly in the Y direction based on Expressions (12) and (13).
 
 Lx =√{square root over (( X 0 e−X 1 e ) 2 +( Y 0 e−Y 1 e ) 2 +( Z 0 e−Z 1 e ) 2 )}{square root over (( X 0 e−X 1 e ) 2 +( Y 0 e−Y 1 e ) 2 +( Z 0 e−Z 1 e ) 2 )}{square root over (( X 0 e−X 1 e ) 2 +( Y 0 e−Y 1 e ) 2 +( Z 0 e−Z 1 e ) 2 )}  [Expression 12]
 
 Ly =√{square root over (( X 1 e−X 2 e ) 2 +( Y 1 e−Y 2 e ) 2 +( Z 1 e−Z 2 e ) 2 )}{square root over (( X 1 e−X 2 e ) 2 +( Y 1 e−Y 2 e ) 2 +( Z 1 e−Z 2 e ) 2 )}{square root over (( X 1 e−X 2 e ) 2 +( Y 1 e−Y 2 e ) 2 +( Z 1 e−Z 2 e ) 2 )}  [Expression 13]
 
     Here, xie, yie, zie (i=0, 1, 2, 3) represent X, Y, and Z components of each vertice p(Pie) (i=0, 1, 2, 3), respectively. 
     Then, the scaling coefficient determination unit  13  perform operations based on Expressions (14) and (15) using the calculated lengths Lx and Ly of the polygon PG in the X direction and the Y direction, respectively, thereby calculating the first scaling coefficient scx and the second scaling coefficient scy.
 
 scx=A*Lx/Zrep (0 &lt;A,Zrep )  [Expression 14]
 
 scy=B*Ly/Zrep (0 &lt;B,Zrep )  [Expression 15]
 
     Here, the value “A” represents a scaling value of the polygon PG in the X direction for performing viewport conversion. The value “B” represents a scaling value of the polygon PG in the Y direction for performing viewport conversion. The value “Zrep” represents a predetermined coefficient. 
     Also, according to the present embodiment, a value “|Z 0 |”, which is an absolute value of the Z component of a vertice p(P 0   e ) of the polygon PG, is used as Zrep. 
     &lt;2-1-3-2&gt; Case where Projection Conversion Matrix P is for Parallel Projection Conversion 
     The scaling coefficient determination unit  13  performs operations based on Expressions (16) and (17) using the calculated lengths Lx and Ly of the polygon PG in the X direction and the Y direction, respectively, thereby calculating the first scaling coefficient scx and the second scaling coefficient scy.
 
 scx=A*Lx (0 &lt;A )  [Expression 16]
 
 scy=B*Ly (0 &lt;B )  [Expression 17]
 
     Here, the value “A” represents a scaling value of the polygon PG in the X direction for performing viewport conversion. The value “B” represents a scaling value of the polygon PG in the Y direction for performing viewport conversion. 
     Note that the scaling values A and B are stored in the scaling coefficient determination unit  13 . 
     &lt;2-1-4&gt; Projection Conversion Unit 
     The projection conversion unit  14  performs projection conversion on the coordinate data Pie constituting the polygon data PD 3 . 
     Here, based on the information relating to the projection conversion matrix projection P input by the projection conversion matrix setup unit  14   a , the projection conversion unit  14  judges whether the projection conversion matrix P is for perspective projection conversion or for parallel projection conversion. Based on a result of the judgment, the projection conversion unit  14  changes the method of performing an operation on the polygon data PD 3 . Note that the projection conversion unit  14  judges a type of the projection conversion matrix P based on whether the fourth row of the projection conversion matrix P is (0001) or not (see the Expressions (16) and (17)). 
     &lt;2-1-4-1&gt; Case where Projection Conversion Matrix P is for Perspective Projection Conversion 
     The projection conversion unit  14  calculates coordinate data (polygon data PD 4 ) (see  FIG. 2 ) representing vertices of a polygon PG on the view coordinate system that is projected onto a projection surface SC, as shown in  FIG. 7 . The projection surface SC is positioned distant by d from the view point and perpendicular to the z axis. 
     Here, the projection conversion unit  14  performs an operation based on an Expression (18) using the polygon data PD 3  and the projection conversion matrix P to obtain the polygon data PD 4 . 
     
       
         
           
             
               
                 
                   
                     
                       
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     Here, the value “Pic” represents coordinate data constituting the polygon data PD 4 . The value “Pie” represents coordinate data constituting the polygon data PD 3 . The value “P” represents a projection conversion matrix on which perspective projection conversion is to be performed. The sign “*” represents multiplication of a matrix and a vector. 
     Then, the projection conversion unit  14  further performs an operation based on an Expression (19) using the polygon data PD 4  to obtain polygon data PD 5  (see  FIG. 2 ). 
     
       
         
           
             
               
                 
                   
                     
                       
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                                       e 
                                       / 
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                                   ) 
                                 
                               
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                               Pic 
                             
                             = 
                           
                         
                       
                     
                   
                   
                     
                       
                           
                         ⁢ 
                         
                           
                             
                               
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                                   / 
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                                 [ 
                                 
                                   
                                     
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                             ⁢ 
                             
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                                     Xie 
                                   
                                 
                                 
                                   
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                                       / 
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                           = 
                           
                             ( 
                             
                               
                                 
                                   
                                     Xie 
                                     / 
                                     
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                                         / 
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                   [ 
                   
                     Expression 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     19 
                   
                   ] 
                 
               
             
             
               
                 
                   ( 
                   
                     
                       i 
                       = 
                       0 
                     
                     , 
                     1 
                     , 
                     2 
                     , 
                     3 
                   
                   ) 
                 
               
               
                 
                     
                 
               
             
           
         
       
     
     Here, the value “Pie” represents coordinate data constituting the polygon data PD 3 . The value “Pid” represents coordinate data constituting the polygon data PD 5 . The value “d” represents a distance between the view point and the projection surface SC. The sign “*” represents multiplication of a matrix and a vector. 
     As shown in the Expression (19), the coordinate data Pid includes a Z component “Zie”, and accordingly varies depending on the distance Zref between the view point and the polygon PG. 
     &lt;2-1-4-2&gt; Case where Projection Conversion Matrix P is for Parallel Projection Conversion 
     The projection conversion unit  14  performs an operation based on an Expression (20) using the polygon data PD 3  and the projection conversion matrix P to obtain the polygon data PD 5  (see  FIG. 2 ). 
     
       
         
           
             
               
                 
                   
                     
                       
                         Pid 
                         = 
                           
                         ⁢ 
                         
                           
                             ( 
                             
                               
                                 
                                   Xid 
                                 
                               
                               
                                 
                                   Yid 
                                 
                               
                               
                                 
                                   Zid 
                                 
                               
                               
                                 
                                   1 
                                 
                               
                             
                             ) 
                           
                           = 
                           
                             
                               P 
                               * 
                               Pie 
                             
                             = 
                             
                               
                                 
                                   [ 
                                   
                                     
                                       
                                         1 
                                       
                                       
                                         0 
                                       
                                       
                                         0 
                                       
                                       
                                         0 
                                       
                                     
                                     
                                       
                                         0 
                                       
                                       
                                         1 
                                       
                                       
                                         0 
                                       
                                       
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                                   ] 
                                 
                                 ⁢ 
                                 
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                               = 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                           
                         ⁢ 
                         
                           ( 
                           
                             
                               
                                 Xie 
                               
                             
                             
                               
                                 Yie 
                               
                             
                             
                               
                                 0 
                               
                             
                             
                               
                                 1 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   [ 
                   
                     Expression 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     20 
                   
                   ] 
                 
               
             
             
               
                 
                   ( 
                   
                     
                       i 
                       = 
                       0 
                     
                     , 
                     1 
                     , 
                     2 
                     , 
                     3 
                   
                   ) 
                 
               
               
                 
                     
                 
               
             
           
         
       
     
     Here, the value “Pid” represents coordinate data constituting the polygon data PD 5 . The value “Pie” represents coordinate data constituting the polygon data PD 3 . The value “Zrep” represents a distance between the view point and the projection surface SC. The value “P” represents a projection conversion matrix on which parallel projection conversion is to be performed. The sign “*” represents multiplication of a matrix and a vector. 
     As shown in the Expression (20), the coordinate data Pid does not include a Z component Zie, and accordingly does not depend on the distance Zref between the view point and the polygon PG. 
     Also, when generating the polygon data PD 5 , the projection conversion unit  14  performs viewport conversion on the generated polygon data PD 5  to generate polygon data PD 6 . 
     This polygon data PD 6  is composed of coordinate data Piw (i=0, 1, 2, 3), which is obtained by converting the coordinate data Pid (i=0, 1, 2, 3) constituting the polygon data PD 5  into a coordinate system on an actual display screen (“screen coordinate system”) as shown in  FIG. 8 . 
     Also, the projection conversion unit  14  calculates brightness of each vertice p(Pid) of the polygon PG based on the Lambert model, the Phong model, or the like, using attribute information BD 2  relating to a normal vector included in the polygon data PD 5 . 
     Then, the projection conversion unit  14  inputs, into the polygon rasterization unit  15 , the polygon data PD 6  including attribute information BD 3  relating to the brightness of each vertice p(Pid). 
     &lt;3-1-5&gt; Polygon Rasterization Unit 
     The polygon rasterization unit  15  rasterizes a contour of the polygon PG using the polygon data PD 6  input by the projection conversion unit  14 , based on the DDA (Digital Differential Analyzer) method. Also, the polygon rasterization unit  15  generates frame data FD representing a raster image of the polygon PG using attribute information BD 1  relating to a color value, the attribute information BD 3  relating to the brightness, and the texture data TD. Note that the frame data FD is composed of color value data of each of pixels constituting the raster image of the polygon PG. 
     Then, the polygon rasterization unit  15  writes the generated frame data FD into the frame buffer  22 . 
     &lt;2-1-6&gt; Image Display Unit 
     The image display unit  10   c  causes a display  100 , which is connected to the outside, to display 3D image based on the frame data FD stored in the frame buffer  22 . 
     &lt;2-2&gt; Projection Conversion Unit 
     The projection conversion matrix setup unit  14   a  is composed of a touch panel and so on. A user can select, as a projection conversion matrix P, which one of a matrix on which perspective projection conversion is to be performed and a matrix on which parallel projection conversion is to be performed. Then, the projection conversion matrix setup unit  14   a  inputs information relating to the projection conversion matrix P selected by the user, into the scaling coefficient determination unit  13  and the projection conversion unit  14 . 
     &lt;2-3&gt; 2D Image Processing Apparatus 
     A 2D image processing apparatus  10   b  includes a processor (not shown) and a memory (not shown). The processor appropriately reads and executes a program, thereby realizing a vector data input reception unit  16 , a vector data conversion unit  18 , a texture generation unit  19 , and a texture mapping unit  20 . 
     &lt;2-3-1&gt; Vector Data Input Reception Unit 
     The vector data input reception unit  16  receives vector data VD input by the user via the vector data input unit  220 . 
     &lt;2-3-3&gt; Vector Data Conversion Unit 
     When vector data VD 1  is input by the vector data input reception unit  16 , the vector data conversion unit  18  performs an operation so as to add  1  as a z component to each of vertice data Vi (xi,yi) (i=0, 1, . . . , 44) constituting the vector data VD 1  and control point data Si (xi,yi) (i=0, 1, . . . , 42). As a result of this conversion, vector data VD 2  is obtained, which is composed of vertice data Vih (xi,yi,1) (i=0, 1, . . . , 44) that is represented by the 3D coordinate system and control point data Sih (xi,yi,1) (i=0, 1, . . . , 42) (see  FIG. 4 ). 
     Also, the vector data conversion unit  18  generates a scaling matrix S represented by an Expression (21), using a value α acquired from the scaling value storage unit  51  as a scaling rate (scaling value) scalex and scaley of the vector data VD 2  in the X direction and the Y direction. 
     
