[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 03:00:27

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=1.83&tr2=1.84&r1=text&r2=text

Patches:
Index: gzz/Documentation/Manuscripts/Irregu/irregu.tex
diff -u gzz/Documentation/Manuscripts/Irregu/irregu.tex:1.83 
gzz/Documentation/Manuscripts/Irregu/irregu.tex:1.84
--- gzz/Documentation/Manuscripts/Irregu/irregu.tex:1.83        Thu Nov 28 
14:02:19 2002
+++ gzz/Documentation/Manuscripts/Irregu/irregu.tex     Fri Nov 29 03:00:27 2002
@@ -378,11 +378,11 @@
 
 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 $y \in B$.
+where $B(y, r)$ is a ball of radius $r$ centered at $y$.
+$A \cap B(y, r) = \emptyset$ should imply $y \notin B$ and 
+$A \cap B(y, r) = B(y,r)$ should imply $y \in B$.
 Additionally, the mapping from $A$'s to $B$'s should be continuous
-so that slow movement of $A$ results into ``rippling'' in $B$.
+so that slow movement of $A$ results into smooth ``rippling'' in $B$.
 Furthermore, the rippling should not be too fast.
 
 %
@@ -401,15 +401,15 @@
 \[
     B = \{\, y \mid \exists x \in A: x R y \,\}.
 \]
-If $d(x,y) \le r$ for all $x R y$, the maximum distortion requirement
+If $\Vert x - y\Vert \le r$ for all $x R y$, the maximum distortion requirement
 is satisfied. 
 
 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 $A$. 
+Furthermore, the inside of $A$ may get displaced to the outside of the 
+image of its border. 
 
 If $R$ is a continuous function in the inverse direction, i.e., 
 $x R y \Leftrightarrow F(y) = x$,
@@ -424,8 +424,8 @@
 texture shading hardware:
 the image of the undistorted shape can be stored in a texture and accessed
 with texture coordinates read from (or offset by) another texture storing the 
-inverse function (or $F(y) - y$). This is called a dependent (or offset)
-texture access.
+inverse function (or the offset $F(y) - y$). 
+This is called a dependent (or offset) texture access.
 
 The forward function cannot be efficiently implemented on fragment level,
 because each fragment may depend on multiple values of the function.
@@ -433,7 +433,7 @@
 but that is likely to not yield good performance if the shape has fine detail.
 Furthermore, intersections in the edge may cause additional problems.
 If the function is bijection, so as to avoid any intersections, we
-can just as well use the invert it and use the inverse mapping.
+can just as well use the inverse mapping.
 
 ---
 
@@ -453,7 +453,7 @@
 \[
     A = \{\, x \mid n \cdot (x - x_0) \le 0 \,\}
 \]
-representing one line segment, where $n$ is the normal of the line.
+representing one line segment, where $n$ is the unit normal of the line.
 
 The offset mapping yields $y \in B$, iff
 \[
@@ -466,17 +466,20 @@
 Thus, offset has only effect in the normal direction and the problem
 is essentially reduced to one dimension.
 We can simply define the ``scattered case'' as: $F(y) = -n f(y)$,
-where the real function $f(y)$ specifies offset in the normal direction.
+where the real function $f(y)$ specifies offset in the (negative) 
+normal direction.
 This yields $y \in B$, iff
 \begin{equation}
     f(y) \ge n \cdot (y - x_0),
 \end{equation}
-The movement is still natural. XXX
+The movement is still natural, because the resulting distortion
+is continuous and local. XXX: rotation?
 
-An anlogy to the edge displacement case, i.e. the distorted edge defined 
-by $y = x + G(x)$ for $x$ in the undistorted edge, by
-limiting the displacement of the shape edge to the normal direction, i.e,
-by defining $G(x) = n f(x)$, where $f(x)$ is the normal displacement.
+Consider then the edge displacement case, i.e. the distorted edge defined 
+by $y = x + G(x)$ for $x$ in the undistorted edge.
+Analogously to the offset function, the displacement can be limited 
+to the normal direction by defining $G(x) = n f(x)$, 
+where $f(x)$ is the normal displacement.
 With one-dimensional displacement, the distorted edge cannot intersect
 itself, and we obtain $y \in B$, iff
 \begin{equation}