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

[Janne V. Kujala]

CVSROOT:        /cvsroot/gzz
Module name:    gzz
Changes by:     Janne V. Kujala <address@hidden>        02/11/29 06:37:45

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

Log message:
        explain more

CVSWeb URLs:
http://savannah.gnu.org/cgi-bin/viewcvs/gzz/gzz/Documentation/Manuscripts/Irregu/irregu.tex.diff?tr1=1.87&tr2=1.88&r1=text&r2=text

Patches:
Index: gzz/Documentation/Manuscripts/Irregu/irregu.tex
diff -u gzz/Documentation/Manuscripts/Irregu/irregu.tex:1.87 
gzz/Documentation/Manuscripts/Irregu/irregu.tex:1.88
--- gzz/Documentation/Manuscripts/Irregu/irregu.tex:1.87        Fri Nov 29 
06:20:40 2002
+++ gzz/Documentation/Manuscripts/Irregu/irregu.tex     Fri Nov 29 06:37:45 2002
@@ -379,10 +379,10 @@
 so as to make the movement of $A$ look natural.
 
 There should be some maximum radius $r$ of distortion such that
-whether a point $y$ is in $B$ depends only on the set $A \cap B(\q, r)$,
-where $B(\q, r)$ is a ball of radius $r$ centered at $\q \in \mathbf{R}^2$.
-$A \cap B(\q, r) = \emptyset$ should imply $\q \notin B$ and 
-$A \cap B(\q, r) = B(y,r)$ should imply $\q \in B$.
+whether a point $\q$ is in $B$ depends only on the set $A \cap B(\q,r)$,
+where $B(\q,r)$ is a ball of radius $r$ centered at $\q \in \mathbf{R}^2$.
+$A \cap B(\q,r) = \emptyset$ should imply $\q \notin B$ and 
+$A \cap B(\q,r) = B(\q,r)$ should imply $\q \in B$.
 Additionally, the mapping from $A$'s to $B$'s should be continuous
 so that slow movement of $A$ results into smooth ``rippling'' in $B$.
 Furthermore, the rippling should not be too fast.
@@ -497,16 +497,22 @@
 ---
 
 To start off, assume that we are drawing the torn edge inside a 
-given \emph{envelope}.
+given smooth \emph{envelope}.
 The envelope is parametrized as a mapping $E(x,y)$ to canvas coordinates
 so that $E(x,0)$ and $E(x,1)$ are the inner and outer edges of the 
-envelope, respectively, and the ripples are contained between these two curves.
+envelope, respectively, and the ripples are contained between these two
+smooth curves.
 An envelope can be defined with a \emph{spine} $E(x,1/2)$ and a normal vector
 $N(x)$ so that $E(x,y) = E(x,1/2) + (y-1/2) N(x)$.
 The envelope should not intersect itself.
 
+This formulation allows us to consider the distortion only
+in the normal ($y$) direction for each point $x$ in the edge.
+On the other hand, we will need to specifically consider the corners
+later.
+
 The edge curve $C(x)$ of a conneccted shape can be obtained by simply 
-displacing the edge in the normal direction of the envelope by a function 
+displacing the edge in the normal direction by a function 
 $0\le f(\p)\le 1$ which only depends on the location $\p$ of the spine:
 $C(x) = E(x, f(E(x,1/2)))$.
 
@@ -521,7 +527,8 @@
 
 These two algorithms correspond to one-dimensional displacement
 and offset distortions, where the one dimension is in the normal direction.
-The corresponding function and inverse function are
+In the relation formulation, the corresponding forward and inverse functions
+are
 \begin{eqnarray}
   G_n(\p) &=& \p + (r N) (2f(\p) - 1), \\
   F_n(\q) &=& \q - (r N) (2f(\q) - 1),