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

[Tuomas J. Lukka]

CVSROOT:        /cvsroot/gzz
Module name:    gzz
Changes by:     Tuomas J. Lukka <address@hidden>        02/11/13 10:14:30

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

Log message:
        Shaping, remove geometric interpretation for a while.

CVSWeb URLs:
http://savannah.gnu.org/cgi-bin/viewcvs/gzz/gzz/Documentation/Manuscripts/Irregu/irregu.tex.diff?tr1=1.40&tr2=1.41&r1=text&r2=text

Patches:
Index: gzz/Documentation/Manuscripts/Irregu/irregu.tex
diff -u gzz/Documentation/Manuscripts/Irregu/irregu.tex:1.40 
gzz/Documentation/Manuscripts/Irregu/irregu.tex:1.41
--- gzz/Documentation/Manuscripts/Irregu/irregu.tex:1.40        Wed Nov 13 
08:26:11 2002
+++ gzz/Documentation/Manuscripts/Irregu/irregu.tex     Wed Nov 13 10:14:30 2002
@@ -282,9 +282,7 @@
 
 In this subsection, we formulate the design criteria of the preceding section 
mathematically
 and discuss
-a simple algorithm for a shape with the desired properties 
-and an interesting graphical explanation
-for the algorithm.
+a simple algorithm for a shape with the desired properties.
 
 To start off, assume that we are drawing the torn edge around a given 
 smooth parametric curve $C = (x(t), y(t))$. The thickness of the roughness of 
the
@@ -304,29 +302,36 @@
 
 Although these algorithms seem different and produce different results, there 
is 
 actually a reasonable generalization which yields to a visual explanation.
-Both algorithms can be represented as
+Both algorithms can be represented as a point being inside the final curve if
 $$
     f(g(p)) < |p-n_C(p)|
 $$
 where $g(p) = p$ for the sprinkled case and $g(p) = n_C(p)$ 
-for the attached case. 
-
-- the function should change slowly enough so that the dot product of movement 
direction and edge normal
-  is visible as the ``rippling speed''
+for the attached case. We can immediately define a family of intermediate forms
+with 
+$$
+    g_\alpha(p) = \alpha p + (1-\alpha) n_C(p),
+$$
+parametrized by $\alpha$
+with usually $0 \le \alpha \le 1$.
 
-The jagged shape is defined by a surface $(x, y, f(x,y))$,
-where $0 \le f(x,y) \le 1$ and $(x, y, 1/2)$ is paper location.
-The shape of an edge torn along a line is obtained by intersecting
-the surface with a plane cutting through the tearing line.
-The intersection is then rotated around the tearing line until
-it is horizontal and stretched so that $z = 0$ and $z = 1$ lines 
-of the cutting plane correspond to the desired maximum positive
-and negative displacements of the edge.
-
-If the cutting plane is vertical, the shape of the torn edge is attached.
-If the angle between $z = 1/2$ plane and the cutting plane is
-small enough, the plane will cut off maxima of the surface,
-producing a sprinkeld shape.
+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}.
+
+% The jagged shape is defined by a surface $(x, y, f(x,y))$,
+% where $0 \le f(x,y) \le 1$ and $(x, y, 1/2)$ is paper location.
+% The shape of an edge torn along a line is obtained by intersecting
+% the surface with a plane cutting through the tearing line.
+% The intersection is then rotated around the tearing line until
+% it is horizontal and stretched so that $z = 0$ and $z = 1$ lines 
+% of the cutting plane correspond to the desired maximum positive
+% and negative displacements of the edge.
+% 
+% If the cutting plane is vertical, the shape of the torn edge is attached.
+% If the angle between $z = 1/2$ plane and the cutting plane is
+% small enough, the plane will cut off maxima of the surface,
+% producing a sprinkeld shape.
 
 \section{Hardware-accelerated implementation}