       
         
           
             
               
                 
                   S 
                   = 
                   
                     
                       [ 
                       
                         
                           
                             scalex 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             scaley 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                     = 
                     
                       [ 
                       
                         
                           
                             α 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             α 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   [ 
                   
                     Expression 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     21 
                   
                   ] 
                 
               
             
           
         
       
     
     In this scaling value determination processing, an optimal scaling value is determined among scaling values included in the scaling value table T stored in the scaling value storage unit  51 . 
     Also, the vector data conversion unit  18  stores therein a determination disable flag F, which is set to “1” when the scaling coefficient cannot be determined. As a result of the scaling value determination processing, when it is judged that the optimal scaling value α is not included in the scaling table T, the vector data conversion unit  18  sets the determination disable flag F to “1”. 
     The vector data conversion unit  18  performs operations (scaling) based on Expressions (22) and (23) using the vector data VD 2  and the scaling matrix S to generate vector data VD 3  (second vector data) (see  FIG. 4 ).
 
 Vit=S*Vih   [Expression 22]
 
 Sit=S*Sih   [Expression 23]
 
     Here, the value “Vih” (i=0, 1, . . . , 44) represents vertice data constituting the vector data VD 2 . The value “Sih” (i=0, 1, . . . , 42) represents control point data constituting the vector data VD 2 . The value “Vit” (i=0, 1, . . . , 44) represents vertice data constituting the vector data VD 3 . The value “Sit” (i=0, 1, . . . , 42) represents control point data constituting the vector data VD 3 . The sign “S” represents a scaling matrix. The sign “*” represents multiplication of a matrix and a vector. 
     Also, the vector data conversion unit  18  performs scaling value determination processing for determining the optimal scaling value α from among scaling values included in the scaling value table T (see  FIG. 9 ) stored in the scaling value storage unit  51 , based on the first scaling coefficient scx and the second scaling coefficient scy input by the scaling coefficient determination unit  13 . The details of the scaling value determination processing are described later. 
     &lt;2-3-4&gt; Texture Generation Unit 
     The texture generation unit  19  performs rasterization processing on the vector data VD 3  to generate a texture to be attached onto the polygon. 
     The texture generation unit  19  acquires the vertice data Vit constituting the vector data VD 3  and the control point data Sit. Then, the texture generation unit  19  approximates a Bezier curve, which is represented by the pex data and the control point data Sit, by an aggregate of minor straight lines to generate outline data composed of an aggregate of pieces of segment data representing the minor straight lines. 
     Then, the texture generation unit  19  generates texture data TD composed of color value data of each of pixels constituting the texture, based on outline data and the attribute information BD 21  relating to a color value included in the vector data VD 1 . Then, the texture generation unit  19  stores the generated texture data TD in the texture buffer  21 . 
     &lt;2-3-5&gt; Texture Mapping Unit 
     The texture mapping unit  20  determines a color value of each pixel of the polygon PG onto which the texture is to be attached, based on the texture data stored in the texture buffer  21 . For example, assume a case where, with respect to the texture represented by the texture data, a pixel on the bottom left is an original point (0,0), the number of pixels in the X direction is TEXW, and the number of pixels in the Y direction is TEXH, as shown in  FIG. 10 . In this case, the color value of each pixel constituting the raster image of the polygon PG is determined such that a pixel located on the coordinate (0,0), a pixel located on the coordinate (TEXW−1,0), a pixel located on the coordinate (TEXW−1,TEXH−1), and a pixel located on the coordinate (0,TEXH−1) are mapped onto the vertice p(P 0   w ), the vertice p(P 1   w ), the vertice p(P 2   w ), and the vertice p(P 3   w ) of the polygon PG, respectively. 
     Here, the texture mapping unit  20  performs perspective correction on the color value of each pixel of the texture, based on the polygon data PD 6  input by the projection conversion unit  14  and the texture data. Also, when the resolution of the raster image of the polygon PG is lower than the resolution of the texture, the texture mapping unit  20  appropriately performs equalization processing to determine the color value of each pixel constituting the raster image of the polygon PG. 
     &lt;3-4&gt; Texture Buffer, Frame Buffer, and Scaling Value Storage Unit 
     The texture buffer  21  and the frame buffer  22  are each composed of a DRAM (Dynamic Random Access Memory) and so on, for example. 
     The texture buffer  21  is a buffer for storing a texture generated by the texture generation unit  19 . 
     Also, the frame buffer  22  is a buffer for writing color data of each pixel generated by the polygon rasterization unit  15 . 
     The scaling value storage unit  51  stores therein the scaling value table T as shown in  FIG. 9 . 
     In the case where a character texture composed of a raster image is scaled, some sort of filtering processing is applied. This tends to cause occurrence of aliasing and character crushing. Compared with this, in the case where processing is performed on vector data without being converted, it is possible to perform scaling with no loss of shape information that is originally included in character data. As a result, by using vector data, even in the case where scaling is performed, it is possible to generate a character texture having a high quality. 
     &lt;3&gt; Operations 
     The following describes, with respect to operations of the graphics rendering apparatus  50  relating to the present embodiment, operations of the 3D image processing apparatus and operations of the 2D image processing apparatus. 
     &lt;3-1&gt; Operations of 3D Image Processing Apparatus 
       FIG. 11  is a flow chart showing the operation of the 3D image processing apparatus relating to the present embodiment. 
     Firstly, when the polygon data input reception unit  11  acquires the polygon data PD 1  (Step S 11 ), the 3D coordinate conversion unit  12  converts the polygon data PD 1  into the 4D coordinate system to generate polygon data PD  2  (Step S 12 ). Furthermore, the polygon data input reception unit  11  performs model view conversion on the polygon data PD  2  to generate polygon data PD 3  (Step S 13 ). 
     Then, the scaling coefficient determination unit  13  determines the first scaling coefficient scx and the second scaling coefficient scy, using the polygon data PD 3  and information relating to projection conversion matrix P input by the projection conversion matrix setup unit  14   a  (Step S 14 ). Then, the scaling coefficient determination unit  13  inputs the determined scx and scy into the vector data conversion unit  18  of the 2D image processing apparatus  10   b.    
     Then, the projection conversion unit  14  performs projection conversion on the polygon data PD 3  to generate polygon data PD 5 , which is composed of coordinate data Pid (i=0, 1, 2, 3) (Step S 15 ), and then performs viewport conversion on the polygon data PD 5  to generate polygon data PD 6 , which is composed of coordinate data Piw (i=0, 1, 2, 3) (Step S 16 ). Then, the projection conversion unit  14  inputs the generated polygon data and PD 6  into the polygon rasterization unit  15 . 
     Lastly, the polygon rasterization unit  15  generates frame data FD using the polygon data PD 6  and texture data TD input by the texture mapping unit  20  (Step S 17 ), and writes the generated frame data FD into the frame buffer  22  (Step S 18 ). 
     Here, before projection conversion is performed by the projection conversion unit  14  (Step S 15 ), the scaling coefficient determination unit  13  calculates the first scaling coefficient scx and the second scaling coefficient scy. Accordingly, compared with the case of calculation of the first scaling coefficient scx and the second scaling coefficient scy using the polygon data PD 5  after projection conversion is performed by the projection conversion unit  14 , it is possible to increase the processing parallelism with the 2D image processing apparatus, thereby increasing the capability of the whole rendering processing. 
     &lt;3-1-1&gt; Scaling Coefficient Calculation 
       FIG. 12  is a flow chart showing operations of scaling coefficient calculation processing. 
     Firstly, the scaling coefficient determination unit  13  calculates, with respect to the polygon PG, the length Lx in the X direction and the length Ly in the Y direction (Step S 41 ). 
     Next, the scaling coefficient determination unit  13  judges whether the information, which relates to the projection conversion matrix P input by the projection conversion matrix setup unit  14   a , relates to a projection conversion matrix (perspective projection conversion matrix) on which perspective projection conversion is to be performed (Step S 42 ). Here, the scaling coefficient determination unit  13  checks whether four components constituting the fourth row of the input projection conversion matrix P are (0,0,0,1). When checking that the four components are (0,0,0,1), the scaling coefficient determination unit  13  judges to perform parallel projection conversion. When checking that the four components are not (0,0,0,1), the scaling coefficient determination unit  13  judges not to perform perspective projection conversion. 
     When judging that the input relation relates to the perspective projection conversion matrix (Step S 42 ), the scaling coefficient determination unit  13  calculates a coefficient Zrep based on the coordinate data Pie constituting the polygon data PD 3  (Step S 43 ). Here, the scaling coefficient determination unit  13  selects coordinate data P 0   e  among the coordinate data Pie constituting the polygon data PD 2 , and adopts a value Z 0   e  of a Z component of the selected coordinate data P 0   e  as the coefficient Zrep. 
     Then, the scaling coefficient determination unit  13  multiplies the length Lpx of the polygon PG in the X direction by a reciprocal of the coefficient Zrep based on the Expression (14), and multiplies the length Lpy of the polygon PG in the Y direction by the reciprocal of the coefficient Zrep based on the Expression (15) (Step S 44 ). Furthermore, the scaling coefficient determination unit  13  multiplies multiplication results of the lengths Lpx and Lpy by the coefficients A and B (A, B&gt;0) to obtain the first scaling coefficient scx and the second scaling coefficient scy, respectively (Step S 45 ). 
     On the contrary, when judging that the information does not relates to the perspective projection conversion matrix (Step S 42 : No), the scaling coefficient determination unit  13  performs multiplication on the coefficient A and the length Lpx of the X direction of the polygon PG based on the Expression (16), and multiplication on the coefficient B and the length Lpy of the Y direction of the polygon PG based on the Expression (17) to calculate the first scaling coefficient scx and the second scaling coefficient scy (Step S 45 ). 
     &lt;3-2&gt; Operations of 2D Image Processing Apparatus 
       FIG. 13  is a flow chart showing the operation of the 2D image processing apparatus relating to the present embodiment. 
     Firstly, the vector data input reception unit  16  acquires vector data VD 1  input by the user via the vector data input unit  220  (Step S 31 ), and inputs the acquired vector data VD 1  into the vector data conversion unit  18 . 
     Next, the vector data conversion unit  18  converts the vector data VD 1  into vector data VD 2  that is represented by the 3D coordinate system (Step S 32 ). 
     Then, the vector data conversion unit  18  performs scaling value determination processing, which is described later, using a value α acquired from the scaling value storage unit  51  (a candidate value for the first scaling value scalex and the second scaling value scaley), the first scaling value scalex, and the second scaling value scaley (Step S 33 ). Then, the vector data conversion unit  18  judges whether the determination disable flag F is set to “1” based on a result of the scaling value determination processing (Step S 34 ). When judging the determination disable flag F is set to “1” (Step S 34 : Yes), the processing ends. 
     On the contrary, when judging that the determination flag F is not set to “1” (Step S 34 : No), the vector data conversion unit  18  generates a scaling matrix S that is represented by an Expression (31), using the determined scaling values scalex and scaley (Step S 35 ). 
     Then, the vector data conversion unit  18  performs operations on the vector data VD 2  based on the Expressions (22) and (23) to convert the vector data VD 2  to generate vector data VD 3  (Step S 36 ). 
     Then, the texture generation unit  19  performs rasterization processing using the vector data VD 3  to generate texture data (Step S 37 ), and stores the generated texture data in the texture buffer  21  (Step S 38 ). 
     Next, the texture mapping unit  20  determines (maps) a color value of each pixel constituting a raster image of the polygon PG, based on the texture data stored in the texture buffer  21  (Step S 39 ). 
     According to the present embodiment, comparatively soon after the 3D image processing apparatus  10   a  starts processing, the first scaling coefficient scx and the second scaling coefficient scy are input into the 2D image processing apparatus  10   b . This enables the 2D image processing apparatus  10   b  to further the processing subsequent to the scaling coefficient determination processing. Accordingly, it is possible to increase the parallelism between the processing of the 3D image processing apparatus  10   a  and the processing of the image processing apparatus  10   b , thereby increasing the processing efficiency of the whole graphics rendering apparatus  50 . 
     &lt;3-2-1&gt; Scaling Value Determination Processing 
       FIG. 14  is a flow chart showing operations of scaling coefficient calculation processing. 
     Firstly, the vector data conversion unit  18  generates gravity coordinate data G (xg,yg) based on 45 pieces of vertice data Vi (xi,yi) (i=0, 1, . . . , 44) constituting the vector data VD 1  (Step S 331 ). 
                   xg   =       (       ∑     i   =   0     44     ⁢           ⁢   xi     )     /   45             [     Expression   ⁢           ⁢   24     ]               yg   =       (       ∑     i   =   0     44     ⁢           ⁢   yi     )     /   45             [     Expression   ⁢           ⁢   25     ]               
The above expressions 24 and 25 are satisfied.
 
