[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 05:43:39

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

Log message:
        Use edges of sets

CVSWeb URLs:
http://savannah.gnu.org/cgi-bin/viewcvs/gzz/gzz/Documentation/Manuscripts/Irregu/irregu.tex.diff?tr1=1.76&tr2=1.77&r1=text&r2=text

Patches:
Index: gzz/Documentation/Manuscripts/Irregu/irregu.tex
diff -u gzz/Documentation/Manuscripts/Irregu/irregu.tex:1.76 
gzz/Documentation/Manuscripts/Irregu/irregu.tex:1.77
--- gzz/Documentation/Manuscripts/Irregu/irregu.tex:1.76        Thu Nov 28 
05:39:16 2002
+++ gzz/Documentation/Manuscripts/Irregu/irregu.tex     Thu Nov 28 05:43:39 2002
@@ -375,8 +375,9 @@
 
 SCRATCH:
 
-Suppose $A \subset \mathbf{R}^2$ is the part of the canvas we want to tear 
off, for example, a rectangle or a smooth ellipse. 
-We want to define the corresponding irregular piece $B \subset \mathbf{R}^2$
+Suppose $A \subset \mathbf{R}^2$ is the edge of the part of the canvas we want 
to tear off, for example, a rectangle or a smooth ellipse. 
+We want to define the corresponding edge $B \subset \mathbf{R}^2$ 
+of the irregular piece 
 so as to satisify the ``natural movement'' propery:
 \[
     A = A_1 \cup A_2 \Rightarrow B = B_1 \cup B_2,
@@ -390,15 +391,21 @@
 \]
 If $R$ is a function, i.e., $x R y \Leftrightarrow y = f(x)$,
 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.
+but the irregular dege may intersect itself.
 If $R$ is a function in the inverse direction, i.e., 
 $x R y \Leftrightarrow g(y) = x$,
-there will be no overlapping, but the irregular shape may be scattered.
+there will be no intersections, but the irregular shape may be scattered.
 If the function is bijection, the topology of the irregular shape
 will match the topology of the original shape.
 
+XXX: in the inverse function case, the region inside the edge $A$
+will map to the region inside $B$.
+
+XXX: in the function case, the inside region may be displaced outside
+the edge.
+
 XXX: relaxing the ``natural movement'' property \dots
+
 
 ---