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revision 1.82 by jvk, Thu Nov 28 18:21:50 2002 UTC revision 1.83 by jvk, Thu Nov 28 19:02:19 2002 UTC
# Line 465  i.e., iff Line 465  i.e., iff
465  \end{equation}  \end{equation}
466  Thus, offset has only effect in the normal direction and the problem  Thus, offset has only effect in the normal direction and the problem
467  is essentially reduced to one dimension.  is essentially reduced to one dimension.
468  We can simply define the ``scattered case'' as:  We can simply define the ``scattered case'' as: $F(y) = -n f(y)$,
469    where the real function $f(y)$ specifies offset in the normal direction.
470    This yields $y \in B$, iff
471  \begin{equation}  \begin{equation}
472      n \cdot (y - x_0) + f(y) \le 0,      f(y) \ge n \cdot (y - x_0),
473  \end{equation}  \end{equation}
 where the real function $f$ specifies offset in the normal direction.  
474  The movement is still natural. XXX  The movement is still natural. XXX
475    
476  An anlogy to the forward function case can be made by  An anlogy to the edge displacement case, i.e. the distorted edge defined
477    by $y = x + G(x)$ for $x$ in the undistorted edge, by
478  limiting the displacement of the shape edge to the normal direction, i.e,  limiting the displacement of the shape edge to the normal direction, i.e,
479  by defining $G(x) = x + n d(x)$, where $d(x)$ is the normal displacement.  by defining $G(x) = n f(x)$, where $f(x)$ is the normal displacement.
480  With one-dimensional displacement, the distorted edge cannot intersect  With one-dimensional displacement, the distorted edge cannot intersect
481  itself, and we can write  itself, and we obtain $y \in B$, iff
 \[  
    B = \{\, y \mid n \cdot (y - x_0) \le d( x_y )\,\},  
 \]  
 where $x_y$ is the projection of $y$ to the undistorted edge.  
 The defining equation can be written as  
482  \begin{equation}  \begin{equation}
483      n \cdot (y - x_0) + f(x_y) \le 0,      f(x_y) \ge n \cdot (y - x_0),
484  \end{equation}  \end{equation}
485  if we define $f(x) = -d(x)$.  where $x_y$ is the projection of $y$ to the undistorted edge.
486    
487  Thus, we have a ``connected case'' in a very similar formulation  Thus, we have a ``connected case'' in a very similar formulation
488  to the above ``scattered case''.  to the above ``scattered case''.
489    

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