     Then, the vector data conversion unit  18  performs translation conversion on the vertice data Vih such that the gravity coordinate G (xg,yg) overlaps an original point (Step S 332 ). Here, Vih=(xi−xg,yi−yg,1) (i=0, . . . , 44) is satisfied. 
     Then, the vector data conversion unit  18  acquires a value (α=0.5) corresponding to an ID “1” included in the scaling value table T (see  FIG. 9 ), and performs an operation based on the Expression (19) to generate vertice data Vit (Step S 333 ). 
     Then, the vector data conversion unit  18  calculates, with respect to the X coordinate of the vertice data Vit, the maximum value xmax and the minimum value xmin, and calculates, with respect to the Y coordinate of the vertice data Vit, the maximum value ymax and the minimum value ymin, using a method described in the item &lt;3-3&gt; later (Step S 334 ). 
     Then, the vector data conversion unit  18  judges whether any of Expressions (26) to (29) is satisfied by the values xmax, xmin, ymax, and ymin, which are calculated using the method described later in the item &lt;4-3&gt;, and the first scaling coefficient scx and the second scaling coefficient scy, which are acquired from the scaling coefficient determination unit  13  (Step S 335 ).
 
 x min&lt;− scx/ 2  [Expression 26]
 
 x max&gt; scx/ 2  [Expression 27]
 
 y min&lt;− scy/ 2  [Expression 28]
 
 y max&gt; scy/ 2  [Expression 29]
 
     Here, when the vector data conversion unit  18  judges that any of the Expressions (26) to (29) is satisfied (Step S 336 : Yes), the scaling value determination processing ends. Here, in the case where any of the Expressions (26) to (29) is satisfied by an ID whose value is smaller by 1 than the value of the ID, which is used for the judgment, corresponding to the scaling value α, the scaling value α corresponding to the ID=5 (α=2.5)) is determined as the optimal value. 
     On the contrary, when judging that all of the Expressions (26) to (29) are not satisfied (Step S 336 : No), the vector data conversion unit  18  further judges whether the value of the ID is smaller than “12”, which is its maximum value (see  FIG. 9 ) (Step S 337 ). 
     When judging that the value of the ID is smaller than “12” (Step S 337 : Yes), the vector conversion unit  18  acquires a scaling value (α=1.0) corresponding to an ID whose value is “2” (Step S 338 ), and then the processing flow proceeds to Step S 334 . 
     In the subsequent processing, the vector conversion unit  18  acquires the value α corresponding to the ID (=“3, 4, . . . , 12”) included in the scaling value table T (see  FIG. 9 ), and performs an operation based on the Expression (19) to generate vertice data Vit and control point data Sit (Step S 334 ). Then, the vector data conversion unit  18  calculates, with respect to the X coordinate of the vertice data Vits, the maximum value xmax and the minimum value xmin, and calculates, with respect to the Y coordinate of the vertice data Vits, the maximum value ymax and the minimum value ymin, using the method described in the item &lt;4-3&gt; later (Step S 335 ). 
     Then, the vector data conversion unit  18  judges whether any of Expressions (45) to (48) is satisfied by the values xmax, xmin, ymax, and ymin, which are calculated using the method described later in the item &lt;4-3&gt;, and the first scaling coefficient scx and the second scaling coefficient scy, which are acquired from the scaling coefficient determination unit  13  (Step S 336 ). When judging that all of the Expressions (45) to (48) are not satisfied (Step S 336 : No), the vector data conversion unit  18  further judges whether the value of the ID is smaller than the value “12” (Step S 337 ). When judging that the value of the ID is smaller than the value “12” (Step S 337 : Yes), the vector data conversion unit  18  acquires a scaling value α corresponding to ID+1. Then, the processing flow proceeds to S 334 . 
     On the contrary, when judging that the value of the ID is equal to or greater than the value “12” (Step S 336 : No), the vector data conversion unit  18  sets the determination disable flag F to “1”, and the processing ends. 
     &lt;3-3&gt; Calculation of Values xmax, xmin, ymax, and ymin 
       FIG. 15  is a flow chart of calculation processing of the values xmax, xmin, ymax, and ymin. Here, Vitx represents an x component of the vertice data Vit, and Vity represents a y component of the vertice data Vit. 
     In the processing of the values xmax, xmin, ymax, and ymin, the vector data conversion unit  18  performs the following processing. 
     Firstly, the vector data conversion unit  18  stores α value V 0   tx , which represents the x component of the vertice data V 0   t  constituting the vector data VD 1  after conversion, in variables xmin and xmax defined on a memory (calculation buffer), and stores a value V 0   ty , which represents a y component of the vertice data V 0   t , in variables ymin and ymax. As a result, the vertice data V 0   t  is initialized (Step S 3341 ). 
     Then, the vector data conversion unit  18  stores a value “1” in a variable “i” defined on the calculation buffer, and then repeatedly performs the following processing until a value “44” is stored in the v variable “i” (Steps S 3342  to S 3351 ). 
     Firstly, the vector data conversion unit  18  compares the variable xmin with the value Vitx (Step S 3343 ). When the variable xmin is greater than the value Vitx, the vector data conversion unit  18  stores the value Vitx in the variable xmin (Step S 3344 ). 
     Next, the vector data conversion unit  18  compares the variable xmax with the value Vitx (Step S 3345 ). When the variable xmax is smaller than the value Vitx, the vector data conversion unit  18  stores the value Vitx in the variable xmax (Step S 3346 ). 
     Next, the vector data conversion unit  18  compares the variable ymin with the value Vity (Step S 3347 ). When the variable ymin is greater than the value Vity, the vector data conversion unit  18  stores the value yi in the variable ymin (Step S 3348 ). 
     Next, the vector data conversion unit  18  compares the variable ymax with the value Vity (Step S 3349 ). When the variable ymax is smaller than the value Vity, the vector data conversion unit  18  stores the value Vity in the variable ymax (Step S 3350 ). 
     Then, the vector data conversion unit  18  increments the variable “i” (Step S 3351 ). 
     The variables xmin, ymin, xmax, and variable eventually held after repetitive performance of the above processing are used for performing Step S 336 . 
     Embodiment 2 
     &lt;1&gt; Data 
     &lt;1-1&gt; Polygon Data 
     According to the present embodiment, in the same way as in the Embodiment 1, polygon data PD 1  is used, which represents a shape of a rectangular polygon PG onto which a texture is to be mapped, as shown in  FIG. 1 . 
     &lt;1-2&gt; Vector Data 
     Vector data VD 1  (first vector data) is data that defines a shape of a character rendered on a 2D surface as shown in  FIG. 2A , in the same way as in the Embodiment 1. As shown in  FIG. 2B , the vector data VD 1  is composed of coordinate data (hereinafter, “vertice data”) Vi (xi,yi) (i=0, 1, . . . , 44) representing a plurality of vertices p(Vi) (45 vertices in the example shown in  FIG. 3B ) on a contour of the character and coordinate data (hereinafter, “control point data”) Si(xi,yi) (i=0, 1, . . . , 42) representing a control point p(Si) that defines a curved line drawn between adjacent vertices p(Vi) and p(Vi+1) along the contour (see  FIG. 4 ). 
     &lt;2&gt; Structure 
     A graphics rendering apparatus  10  relating to the present embodiment includes, as shown in  FIG. 16 , a 3D image processing apparatus  10   a  for processing polygon data PD 1  input by a polygon data input unit  210 , a 2D image processing apparatus  10   b  for processing vector data VD 1  (first vector data) input by a vector data input unit  220 , a frame buffer  22 , a texture buffer  21 , a projection conversion matrix setup unit  14   a  for setting up a parameter relating to a projection conversion matrix P for use by the 3D image processing apparatus  10   a . Note that the projection conversion matrix setup unit  14   a , the texture buffer  21 , and the frame buffer  22  in the present embodiment have the same structure as those in the Embodiment 1, and accordingly descriptions thereof are omitted here. 
     &lt;2-1&gt; 3D Image Processing Apparatus 
     The 3D image processing apparatus  10   a  includes a processor (not shown) and a memory (not shown). The processor appropriately reads and executes a program, thereby realizing a polygon data input reception unit  11 , a 3D coordinate conversion unit  12 , a scaling coefficient determination unit  13 , a projection conversion unit  14 , a polygon rasterization unit  15 , and an image display unit  10   c  for causing a display  100 , which is connected to the outside, to display 3D image based on 3D image data stored in the frame buffer  22 . Note that the 3D coordinate conversion unit  12 , the projection conversion unit  14 , the scaling coefficient determination unit  13 , and the polygon rasterization unit  15  in the present embodiment have the same structure as those in the Embodiment 1, and accordingly descriptions thereof are omitted here. 
     The polygon data input reception unit  11  relating to the present embodiment has substantially the same structure as the polygon data input unit  11  described in the Embodiment 1, and inputs polygon data PD 1  into a virtual plate generation unit  17  which is described later, in addition to the 3D coordinate conversion unit  12 . 
     &lt;2-2&gt; 2D Image Processing Apparatus 
     A 2D image processing apparatus  10   b  includes a processor (not shown) and a memory (not shown). The processor appropriately reads and executes a program, thereby realizing a vector data input reception unit  16 , a virtual plate generation unit  17 , a vector data conversion unit  18 , a texture generation unit  19 , and a texture mapping unit  20 . Note that the vector data input reception unit  16 , the texture generation unit  19 , and the texture mapping unit  20  in the present embodiment have the same structure as those in the Embodiment 1, and accordingly descriptions thereof are omitted here. 
     &lt;2-2-1&gt; Virtual Plate Generation Unit 
     The virtual plate generation unit  17  generates virtual plate data (see  FIGS. 7A-17D ) based on vector data VD 1  and polygon data PD 1 . The virtual plate data is composed of a length Lplatex in the x direction (a length of a third side) and a length Lplatey in the y direction (a length of a fourth side) of a rectangular virtual plate Plate ( FIG. 17B ) and coordinate data Vplatei (i=0, 1, 2, 3) of a vertice p(Vplatei) (i=0, 1, 2, 3) of the virtual plate Plate. 
     Here, the virtual plate Plate includes a vector image represented by the vector data VD 1 . 
                     Lplatex   Lplatey     =     Lx   Ly             [     Expression   ⁢           ⁢   30     ]               
The above expression is satisfied by the length Lplatex in the x direction and the length Lplatey of the length in the y direction of the virtual plate Plate and the length Lx in the x direction and the length Ly in the y direction of the polygon PG.
 
