/[gzz]/gzz/Documentation/Manuscripts/Irregu/irregu.tex
ViewVC logotype

Diff of /gzz/Documentation/Manuscripts/Irregu/irregu.tex

Parent Directory Parent Directory | Revision Log Revision Log | View Patch Patch

revision 1.79 by jvk, Thu Nov 28 15:22:14 2002 UTC revision 1.80 by jvk, Thu Nov 28 16:44:45 2002 UTC
# Line 372  and discuss Line 372  and discuss
372  a simple algorithm for a shape with the desired properties.  a simple algorithm for a shape with the desired properties.
373    
374  Suppose $A \subset \mathbf{R}^2$ is the part of the canvas we want to tear off,  Suppose $A \subset \mathbf{R}^2$ is the part of the canvas we want to tear off,
375  for example, a rectangle or a smooth ellipse.  for example, a rectangle or an ellipse.
376  We want to define the corresponding irregular piece $B \subset \mathbf{R}^2$  We want to define the corresponding irregular piece $B \subset \mathbf{R}^2$
377  so as to satisify the ``natural movement'' property:  so as to make the movement of $A$ look natural.
378    There should be some maximum radius $r$ of rippling, so that
379    whether $y \in B$ depends only on $A \cap B(y, r)$. Furthermore
380    $A \cap B(y, r) = \emptyset$ should imply $y \notin B$ and
381    $A \cap B(y, r) = B(y,r)$ should imply $y \in B$.
382    
383    One way of defining such properties is to start with the requirement that:
384  \[  \[
385      A = A_1 \cup A_2 \Rightarrow B = B_1 \cup B_2,      A = A_1 \cup A_2 \Rightarrow B = B_1 \cup B_2,
386  \]  \]
387  for all $A, A_1, A_2 \subset \mathbf{R}^2$ and  for all $A, A_1, A_2 \subset \mathbf{R}^2$ and
388  matching irregular shapes $B, B_1, B_2 \subset \mathbf{R}^2$.  matching irregular shapes $B, B_1, B_2 \subset \mathbf{R}^2$.
389  From this porperty, it follows that the mapping from $A$'s to $B$'s  From this porperty, it follows that the mapping from $A$'s to $B$'s
390  is actually defined by a point relation $R \in \mathbf{R}^2$:  is actually defined by a point relation $R \subset \mathbf{R}^2$:
391  \[  \[
392      B = \{\, y \mid \exists x \in A: x R y \,\}.      B = \{\, y \mid \exists x \in A: x R y \,\}.
393  \]  \]
# Line 431  We split the edge of the original shape Line 437  We split the edge of the original shape
437  each assumed to be a straight line segment, and then consider  each assumed to be a straight line segment, and then consider
438  the problem of distorting each one separately.  the problem of distorting each one separately.
439    
440  Consider a half-plane shape \[  Consider a half-plane shape
441    A = \{\, x \mid n \cdot (x - x_0) \le 0 \,\},  \[
442        A = \{\, x \mid n \cdot (x - x_0) \le 0 \,\}
443  \]  \]
444  where $n$ is the normal of the edge line.  representing one line segment, where $n$ is the normal of the line.
445    
446  The offset mapping yields $y \in B$, iff  The offset mapping yields $y \in B$, iff
447  \[  \[
448      y + F(y) \in A,      y + F(y) \in A,
449  \]  \]
450  i.e.,  i.e., iff
451  \begin{equation}  \begin{equation}
452      n \cdot (y - x_0) + n \cdot F(y) \le 0,      n \cdot (y - x_0) + n \cdot F(y) \le 0,
453  \end{equation}  \end{equation}
454  Thus, offset has only effect in the normal direction and the problem  Thus, offset has only effect in the normal direction and the problem
455  is essentially reduced to one dimension.  is essentially reduced to one dimension.
456  Simply define the ``scattered case'' as:  We can simply define the ``scattered case'' as:
457  \begin{equation}  \begin{equation}
458      n \cdot (y - x_0) + f(y) \le 0,      n \cdot (y - x_0) + f(y) \le 0,
459  \end{equation}  \end{equation}
460  where the real function $f$ specifies offset in the normal direction.  where the real function $f$ specifies offset in the normal direction.
461  The movement is still natural.  The movement is still natural. XXX
462    
463  An anlogy to the forward function case can be made by  An anlogy to the forward function case can be made by
464  limiting the displacement of the shape edge to the normal direction, i.e,  limiting the displacement of the shape edge to the normal direction, i.e,
# Line 470  if we define $f(x) = -d(x)$. Line 477  if we define $f(x) = -d(x)$.
477  Thus, we have a ``connected case'' in a very similar formulation  Thus, we have a ``connected case'' in a very similar formulation
478  to the above ``scattered case''.  to the above ``scattered case''.
479    
   
 - consider vector valued ripple function  
   
 XXX: relaxing the ``natural movement'' property \dots  
   
   
480  ---  ---
481    
482  To start off, assume that we are drawing the torn edge inside a  To start off, assume that we are drawing the torn edge inside a
# Line 502  A suitable choice for $f$ would be Line 503  A suitable choice for $f$ would be
503  a function with noise at different frequencies, but with lower frequencies  a function with noise at different frequencies, but with lower frequencies
504  emphasized more, such as turbulence\cite{perlin-noise-intro}.  emphasized more, such as turbulence\cite{perlin-noise-intro}.
505    
   
506  Although these algorithms seem different and produce different results, there is  Although these algorithms seem different and produce different results, there is
507  actually a general formulation which yields to a visual explanation.  actually a general formulation which yields to a visual explanation.
508  Both algorithms can be seen as computing the intersection of a  Both algorithms can be seen as computing the intersection of a

Legend:
Removed from v.1.79  
changed lines
  Added in v.1.80

savannah-hackers-public@gnu.org
ViewVC Help
Powered by ViewVC 1.1.26