[Gzz-commits] gzz/Documentation/Manuscripts/Irregu irregu.tex

[Janne V. Kujala]

CVSROOT:        /cvsroot/gzz
Module name:    gzz
Changes by:     Janne V. Kujala <address@hidden>        02/11/28 14:02:19

Modified files:
        Documentation/Manuscripts/Irregu: irregu.tex 

Log message:
        cleanup

CVSWeb URLs:
http://savannah.gnu.org/cgi-bin/viewcvs/gzz/gzz/Documentation/Manuscripts/Irregu/irregu.tex.diff?tr1=1.82&tr2=1.83&r1=text&r2=text

Patches:
Index: gzz/Documentation/Manuscripts/Irregu/irregu.tex
diff -u gzz/Documentation/Manuscripts/Irregu/irregu.tex:1.82 
gzz/Documentation/Manuscripts/Irregu/irregu.tex:1.83
--- gzz/Documentation/Manuscripts/Irregu/irregu.tex:1.82        Thu Nov 28 
13:21:50 2002
+++ gzz/Documentation/Manuscripts/Irregu/irregu.tex     Thu Nov 28 14:02:19 2002
@@ -465,27 +465,25 @@
 \end{equation}
 Thus, offset has only effect in the normal direction and the problem
 is essentially reduced to one dimension.
-We can simply define the ``scattered case'' as:
+We can simply define the ``scattered case'' as: $F(y) = -n f(y)$,
+where the real function $f(y)$ specifies offset in the normal direction.
+This yields $y \in B$, iff
 \begin{equation}
-    n \cdot (y - x_0) + f(y) \le 0,
+    f(y) \ge n \cdot (y - x_0),
 \end{equation}
-where the real function $f$ specifies offset in the normal direction.
 The movement is still natural. XXX
 
-An anlogy to the forward function case can be made by
+An anlogy to the edge displacement case, i.e. the distorted edge defined 
+by $y = x + G(x)$ for $x$ in the undistorted edge, by
 limiting the displacement of the shape edge to the normal direction, i.e,
-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.
 With one-dimensional displacement, the distorted edge cannot intersect
-itself, and we can write
-\[
-   B = \{\, y \mid n \cdot (y - x_0) \le d( x_y )\,\}, 
-\]
-where $x_y$ is the projection of $y$ to the undistorted edge.
-The defining equation can be written as
+itself, and we obtain $y \in B$, iff
 \begin{equation}
-    n \cdot (y - x_0) + f(x_y) \le 0,
+    f(x_y) \ge n \cdot (y - x_0),
 \end{equation}
-if we define $f(x) = -d(x)$.
+where $x_y$ is the projection of $y$ to the undistorted edge.
+
 Thus, we have a ``connected case'' in a very similar formulation
 to the above ``scattered case''.