     Also, the virtual plate generation unit  17  inputs, into the vector data conversion unit  18 , the calculated coordinate data Vplatek (k=0, 1, 2, 3) of the four vertices of the rectangular virtual plate Plate and the length Lplatex in the x direction and the length Lplatey in the y direction of the virtual plate Plate. 
     &lt;2-2-2&gt; Vector Data Conversion Unit 
     The vector data conversion unit  18  converts, into data represented by the three dimension coordinate system, the vertice data Vi (i=0, 1, . . . , 44) and the control point data Si (i=0, 1, . . . , 42) constituting the vector data VD 1  and the coordinate data Vplatek (k=0, 1, 2, 3) of the vertices of the virtual plate Plate. As a result of the conversion, vector data VD 2  (see  FIG. 4 ), which is composed of vertice data Vih (i=0, 1, . . . , 44) and control point data Sih (i=0, 1, . . . , 42), and coordinate data Vplatekh (k=0, 1, 2, 3) (see  FIG. 18 ) are generated. 
     Also, the vector data conversion unit  18  calculates a first scaling ratio (first scaling value) scalex and a second scaling ratio (second scaling value) scaley based on Expressions (31) and (32), using the length Lplatex in the x direction and the length Lplatey in the y direction of the virtual plate Plate, and the first scaling coefficient scx and the second scaling coefficient scy input by the scaling coefficient determination unit  13 .
 
scale x=scx/L plate x   [Expression 31]
 
scale y=scy/L plate y   [Expression 32]
 
     Then, the vector data conversion unit  18  generates a scaling matrix S represented by an Expression (33) using the calculated scaling values scalex and scaley. 
     
       
         
           
             
               
                 
                   
                     
                       
                         S 
                         = 
                         
                           [ 
                           
                             
                               
                                 scalex 
                               
                               
                                 0 
                               
                               
                                 0 
                               
                             
                             
                               
                                 0 
                               
                               
                                 scaley 
                               
                               
                                 0 
                               
                             
                             
                               
                                 0 
                               
                               
                                 0 
                               
                               
                                 1 
                               
                             
                           
                           ] 
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           [ 
                           
                             
                               
                                 
                                   scx 
                                   / 
                                   Lplatex 
                                 
                               
                               
                                 0 
                               
                               
                                 0 
                               
                             
                             
                               
                                 0 
                               
                               
                                 
                                   scy 
                                   / 
                                   Lplatey 
                                 
                               
                               
                                 0 
                               
                             
                             
                               
                                 0 
                               
                               
                                 0 
                               
                               
                                 1 
                               
                             
                           
                           ] 
                         
                       
                     
                   
                 
               
               
                 
                   [ 
                   
                     Expression 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     33 
                   
                   ] 
                 
               
             
           
         
       
     
     The vector data conversion unit  18  performs operations (scaling) with respect to vertice data Vih (i=0, 1, . . . , 44) and control point data Sih (i=0, 1, . . . , 42) constituting the vector data VD 2  and the coordinate data Vplatekh (k=0, 1, 2, 3) of the four vertices of the virtual plate Plate, based on the Expressions (34), (35), and (36), respectively. As a result, vector data VD 3  (second vector data), which is composed of vertice data Vit (i=0, 1, . . . , 44) and control point data Sit (i=0, 1, . . . , 42), and coordinate data Vplatekt are generated (see  FIG. 4  and  FIG. 18 ).
 
 Vit=S*Vih   [Expression 34]
 
 Sit=S*Sih   [Expression 35]
 
 V plate kt=S*V plate kh   [Expression 36]
 
