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                                              322
CHAPTER 4                                                                      Graphics




                                                                   C2




                                                                                  D2

   v                                                      D1




                                                                        C1

                  u



       FIGURE 4.20 Coordinate mapping from a unit square to a four-sided Coons patch


Two surfaces can be described that are linear interpolations between the bound-
ary curves. Along the u axis, the surface SC is defined by
S C (u, v ) = ( 1 – v ) × C 1 (u) + v × C 2 (u)

Along the v axis, the surface SD is given by
S D (u, v ) = ( 1 – u ) × D 1 (v) + u × D 2 (v)

A third surface is the bilinear interpolation of the four corners:
S B (u, v ) = ( 1 – v ) × [ ( 1 – u ) × C 1 (0) + u × C 1 (1) ]
                  + v × [ ( 1 – u ) × C 2 (0) + u × C 2 (1) ]

The coordinate mapping for the shading is given by the surface S, defined as
S = SC + SD – SB

This defines the geometry of each patch. A patch mesh is constructed from a
sequence of one or more such colored patches.

Patches can sometimes appear to fold over on themselves—for example, if a
boundary curve intersects itself. As the value of parameter u or v increases in
parameter space, the location of the corresponding pixels in device space may
change direction so that new pixels are mapped onto previous pixels already

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