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

[Janne V. Kujala]

CVSROOT:        /cvsroot/gzz
Module name:    gzz
Changes by:     Janne V. Kujala <address@hidden>        02/11/30 03:16:45

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

Log message:
        some How

CVSWeb URLs:
http://savannah.gnu.org/cgi-bin/viewcvs/gzz/gzz/Documentation/Manuscripts/Irregu/irregu.tex.diff?tr1=1.98&tr2=1.99&r1=text&r2=text

Patches:
Index: gzz/Documentation/Manuscripts/Irregu/irregu.tex
diff -u gzz/Documentation/Manuscripts/Irregu/irregu.tex:1.98 
gzz/Documentation/Manuscripts/Irregu/irregu.tex:1.99
--- gzz/Documentation/Manuscripts/Irregu/irregu.tex:1.98        Sat Nov 30 
02:59:28 2002
+++ gzz/Documentation/Manuscripts/Irregu/irregu.tex     Sat Nov 30 03:16:45 2002
@@ -540,31 +540,29 @@
 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.
+in the normal direction ($y$-axis) 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 by a function 
-$0\le f(\p)\le 1$ which only depends on the location $\p$ of the spine:
-$C(x) = E(x, f(E(x,1/2)))$.
+$f(\p)$: $C(x) = E(x, 1/2 + f(E(x,1/2)))$.
 
-The scattered case, on the other hand, can be obtained through a decision 
process:
-again using a function $f(\p)$ of location $\p$, 
+The scattered case, on the other hand, can be obtained by offsetting
+in the normal direction: using an offset function $f(\p)$ of location $\p$:
 a given point $E(x,y)$ is {\em inside} the tear-out,
-iff $f(E(x,y)) > y$.
+iff $y - f(E(x,y)) < 1/2$. Here, the offset function is negated
+for reasons of other formulations. XXX
 
 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}.
 
-These two algorithms correspond to one-dimensional displacement
-and offset distortions, where the one dimension is in the normal direction.
-In the relation formulation, the corresponding forward and inverse functions
-are
+These one-dimensional displacement and offset mappings correspond
+to the following 2D forward and inverse functions:
 \begin{eqnarray}
-  G_n(\p) &=& \p + (r N) (2f(\p) - 1), \\
-  F_n(\q) &=& \q - (r N) (2f(\q) - 1),
+  G_N(\p) &=& \p + (r N) f(\p), \\
+  F_N(\q) &=& \q - (r N) f(\q),
 \end{eqnarray}
 where $N$ is the unit normal, $r$ is the distortion radius,
 and the undistorted shape $A$ is the region inside the spine $E(x,1/2)$.