     Here, the sign “*” represents multiplication of a matrix and a vector. 
     &lt;3&gt; Operations 
     The following describes operations of the graphics rendering apparatus relating to the present embodiment. Note that the operations of the 3D image processing apparatus  10   a  are the same as those relating to the Embodiment 1, and accordingly description thereof is omitted here. 
     &lt;3-1&gt; Operations of 2D Image Processing Apparatus 
       FIG. 19  is a flow chart showing operation of the 2D image processing apparatus relating to the present embodiment. 
     Firstly, when the vector data input reception unit  16  acquires vector data VD 1  (Step S 21 ), the virtual plate generation unit  17  performs virtual plate generation processing, which is described later, using the vector data VD 1  and polygon data PD 1  input by the polygon data input reception unit  11  (Step S 22 ). Then, the virtual plate generation unit  17  inputs, into the vector data conversion unit  18 , the calculated coordinate data Vplatek (k=0, 1, 2, 3) of the vertices of the virtual plate Plate and the length Lplatex in the x direction and the length Lplatey in the y direction of the virtual plate Plate. 
     Then, the vector data conversion unit  18  generates vector data VD 2  represented by the 3D coordinate system and coordinate data Vplatekh (k=0, 1, 2, 3) using the vector data VD 1  and the coordinate data Vplatek (k=0, 1, 2, 3) of the vertices of the virtual plate Plate (Step S 23 ). 
     Then, based on the Expressions (31) and (32), the vector data conversion unit  18  calculates the first scaling value scalex and the second scaling value scaley, using the length Lplatex in the x direction and the length Lplatey in the y direction of the virtual plate Plate and the first scaling coefficient scx and the second scaling coefficient scy. Then, the vector data conversion unit  18  generates a scaling matrix S represented by the Expression (33) (Step S 24 ). 
     Then, the vector data conversion unit  18  performs operations on the vector data VD 2  and the coordinate data Vplateih (i=0, 1, 2, 3) of the vertices of the virtual plate Plate based on the Expressions (24) to (26). The calculations result in generation of vector data VD 3 , which is composed of vertice data Vit (i=0, 1, . . . , 44) and control point data Sit (i=0, 1, . . . , 42), and coordinate data Vplatekt (k=0, 1, 2, 3) (Step S 25 ). The vector data conversion unit  18  inputs the generated vector data VD 3  and coordinate Vplatekt into the texture generation unit  19  (k=0, 1, 2, 3). 
     Then, the texture generation unit  19  performs rasterization processing using the vector data VD 3  and the coordinate data Vplatekt (k=0, 1, 2, 3) representing the vertices of the virtual plate Plate to generate texture data TD (Step S 26 ), and then writes the generated texture data into the texture buffer  21  (Step S 27 ). 
     Next, the texture mapping unit  20  determines (maps) a color value of each pixel constituting a raster image of the polygon PG, based on texture data stored in the texture buffer  21  (Step S 28 ). 
     &lt;3-2-1&gt; Virtual Plate Generation Processing 
       FIG. 20  is a flow chart showing operations of scaling coefficient calculation processing. 
     Firstly, the virtual plate generation unit  17  performs bounding box calculation processing to calculate a bounding box, which includes an aggregate of pieces of vertice data input by the vector data input reception unit  16  (Step S 81 ). 
     In the bounding box calculation processing, the virtual plate generation unit  17  performs the following processing based on the method which is described in the item &lt;4-3&gt; in the Embodiment 1 with reference to  FIG. 15 . As a result, variables xmin, xmax, ymin, and ymax are obtained. 
     Firstly, the virtual plate generation unit  17  stores a value V 0   tx , which represents an x component of the vertice data V 0   t  constituting the vector data VD 1  after conversion, in variables xmin and xmax defined on the memory (calculation buffer), and stores a value V 0   ty , which represents a y component of the vertice data V 0   t , in variables ymin and ymax. As a result, the vertice data V 0   t  is initialized (Step S 3341  in  FIG. 15 ). 
     Then, the virtual plate generation unit  17  stores a value “1” in a variable “i” defined on the calculation buffer, and then repeatedly performs the following processing until a value “44” is stored in the variable “i” (Steps S 3342  to S 3351  in  FIG. 15 ). 
     Firstly, the virtual plate generation unit  17  compares the variable xmin with a value Vitx (Step S 3343  in  FIG. 15 ). When the variable xmin is greater than the value Vitx, the virtual plate generation unit  17  stores the value Vitx in the variable xmin (Step S 3344  in  FIG. 15 ). 
     Then, the virtual plate generation unit  17  compares the variable xmax with the value Vitx (Step S 3345  in  FIG. 15 ). When the variable xmax is smaller than the value Vitx, the virtual plate generation unit  17  stores the value Vitx in the variable xmax (Step S 3346  in  FIG. 15 ). 
     Next, the virtual plate generation unit  17  compares the variable ymin with the value Vity (Step S 3347  in  FIG. 15 ). When the variable ymin is greater than the value Vity, the virtual plate generation unit  17  stores the value Vity in the variable ymin (Step S 3348  in  FIG. 15 ). 
     Next, the virtual plate generation unit  17  compares the variable ymax with the value Vity (Step S 3349  in  FIG. 15 ). When the variable ymax is smaller than the value Vity, the virtual plate generation unit  17  stores the value Vity in the variable ymax (Step S 3350  in  FIG. 15 ). 
     Then, the virtual plate generation unit  17  increments the variable “i” by 1 (Step S 3351  in  FIG. 15 ). 
     After repeatedly performing the above processing, the virtual plate generation unit  17  determines, as shown in  FIG. 17A , as a bounding box VB, a rectangular area (hatched part in  FIG. 17A ) specified by four vertices p(VB 0 ) (xmin,ymin), p(VB 1 ) (xmax,ymin), p(VB 2 ) (xmax,ymax), and p(VB 3 ) (xmin,ymax), using the variables xmin, ymin, xmax, and ymax, which are eventually stored. 
     Next, the virtual plate generation unit  17  sets a margin mx in the x direction of the virtual plate Plate with respect to the bounding box VB to δ*VBx (Step S 82 ). The value “VBx” represents the length in the x direction of the bounding box VB, the value “δ” represents a constant that falls within a range of 0 to 1 inclusive. 
     Then, the virtual plate generation unit  17  calculates the length Lplatex in the x direction and the length Lplatey in the y direction of the rectangular virtual plate Plate, using the length VBx in the x direction, the length VBy in the y direction, and the margin mx in the x direction of the bounding box VB, and the polygon data PD 1  (Step S 83 ). 
     Here, firstly, the virtual plate generation unit  17  calculates the length Lx in the x direction and the length Ly in the y direction of the polygon PG, using the polygon data PD 1 . Here, the polygon data PD 1  is rearranged by the polygon data input reception unit  11 . Accordingly, the virtual plate generation unit  17  calculates, as the length Lx in the x direction of the polygon, a distance between a point p(P 0 ) and a point p(P 1 ) (or a distance between point p(P 2 ) and a point p(P 3 )). Also, the virtual plate generation unit  17  calculates, as the length Ly in the y direction of the polygon, a distance between the point p(P 1 ) and the point p(P 2 ) (or a distance between the point p(P 0 ) and the point p(P 3 )). In other words, the virtual plate generation unit  17  calculates the length Lx in the x direction and the length Ly in the y direction of the polygon based on Expressions (37) and (38), respectively.
 
 Lx=|X 0 −X 1|  [Expression 37]
 
 Ly=|Y 1 −Y 2|  [Expression 38]
 
     Then, the virtual plate generation unit  17  determines the length Lplatex in the x direction and the length Lplatey in the y direction of the virtual plate Plate such that Expressions (39) and (40) are satisfied, respectively, as shown in  FIG. 17B .
 
 L plate x=VBx+mx=VBx (1+δ)  [Expression 39]
 
 L plate y=L plate x*Ly/Lx=VBx (1+δ)* Ly/Lx   [Expression 40]
 
     The virtual plate generation unit  17  judges whether the length VBy (=ymax−ymin) in the y direction of the bounding box VB is longer than the length Lplatey in the y direction of the virtual plate Plate (Step S 84 ). Here, the judgment is performed by judging whether the virtual plate Plate is large enough to include the vector graphics represented by the vector data VD 1 . 
     When judging that the length VBy (=ymax−ymin) in the y direction of the bounding box VB is shorter than the length Lplatey in the y direction of the virtual plate Plate (Step S 84 : No), the virtual plate generation unit  17  generates pieces of vertice data of Vplate 0  (xmin−mx/2,ymin−my/2), Vplate 1  (xmax+mx/2,ymin−my/2), Vplate 2  (xmax+mx/2,ymax+my/2), and Vplate 3  (xmin−mx/2,ymax+my/2), which correspond to four vertices p(Vplatei) (i=0, 1, 2, 3) of the virtual plate Plate, respectively (Step S 87 ). 
     On the contrary, when judging that the length VBy (=ymax−ymin) in the y direction of the bounding box VB is equal to or longer than the length Lplatey in the y direction of the virtual plate Plate (Step S 84 : Yes), the virtual plate generation unit  17  sets a margin my in the y direction of the virtual plate Plate to the bounding box VB to p*VBy (Step S 85 ). The value VBy represents the length of the bounding box VB in the y direction, and the value ρ represents a constant that falls within a range of 0 to 1 inclusive. 
     Then, the virtual plate generation unit  17  sets the length Lplatex in the x direction and the length Lplatey in the y direction of the virtual plate Plate such that Expressions (41) and (42) are satisfied, as shown in  FIG. 17D  (Step S 86 ).
 
 L plate x=L plate y*Lx/Ly=VBy (1+ρ)* Lx/Ly   [Expression 41]
 
 L plate y=VBy+mx=VBy (1+ρ)  [Expression 42]
 
