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 |
|
|