[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 13:03:08

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=1.80&tr2=1.81&r1=text&r2=text

Patches:
Index: gzz/Documentation/Manuscripts/Irregu/irregu.tex
diff -u gzz/Documentation/Manuscripts/Irregu/irregu.tex:1.80 
gzz/Documentation/Manuscripts/Irregu/irregu.tex:1.81
--- gzz/Documentation/Manuscripts/Irregu/irregu.tex:1.80        Thu Nov 28 
11:44:45 2002
+++ gzz/Documentation/Manuscripts/Irregu/irregu.tex     Thu Nov 28 13:03:08 2002
@@ -375,30 +375,44 @@
 for example, a rectangle or an ellipse. 
 We want to define the corresponding irregular piece $B \subset \mathbf{R}^2$ 
 so as to make the movement of $A$ look natural.
-There should be some maximum radius $r$ of rippling, so that
-whether $y \in B$ depends only on $A \cap B(y, r)$. Furthermore
-$A \cap B(y, r) = \emptyset$ should imply $y \notin B$ and 
-$A \cap B(y, r) = B(y,r)$ should imply $y \in B$.
 
-One way of defining such properties is to start with the requirement that:
-\[
-    A = A_1 \cup A_2 \Rightarrow B = B_1 \cup B_2,
-\]
-for all $A, A_1, A_2 \subset \mathbf{R}^2$ and 
-matching irregular shapes $B, B_1, B_2 \subset \mathbf{R}^2$.
-From this porperty, it follows that the mapping from $A$'s to $B$'s 
-is actually defined by a point relation $R \subset \mathbf{R}^2$:
+There should be some maximum radius $r$ of distortion such that
+whether a point $y$ is in $B$ depends only on the set $A \cap B(y, r)$,
+where
+$A \cap B(y, r) = \emptyset$ implies $y \notin B$ and 
+$A \cap B(y, r) = B(y,r)$ implies $y \in B$.
+Additionally, slow movement of $A$ should result into small
+changes in $B$ so that the rippling is not too fast.
+XXX: continuous mapping?
+
+%
+%to start with the requirement that:
+%\[
+%    A = A_1 \cup A_2 \Rightarrow B = B_1 \cup B_2,
+%\]
+%for all $A, A_1, A_2 \subset \mathbf{R}^2$ and 
+%matching irregular shapes $B, B_1, B_2 \subset \mathbf{R}^2$.
+%From this porperty, it follows that the mapping from $A$'s to $B$'s 
+%is actually defined by a point relation $R \subset (\mathbf{R}^2)^2$:
+%
+One way of defining such mapping is a relation 
+$R \subset (\mathbf{R}^2)^2$ that identifies the points of $A$ 
+with points in $B$:
 \[
     B = \{\, y \mid \exists x \in A: x R y \,\}.
 \]
-If $R$ is a function, i.e., $x R y \Leftrightarrow y = G(x)$,
+If $d(x,y) \le r$ for all $x R y$, the maximum distortion requirement
+is satisfied. 
+Furthermore, $R$ should be continuous to create natural rippling.
+
+If $R$ is a continuous function, i.e., $x R y \Leftrightarrow y = G(x)$,
 the irregular shapes will be connected (if the original shape is connected), 
 but some parts of original 
 shape may map to overlapping parts in the irregular shape.
 Furthermore, the inside of $A$ may get displaced to outside of the 
 image of the border of $A$. 
 
-If $R$ is a function in the inverse direction, i.e., 
+If $R$ is a continuous function in the inverse direction, i.e., 
 $x R y \Leftrightarrow F(y) = x$,
 there will be no overlapping.
 Specifically, the edge of $A$ will map to the edge of $B$.