     Lastly, the virtual plate generation unit  17  generates four pieces of vertice data Vplate 0  (xmin−mx/2,ymin−my/2), Vplate 1  (xmax+mx/2,ymin−my/2), Vplate 2  (xmax+mx/2,ymax+my/2), and Vplate 3  (xmin−mx/2,ymax+my/2) corresponding to the four vertices p(Vplatei), respectively (Step S 87 ). 
     Embodiment 3 
     &lt;1&gt; Data 
     &lt;1-1&gt; Polygon Data 
     The polygon data PD 1  relating to the present embodiment is composed of 11 pieces of coordinate data Pi (i=0, 1, . . . , 10) (see  FIG. 22 ) representing a polygon having 11 vertices p(Pi) (i=0, 1, . . . , 10), as shown in  FIG. 21A . 
     &lt;1-2&gt; Vector Data 
     Vector data VD 1  (first vector data) defines a shape of a character rendered on a two-dimensional surface as shown in  FIG. 3A , in the same way as in the Embodiment 1. As shown in  FIG. 3B , the vector data VD 1  is composed of a plurality of pieces (45 in the example in  FIG. 3B ) of coordinate data Vi (xi,yi) (i=0, 1, . . . , 44) of a vertice p(Vi) (hereinafter, “vertice data”) on a contour of a character and a plurality of pieces of coordinate data Si(xi,yi) (i=0, 1, . . . , 42) of a control point p(Si) (hereinafter, “control point data”) defining a curved line drawn between adjacent vertices p(Vi) and p(Vi+1) along the contour (see  FIG. 4 ). 
     &lt;2&gt; Structure 
     A graphics rendering apparatus  30  relating to the present embodiment has substantially the same structure as in the Embodiment 2, as shown in  FIG. 23 , and differs from that in the Embodiment 2 in structure of a 3D image processing apparatus  10   a  for processing polygon data PD 1  input by the polygon data input unit  210 . Note that the compositional elements of the graphics rendering apparatus  30  that are the same those in the Embodiment 2 have the same referential numerals and descriptions thereof are omitted. 
     &lt;2-1&gt; 3D Image Processing Apparatus 
     The 3D image processing apparatus  10   a  has substantially the same structure as that in the Embodiment 2. A processor (not shown) appropriately reads a program into a memory (not shown) and executes the read program, thereby realizing a bounding box generation unit  31 , a 3D coordinate conversion unit  12 , a scaling coefficient determination unit  13 , a projection conversion unit  14 , and so on of the 3D image processing apparatus  10   a . With respect to the structure of the 3D image processing apparatus  10   a  that is the same as that in the Embodiment 2, description is appropriately omitted. 
     &lt;2-1-1&gt; Bounding Box Generation Unit 
     The bounding box generation unit  31  generates bounding box data PBi (i=0, 1, 2, 3) (see  FIG. 22 ) representing vertices p(PBi) of a rectangular bounding box PB including a polygon having 11 vertices represented by polygon data PD 1 , as shown in  FIG. 21B . 
     &lt;2-1-2&gt; 3D Image Conversion Unit 
     The 3D coordinate conversion unit  12  converts the bounding box data PBi into data represented by a 4D coordinate system as well as the polygon data PD 1  to generate bounding box data PBim (i=0, 1, 2, 3), and performs model view conversion on the bounding box data PBim to generate bounding box data PBie (i=0, 1, 2, 3) (see  FIG. 22 ). Also, the 3D coordinate conversion unit  12  inputs the bounding box data PBie into the scaling coefficient determination unit  13 . 
     &lt;2-1-3&gt; Scaling Coefficient Determination Unit 
     The scaling coefficient determination unit  13  determines a scaling coefficient in the same method as in the Embodiment 1, using third bounding box data PBie input by the 3D coordinate conversion unit  12 . Also, the scaling coefficient determination unit  13  inputs the first scaling coefficient scx and the second scaling coefficient scy into the vector data conversion unit  18 . 
     &lt;2-1-4&gt; Projection Conversion Unit 
     The projection conversion unit  14  performs coordinate conversion on the bounding box data PBie as well as the polygon data PD 3  to calculate coordinate data of a vertice p(PBiw) (i=0, 1, 2, 3) on the screen coordinate system, and inputs the calculated coordinate data into the polygon rasterization unit  15 . 
     &lt;2-2&gt; 2D Image Processing Apparatus 
     The 2D image processing apparatus  10   b  has substantially the same structure as in the Embodiment 2. A processor (not shown) appropriately reads a program into a memory (not shown) and executes the read program, thereby realizing a virtual plate generation unit  17 , a texture mapping unit  20 , and so on of the 2D image processing apparatus  10   b . With respect to the structure of the 2D image processing apparatus  10   b  that is the same as that in the Embodiment 2, description is appropriately omitted. 
     &lt;2-2-1&gt; Virtual Plate Generation Unit 
     The virtual plate generation unit  17  generates a virtual plate in the same procedure as in the Embodiment 2, using the bounding box data PBi. Also, the virtual plate generation unit  17  inputs the virtual plate data into the 2D coordinate conversion unit  18 . 
     &lt;2-2-2&gt; Texture Mapping Unit 
     The texture mapping unit  20  determines a color value of each pixel of the polygon PG onto which a texture is to be attached, based on texture data stored in the texture buffer  21 . For example, assume a case where, with respect to the texture represented by the texture data, a pixel on the bottom left is an original point (0,0), the number of pixels in the X direction is TEXW, and the number of pixels in the Y direction is TEXH, as shown in  FIG. 24 . In this case, the color value of each pixel constituting the raster image of the polygon PG is determined such that a pixel located on the coordinate (0,0), a pixel located on the coordinate (TEXW−1,0), a pixel located on the coordinate (TEXW−1,TEXH−1), and a pixel located on the coordinate (0,TEXH−1) are mapped onto the vertice p(PB 0   w ), the vertice p(PB 1   w ), the vertice p(PB 2   w ), and the vertice p(PB 3   w ) of the polygon PG, respectively. 
     &lt;3&gt; Operations 
     The operations of the graphics rendering apparatus  30  relating to the present embodiment are substantially the same as those in the Embodiment 2. The operations of the 3D image processing apparatus  10   a  relating to the present embodiment differ from those in the Embodiment 2. 
     &lt;3-1&gt; Operations of 3D Image Processing Apparatus 
       FIG. 25  is a flow chart showing operations of the 3D image processing apparatus relating to the present embodiment. 
     Firstly, when the polygon data input reception unit  11  acquires polygon data PD 1  (Step S 311 ), the bounding box generation unit  31  calculates bounding box data PBi (i=0, 1, 2, 3) representing a bounding box PB including a polygon PG represented by the polygon data PD 1  (Step S 312 ). The calculation processing of the bounding box data PBi performed by the bounding box generation unit  31  is described in detail later in the &lt;3-2&gt; Bounding Box Calculation. 
     Next, the 3D coordinate conversion unit  12  converts the bounding box data PBi into data represented by the 4D coordinate system to generate bounding box data PBim (i=0, 1, 2, 3) as shown in  FIG. 22  (Step S 312 ). Furthermore, the 3D coordinate conversion unit  12  performs model view conversion on the bounding box data PBi to generate bounding box data PBie (i=0, 1, 2, 3) (Step S 313 ). 
     Then, the scaling coefficient determination unit  13  determines the first scaling coefficient scx and the second scaling coefficient scy, using the bounding box data PBie and information relating to a projection conversion matrix P input by the projection conversion matrix setup unit  14   a  (Step S 314 ). Then, the scaling coefficient determination unit  13  inputs the determined scx and scy into the vector data conversion unit  18 . 
     Then, the projection conversion unit  14  performs projection conversion or the like on the bounding box data PBie to generate bounding box data PBid (i=0, 1, 2, 3) (see  FIG. 22 ) (Step S 315 ). Then, the projection conversion unit  14  performs viewport conversion on the viewport bounding box data PBid to generate bounding box data PBiw (i=0, 1, 2, 3) (see  FIG. 22 ) (Step S 316 ), and inputs the generated bounding box data PBiw (i=0, 1, 2, 3) into the polygon rasterization unit  15 . 
     Lastly, the polygon rasterization unit  15  generates frame data FD using the polygon data PD 6  and texture data TD input by the texture mapping unit  20  (Step S 317 ), and writes the generated frame data FD into the frame buffer  22  (Step S 318 ). 
     &lt;3-2&gt; Bounding Box Calculation 
     The following describes the operations of the bounding box calculation processing performed by the graphics rendering apparatus  30  relating to the present embodiment, with reference to the flow chart shown in  FIG. 26 . 
     Firstly, the polygon data input reception unit  11  acquires polygon data PD 1  input by the user, and inputs the polygon data PD 1  into the bounding box generation unit  31 . 
     The bounding box generation unit  31  stores a value x0 on variables xmin and xmax defined on a calculation buffer, stores a value y0 on variables ymin and ymax, and stores a value “1” on a variable i to initialize the values (Step S 811 ). 
     Next, the bounding box generation unit  31  judges whether a value “11” that is the total number of vertices p(Pi) is stored in the variable “i” (Step S 812 ). 
     If the bounding box generation unit  31  judges that an integer “10” is stored in the variable “i” (the processing of determining four vertices of the rectangular bounding box PB completes) (Step S 812 : Yes), the bounding box calculation processing ends. 
     On the contrary, when judging that integer “10” is not stored in the variable “i” (the processing of determining four vertices of the rectangular bounding box PB does not complete) (Step S 812 : No), the bounding box generation unit  31  compares the variable xmin with the value xi (Step S 813 ). When judging that the value xi is smaller than the variable xmin (Step S 813 : Yes), the bounding box generation unit  31  updates the variable xmin with the value xi (Step S 814 ). On the contrary, when judging that the value xi is equal to or greater than the variable xmin (Step S 813 : No), the bounding box generation unit  31  does not updates the variable xmin 
     Next, the bounding box generation unit  31  compares the variable xmax with the value xi (Step S 815 ). When judging that the value xi is greater than the variable xmax (Step S 815 : Yes), the bounding box generation unit  31  updates the variable xmax with the value xi (Step S 816 ). On the contrary, when judging that the value xi is equal to or smaller than the variable xmax (Step S 815 : No), the bounding box generation unit  31  does not updates the variable xmax. 
     Next, in Steps  817  to  820 , the bounding box generation unit  31  replaces the variable xmin, the variable xmax, and the value xi with the variable ymin, the variable ymax, and the value yi, respectively, and performs processing in the same way as in Steps S 813  to S 816 . 
     Then, the bounding box generation unit  31  increments the value “i” (Step S 821 ). 
     After repeatedly performing the above processing, the bounding box generation unit  31  repeats the processing of Steps S 812  to S 821  until the variable “i” is set to the integer “10”, using the variables xmin, ymin, xmax, and ymax that are eventually held therein. This results in calculation of pieces of coordinate data PB 0  (xmin,ymin), PB 1  (xmax,ymin), PB 2  (xmax,ymax), and PB 3  (xmin,ymax) of four vertices p(PBi) (i=0, 1, 2, 3) constituting the bounding box. Also, the bounding box generation unit  31  inputs the calculated coordinate data Bi (i=0, 1, 2, 3) of four vertices p(PBi) (i=0, 1, 2, 3) into the 3D coordinate conversion unit  12  and the virtual plate generation unit  17 . 
     Next, the virtual plate generation unit  17  generates a virtual plate Plate in the same method as in the Embodiment 1, using bounding box data PBi (i=0, 1, 2, 3) instead of the vertice data Pi (i=0, 1, 2, 3) of the polygon data  23  (see &lt;3-2-1&gt; Virtual Plate Generation Processing in the Embodiment 1 and  FIG. 20 ). 
     Embodiment 4 
     &lt;1&gt; Data 
     &lt;1-1&gt; Polygon Data 
     According to the present embodiment, in the same way as in the Embodiment 1, polygon data PD 1  (see  FIG. 2 ) is used, which represents a shape of a rectangular polygon PG as shown in  FIG. 1 . 
     &lt;1-2&gt; Vector Data 
     Vector data VD 1  (first vector data) defines a shape of a character rendered on a two-dimensional surface as shown in  FIG. 3A , in the same way as in the Embodiment 1. As shown in  FIG. 3B , the vector data VD 1  is composed of a plurality of pieces (45 in the example in  FIG. 3B ) of coordinate data Vi (xi,yi) (i=0, 1, . . . , 44) of a vertice p(Vi) on a contour of a character and a plurality of pieces of coordinate data Si(xi,yi) (i=0, 1, . . . , 42) of a control point p(Si) defining a curved line drawn between adjacent vertices p(Vi) and p(Vi+1) along the contour (see  FIG. 4 ). 
     &lt;2&gt; Structure 
     A graphics rendering apparatus  40  relating to the present embodiment includes, as shown in  FIG. 27 , a 3D image processing apparatus  10   a  for processing polygon data PD 1  input by a polygon data input unit  210 , a 2D image processing apparatus  10   b  for processing vector data VD 1  (first vector data) input by a vector data input unit  220 , a frame buffer  22 , a texture buffer  21 , a projection conversion matrix setup unit  14   a  for setting up a parameter relating to a projection conversion matrix P for use by the 3D image processing apparatus  10   a . Note that a projection conversion matrix setup unit  14   a , a texture buffer  21 , and a frame buffer  22  relating to the present embodiment have the same structure as those in the Embodiment 1, and accordingly descriptions thereof are omitted. 
     &lt;2-1&gt; 3D Image Processing Apparatus 
     The 3D image processing apparatus  10   a  have substantially the same structure as that in the Embodiment 2. A processor (not shown) appropriately reads a program into a memory (not shown) and executes the read program, thereby realizing a polygon data input reception unit  11 , a 3D coordinate conversion unit  12 , a scaling coefficient determination unit  13 , a projection conversion unit  14 , a polygon rasterization unit  15 , an image display unit  10   c  for causing a display  100 , which is connected to the outside, to display 3D image based on 3D image data stored in the frame buffer  22 , a data count monitor unit  41 , and a processing method setup unit  42 . With respect to the structure of the 3D image processing apparatus  10   a  that is the same as that in the Embodiment 2, description is appropriately omitted. 
     &lt;2-1-1&gt; 3D Coordinate Conversion Unit 
     The 3D coordinate conversion unit  12  calculates a matrix T 0 , a scaling matrix SC, and a matrix T 1 . The matrix T 0  is for performing translation conversion on the virtual plate Plate such that the vertice p(Vplate 0   m ) of the virtual plate Plate coincides with the original point. The scaling matrix SC is for scaling the virtual plate Plate so as to coincide in size with the polygon PG. The matrix T 1  is for performing translation conversion on the virtual plate Plate so as to be back to the original position. Note that the scaling matrix SC is represented by an Expression (43). 
                   SC   =     [           Lx   /   Lplatex         0       0       0           0         Ly   /   Lplatey         0       0           0       0       1       0           0       0       0       1         ]             [     Expression   ⁢           ⁢   43     ]               
&lt;2-1-2&gt; Data Count Monitor Unit
 
