[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 11:44:45

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

Log message:
        Fix typos and more How

CVSWeb URLs:
http://savannah.gnu.org/cgi-bin/viewcvs/gzz/gzz/Documentation/Manuscripts/Irregu/irregu.tex.diff?tr1=1.79&tr2=1.80&r1=text&r2=text

Patches:
Index: gzz/Documentation/Manuscripts/Irregu/irregu.tex
diff -u gzz/Documentation/Manuscripts/Irregu/irregu.tex:1.79 
gzz/Documentation/Manuscripts/Irregu/irregu.tex:1.80
--- gzz/Documentation/Manuscripts/Irregu/irregu.tex:1.79        Thu Nov 28 
10:22:14 2002
+++ gzz/Documentation/Manuscripts/Irregu/irregu.tex     Thu Nov 28 11:44:45 2002
@@ -372,16 +372,22 @@
 a simple algorithm for a shape with the desired properties.
 
 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. 
+for example, a rectangle or an ellipse. 
 We want to define the corresponding irregular piece $B \subset \mathbf{R}^2$ 
-so as to satisify the ``natural movement'' property:
+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 $R \in \mathbf{R}^2$:
+is actually defined by a point relation $R \subset \mathbf{R}^2$:
 \[
     B = \{\, y \mid \exists x \in A: x R y \,\}.
 \]
@@ -431,27 +437,28 @@
 each assumed to be a straight line segment, and then consider
 the problem of distorting each one separately.
 
-Consider a half-plane shape \[
-  A = \{\, x \mid n \cdot (x - x_0) \le 0 \,\},
+Consider a half-plane shape 
+\[
+    A = \{\, x \mid n \cdot (x - x_0) \le 0 \,\}
 \]
-where $n$ is the normal of the edge line.
+representing one line segment, where $n$ is the normal of the line.
 
 The offset mapping yields $y \in B$, iff
 \[
     y + F(y) \in A,
 \]
-i.e.,
+i.e., iff
 \begin{equation}
     n \cdot (y - x_0) + n \cdot F(y) \le 0,
 \end{equation}
 Thus, offset has only effect in the normal direction and the problem
 is essentially reduced to one dimension.
-Simply define the ``scattered case'' as:
+We can simply define the ``scattered case'' as:
 \begin{equation}
     n \cdot (y - x_0) + f(y) \le 0,
 \end{equation}
 where the real function $f$ specifies offset in the normal direction.
-The movement is still natural.
+The movement is still natural. XXX
 
 An anlogy to the forward function case can be made by
 limiting the displacement of the shape edge to the normal direction, i.e,
@@ -470,12 +477,6 @@
 Thus, we have a ``connected case'' in a very similar formulation
 to the above ``scattered case''.
 
-
-- consider vector valued ripple function
-
-XXX: relaxing the ``natural movement'' property \dots
-
-
 ---
 
 To start off, assume that we are drawing the torn edge inside a 
@@ -501,7 +502,6 @@
 A suitable choice for $f$ would be
 a function with noise at different frequencies, but with lower frequencies
 emphasized more, such as turbulence\cite{perlin-noise-intro}.
-
 
 Although these algorithms seem different and produce different results, there 
is 
 actually a general formulation which yields to a visual explanation.