     The data count monitor unit  41  includes a first counter (not shown) for counting the number of pieces of data constituting the polygon data PD 1  input by the polygon data input reception unit  11  and a second counter (not shown) for counting the number of data constituting the vector data VD 1  (first vector data) input by the vector data input reception unit  16 . Also, the data count monitor unit  41  inputs a count value CP of the first counter and a count value CV of the second counter into the processing method setup unit  42 . 
     &lt;2-1-3&gt; Processing Method Setup Unit 
     The processing method setup unit  42  changes a processing method based on the number of pieces of data CP constituting the polygon data PD 1  input by the data count monitor unit  41  and the number of pieces of data CV constituting the vector data VD 1  (first vector data). 
     The processing method setup unit  42  reads the counter values CP and CV at predetermined intervals. Here, the processing method setup unit  42  determines to adopt which of two processing methods depending on whether an Expression (44) is satisfied by the counter values CP and CV.
 
β* CP&lt;CV   [Expression 44]
 
     Here, the value CP represents the number of pieces of data constituting the polygon data PD. The value CV represents the number of pieces of data constituting the vector data VD 1  (first vector data). The value β represents a constant that is equal to or greater than 1. The processing methods are described in detail later in &lt;3&gt; Operations. Also, the processing method setup unit  42  notifies the compositional elements included in the 3D image processing apparatus  10   a  and the 2D image processing apparatus  10   b  of the determined processing method. 
     &lt;2-2&gt; 2D Image Processing Apparatus 
     The 2D image processing apparatus  10   b  has substantially the same structure as that in the Embodiment 2. A processor (not shown) appropriately reads a program into a memory (not shown) and executes the read program, thereby realizing a vector data input reception unit  16 , a virtual plate generation unit  17 , a vector data conversion unit  18 , a texture generation unit  19 , and a texture mapping unit  20  of the 2D image processing apparatus  10   b . With respect to the structure of the 2D image processing apparatus  10   b  that is the same as that in the Embodiment 2, description is appropriately omitted. 
     &lt;3&gt; Operations 
     &lt;3-1&gt; Operations of 3D Image Processing Apparatus 
       FIG. 28  is a flow chart showing operations of the 3D image processing apparatus relating to the present embodiment. 
     Firstly, the polygon data input reception unit  11  acquires polygon data PD 1  input by the user via the polygon data input unit  210  (Step S 11 ). 
     Next, the processing method setup unit  42  determines whether the Expression (44) is satisfied by counter values CP and CV. 
     When the 3D coordinate conversion unit  12  judges that the processing method setup unit  42  judges that the Expression (44) is not satisfied by the counter values CP and CV (Step S 41 : No), the 3D coordinate conversion unit  12  converts the polygon data PD 1  into data represented by the 4D coordinate system (Step S 12 ). The subsequent processing in Steps S 13  to S 18  is the same as that in the Embodiment 1, and accordingly description thereof is omitted. 
     On the contrary, when the processing method setup unit  42  judges that the Expression (44) is satisfied by the counter values CP and CV (Step S 41 : Yes), the 3D coordinate conversion unit  12  converts the polygon data PD 1  into data represented by the 4D coordinate system to generate polygon data PD 2 . 
     Here, the 3D coordinate conversion unit  12  converts, into data represented by the 4D coordinate system, the vertice data Vi and the control point data Si (see  FIG. 4  and  FIG. 29 ) that constitute the vector data VD 1  and the coordinate data Vplatek (k=0, 1, 2, 3) (see  FIG. 30 ) of the vertice of the virtual plate Plate. This results in generation of vector data VD 22  (see  FIG. 29 ) composed of vertice data Vim (i=0, 1, . . . , 44) and control point data Sim (i=0, 1, . . . , 42), and coordinate data Vplatekm (k=0, 1, 2, 3) (see  FIG. 30 ) (Step S 42 ). 
     Next, the 3D coordinate conversion unit  12  calculates matrices T 0 , SC, and T 1  that are to be used for performing conversion for superimposing the vertice Vplatekm (k=0, 1, 2, 3) of the virtual plate Plate on the vertice Pim (i=0, 1, 2, 3) of the polygon PG (Step S 43 ). 
     Then, the 3D coordinate conversion unit  12  performs model view conversion on the polygon data PD 2  and performs an operation for dividing each of the x, y, and z components by the w component (see Expression (17)), to generate polygon data PD 5 . The 3D coordinate conversion unit  12  performs vector data conversion by multiplying each of the coordinate data Vplatekm (k=0, 1, 2, 3) and the vector data VD 2  by each of the matrices T 0 , SC, and T 1  (Step S 44 ), and then performs model view conversion on each of the coordinate data Vplatekm (k=0, 1, 2, 3) and the vector data VD 2  (Step S 45 ). Then, the 3D coordinate conversion unit  12  performs an operation for dividing each of the x, y, and z components by the w component (see Expression (17)), to generate vector data VD 23  (see  FIG. 29 ) composed of the vertice data Vie (i=0, 1, . . . , 44) and the control point data Sie (i=0, 1, . . . , 42), and coordinate data Vplateie (i=0, 1, 2, 3) (see  FIG. 30 ) (Step S 46 ). 
     Then, the projection conversion unit  14  performs viewport conversion on each of the polygon data PD 5 , the vector data VD 3 , and the virtual plate data Vplateie to generate polygon data PD 6 . Also, the projection conversion unit  14  generates vecor data VD 24  (see  FIG. 29 ) composed of vertice data Viw (i=0, 1, . . . , 44) and control point data Siw (i=0, 1, . . . , 42), and coordinate data Vplateiw (i=0, 1, 2, 3) (see  FIG. 30 ) represented by the screen coordinate system (Step S 47 ). 
     Then, the projection conversion unit  14  extracts x and y components with respect to each of the vector data VD 4  and the coordinate data Vplateiw (i=0, 1, 2, 3) to generate vector data VD 25  (see  FIG. 29 ) composed of vertice data Viwv (xiw,yiw) (i=0, 1, . . . , 44) and Siwv (xiw,yiw) (i=0, 1, . . . , 42) and coordinate data Vplateiwv (xplateiw,yplateiw) (i=0, 1, 2, 3) (see  FIG. 30 ). Then, the projection conversion unit  14  inputs the generated vector data VD 25  and coordinate data Vplateiwv into the texture generation unit  19  (Step S 48 ). 
     Lastly, the polygon rasterization unit  15  generates frame data FD using the polygon data PD 6  and texture data TD input by the texture mapping unit  20  (Step S 17 ), and writes the generated frame data FD into the frame buffer  22  (Step S 18 ). 
     &lt;3-2&gt; Operations of 2D Image Processing Apparatus 
       FIG. 31  is a flow chart showing operations of the 2D image processing apparatus relating to the present embodiment. 
     Firstly, the vector data input reception unit  16  acquires vector data VD 1  (Step S 21 ). Then, the virtual plate generation unit  17  performs virtual plate generation processing using the vector data VD 1  and the polygon data PD 1  to calculate the coordinate data Vplatek (k=0, 1, 2, 3) of the vertice of the virtual plate Plate and the length Lplatex in the x direction and the length Lplatey in the y direction of the virtual plate Plate (Step S 22 ). The virtual plate generation processing relating to the present embodiment is the same as that in the Embodiment 1, and accordingly the description thereof is omitted. 
     Next, the processing method setup unit  42  determines whether the relational expression (44) is satisfied by counter values CP and CV (Step S 41 ). 
     When the processing method setup unit  42  judges that the relational expression (44) is not satisfied by the counter values CP and CV (Step S 41 : No), the 3D coordinate conversion unit  12  converts the vector data VD 1  into data represented by the 4D coordinate system (Step S 23 ). The subsequent processing in Steps  24  to S 27  is the same as that in the Embodiment 1, and accordingly description thereof is omitted. 
     On the contrary, when the processing method setup unit  42  judges that the relational expression (44) is satisfied by the counter values CP and CV (Step S 41 : Yes), the virtual plate generation unit  17  inputs, into the 3D coordinate conversion unit  12 , the coordinate data Vplatei (i=0, 1, 2, 3) of the vertice of the virtual plate Plate, the length Lplatex in the x direction and the length Lplatey in the y direction of the virtual plate Plate, and the length Lx in the x direction and the length Ly in the y direction of the polygon PG (Step S 51 ). 
     Then, when the coordinate data of the vertice of the virtual plate Plate Vplateiwv (xplateiwv,yplateiwv) (i=0, 1, 2, 3) and the vector data VD 5  are input by the projection conversion unit  14 , the virtual plate generation unit  17  generates texture data using the input data (Step S 52 ), and writes the generated texture data into the texture buffer  21  (Step S 27 ). 
     Modification Examples 
     (1) The above Embodiments 1 to 4 have described the example where the polygon data PD 1 , which is represented by the 2D coordinate system, is input by the polygon data input reception unit  11 . The present invention is not limited to this example. Alternatively, the polygon data PD 2 , which is composed of coordinate data on the 4D coordinate system, may be directly input by the polygon data input reception unit  11 . 
     In the present modification example, the polygon data input reception unit  11  extracts only x and y components from each of piece of coordinate data constituting the polygon data PD 2 , and inputs the extracted x and y components into the virtual plate generation unit  17 . 
     According to the present modification example, the 3D coordinate conversion unit  12  does not need to perform processing for converting the polygon data PD 1  represented by the 2D coordinate system into the polygon data PD 2  represented by the 4D coordinate system. 
     (2) The above Embodiments 2 to 4 have described the example where the vector data VD 1 , which is represented by the 2D coordinate system, is input by the vector data input reception unit  16 . The present invention is not limited to this example. Alternatively, the vector data VD 2 , which is composed of coordinate data on the 3D coordinate system, may be directly input by the vector data input reception unit  16 . 
     In the present modification example, the vector data input reception unit  11  extracts only x and y components from each of piece of vertice data Vih constituting the vector data VD 2 , and inputs the extracted x and y components into the virtual plate generation unit  17 . 
     According to the present modification example, the vector data conversion unit  18  does not need to perform processing for converting the vector data VD 1  represented by the 2D coordinate system into the vector data VD 2  represented by the 3D coordinate system. 
     (3) The above Embodiments 1 to 4 have described the example where Z 0   e  that is the Z component of the vertice p(P 0   e ) of the polygon PG is adopted as Zrep. The present invention is not limited to this example. Alternatively, a Z component of the coordinate data Pie corresponding to the vertice p(Pie), which is nearest to the view point in the Z axis direction among the four vertices constituting the polygon PG, may be adopted as Zrep. 
     Also, in the present modification example, it may be possible to employ an example where an average value of Z components of the pieces of the coordinate data Pie (i=0, 1, 2, 3) one-to-one corresponding to the four vertices constituting the polygon PG is adopted as Zrep. 
                   Zrep   =         Z   ⁢           ⁢   0   ⁢   e     +     Z   ⁢           ⁢   1   ⁢   e     +     Z   ⁢           ⁢   2   ⁢   e     +     Z   ⁢           ⁢   3   ⁢   e       4             [     Expression   ⁢           ⁢   45     ]               
In this case, the above Expression 45 is satisfied.
 
     Furthermore, the present modification example has provided the case where the polygon PG is rectangular. However, the present invention is not limited to this. Alternatively, assume that the number of vertices of the polygon PG is K. 
                   Zrep   =         ∑     i   =   0       K   -   1       ⁢           ⁢   Zie     K             [     Expression   ⁢           ⁢   46     ]               
In this case, the above Expression 46 is satisfied.
 
(4) The above Embodiment 2 has provided the example where the virtual plate generation unit  17  calculates the bounding box VB using one piece of vector data VD 1 . However, the present invention is not limited to this example. Alternatively, one bounding box may be calculated using a plurality of pieces of vector data.
 
(5) The above Embodiment 2 has provided the example where the virtual plate generation unit  17  sets the margin mx in the x direction to δ*PBx (PBx: the length of the bounding box in the x direction). However, the present invention is not limited to this example. Alternatively, where the margin mx may be set to a fixed value ε 1 , regardless of the length PBx in the x direction of the bounding box. Further alternatively, another method may be appropriately employed.
 
Further alternatively, the margin my in the y direction also may be set to a fixed value ε 2 , regardless of the length PBy in the y direction of the bounding box.
 
(6) The above Embodiment 2 has provided the example where the bounding box is calculated based on the vector data VD 1 . However, the present invention is not limited to this example. Alternatively, when information corresponding to the bounding box relating to the Embodiment 2 is included beforehand in the vector data VD 1  as attribute information of a font, the data may be used as the bounding box without performing any conversion.
 
     As a result, it is possible to omit the bounding box calculation processing in the virtual plate generation processing. This can improve the processing efficiency. 
     (7) The above Embodiment 2 has provided the example where the bounding box is calculated based on the vector data VD 1  representing one character. However, the present invention is not limited to this example. Alternatively, as shown in  FIG. 32 , a bounding box may be generated with respect to each of a plurality of pieces of vector data representing a string of characters, to generate a virtual plate Plate including the generated plurality of bounding boxes. 
     In the present modification example, the following may be employed. With respect to each of the pieces of vector data representing a character, the length Lx in the x direction and the length Ly in the y direction of the bounding box are calculated. Then, the sum LxA of the lengths in the x direction with respect to the plurality of bounding boxes and the sum LyA of the lengths in the y direction with respect to the plurality of bounding boxes are calculated. A rectangular virtual plate having the length LxA in the x direction and the length LyA in the y direction is determined as a bounding box. 
     Here, there is a case where each of the pieces of vector data constituting the string of characters includes attribute information relating to an offset length (Escapement or Advanced Width) between adjacent characters. In this case, it is only necessary to add the sum of Escapement included in the pieces of vector data to one of the length LxA in the x direction and the length LyA in the y direction of the bounding box. 
     (8) The above Embodiment 3 has provided the example where the polygon data PD 1  represents a polygonal polygon composed of 11 vertices p(Pi) (i=0, 1, . . . , 10) as shown in  FIG. 21 . However, the present invention is not limited to this example. Alternatively, the polygon data PD 1  may represent a polygon composed of a plurality of vertices such as a circular polygon, a polygonal polygon having less than 10 vertices, or a polygonal polygon having more than 11 vertices.
 
(9) The above Embodiment 1 has provided the example where the discrete scaling rate (scaling value) α is stored in the scaling value storage unit  51  to determine the optimal scaling value. However, the present invention is not limited to this example. Alternatively, it may be possible to set an objective function such as shown in an Expression (47) and continuously vary the scaling value, and set the scaling value α corresponding to the least value of the objective function.
 
obj(α)=− f (α)+Penalty  [Expression 47]
 
     Here, the value obj(α) represents an objective function. The value f(α) represents a function of an area of a rectangular bounding box VBA including a vector graphics represented by the vector data VD 2  on which scaling has been performed, as shown in  FIGS. 3A and 33B . Here, scaling is performed using the first scaling value scalex and the second scaling value scaley as the value α. 
     The value Penalty represents a function (penalty function) as shown in  FIGS. 3A and 33B . A bounding box VBA and a rectangular region Plate, which has lengths in the x direction and the y direction that are equal to the first scaling coefficient scx and the second scaling coefficient scy, respectively, are overlapped with each other, such that the bounding box VBA and the rectangular region Plate have the same center. The penalty function is proportional to an area of a region where the bounding box VBA and the rectangular region Plate are not overlapped with each other (hatched part). 
     As described in the Embodiment 1, the vector data conversion unit  18  performs translation conversion such that the gravity coordinate G of the vertice data Vi (i=0, 1, . . . , 44) overlaps with the view point (original point), thereby to generate vertice data Vih (i=0, 1, . . . , 44). Then, the vector data conversion unit  18  calculates a value of the objective function obj(α) based on the Expression (47) using the vector data composed of the vertice data Vih and the value α. 
     Then, the vector data conversion unit  18  determines, as the optimal value, the value α corresponding to the minimum value of the value obj(α) that is continuously varied within a predetermined range (such as a range of 0.5 to 100 inclusive). 
     The steepest descent method or the like may be employed for calculating the minimum value of the value obj(α). 
     (10) The Embodiment 4 has provided the example where the processing method setup unit  42  changes the processing method based on information input by the data count monitor unit  41 . However, the present invention is not limited to this. 
     &lt;Supplementary Explanations&gt; 
     (1) The graphics rendering apparatus relating to the present invention is typically realized as an LSI that is a semiconductor integrated circuit. The compositional elements of the graphics rendering apparatus may be separately integrated into one chip, or integrated into one chip including part or all of the compositional elements. Although the system LSI is used here, the system LSI may be called an IC, a system LSI, a super LSI, or an ultra LSI, depending on the integration degree. 
     Also, a method of forming integrated circuits is not limited to LSIs, and may be realized using a dedicated circuit or a general-purpose processor. Furthermore, it may be possible to use an FPGA (Field Programmable Gate Array) programmable after manufacturing LSIs or a reconfigurable processor in which connection and setting of a circuit cell inside an LSI can be reconfigured. 
     Furthermore, when new technology for forming integrated circuits that replaces LSIs becomes available as a result of progress in semiconductor technology or semiconductor-derived technologies, functional blocks may be integrated using such technology. One possibility lies in adaptation of biotechnology. 
     Furthermore, by combining a semiconductor chip which is formed by integrating the graphics rendering apparatus relating to the present invention and a display for rendering images, it is possible to structure a rendering apparatus applicable to various purposes. It is possible to utilize the present invention as an information rendering unit for use in a portable phone, a TV, a digital video recorder, a digital video camera, a car navigation system, and so on. In order to realize such a display, it is possible to combine, with the present invention, a various types of flat displays such as cathode-ray tube (CRT), liquid crystal, PDP (plasma display panel), and organic EL displays, and a projection display exemplified by a projector, and so on. 
     The graphics rendering apparatus relating to the present invention is available for various purposes. The graphics rendering apparatus has a high utility value as an information display unit for menu display, Web browser, editor, EPG, and map display in a battery-driven portable display terminal such as a portable phone, a portable music player, a digital camera, and a digital video camera, and a high-resolution information display apparatus such as a TV, a digital video recorder, and a car navigation system, and so on. 
     
       
         
           
               
             
               
                   
               
               
                 Reference Signs List 
               
               
                   
               
             
            
               
                   
               
            
           
           
               
               
            
               
                 10, 30, 40, and 50: 
                 graphics rendering apparatus 
               
               
                 10a: 
                 3D image processing apparatus 
               
               
                 10b: 
                 2D image processing apparatus 
               
               
                 10c: 
                 image display unit 
               
               
                 11: 
                 polygon data input reception unit 
               
               
                 12: 
                 3D coordinate conversion unit 
               
               
                 13: 
                 scaling coefficient determination unit 
               
               
                 14: 
                 projection conversion unit 
               
               
                 15: 
                 polygon rasterization unit 
               
               
                 16: 
                 vector data input reception unit 
               
               
                 17: 
                 virtual plate generation unit 
               
               
                 18: 
                 vector data conversion unit 
               
               
                 19: 
                 texture generation unit 
               
               
                 20: 
                 texture mapping unit 
               
               
                 21: 
                 texture buffer 
               
               
                 22: 
                 frame buffer 
               
               
                 31: 
                 bounding box generation unit 
               
               
                 41: 
                 data count monitor unit 
               
               
                 42: 
                 processing method setup unit 
               
               
                 51: 
                 scaling value storage unit 
               
               
                 PD1, PD2, PD3, PD4, PD5, and  
                 polygon data 
               
               
                 PD6: 
                   
               
               
                 T: 
                 scaling value table 
               
               
                 VD1, VD2, VD3, VD4, and VD5: 
                 vector data