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revision 1.87 by jvk, Fri Nov 29 11:20:40 2002 UTC revision 1.88 by jvk, Fri Nov 29 11:37:45 2002 UTC
# Line 379  We want to define the corresponding irre Line 379  We want to define the corresponding irre
379  so as to make the movement of $A$ look natural.  so as to make the movement of $A$ look natural.
380    
381  There should be some maximum radius $r$ of distortion such that  There should be some maximum radius $r$ of distortion such that
382  whether a point $y$ is in $B$ depends only on the set $A \cap B(\q, r)$,  whether a point $\q$ is in $B$ depends only on the set $A \cap B(\q,r)$,
383  where $B(\q, r)$ is a ball of radius $r$ centered at $\q \in \mathbf{R}^2$.  where $B(\q,r)$ is a ball of radius $r$ centered at $\q \in \mathbf{R}^2$.
384  $A \cap B(\q, r) = \emptyset$ should imply $\q \notin B$ and  $A \cap B(\q,r) = \emptyset$ should imply $\q \notin B$ and
385  $A \cap B(\q, r) = B(y,r)$ should imply $\q \in B$.  $A \cap B(\q,r) = B(\q,r)$ should imply $\q \in B$.
386  Additionally, the mapping from $A$'s to $B$'s should be continuous  Additionally, the mapping from $A$'s to $B$'s should be continuous
387  so that slow movement of $A$ results into smooth ``rippling'' in $B$.  so that slow movement of $A$ results into smooth ``rippling'' in $B$.
388  Furthermore, the rippling should not be too fast.  Furthermore, the rippling should not be too fast.
# Line 497  to the above ``scattered case''. Line 497  to the above ``scattered case''.
497  ---  ---
498    
499  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
500  given \emph{envelope}.  given smooth \emph{envelope}.
501  The envelope is parametrized as a mapping $E(x,y)$ to canvas coordinates  The envelope is parametrized as a mapping $E(x,y)$ to canvas coordinates
502  so that $E(x,0)$ and $E(x,1)$ are the inner and outer edges of the  so that $E(x,0)$ and $E(x,1)$ are the inner and outer edges of the
503  envelope, respectively, and the ripples are contained between these two curves.  envelope, respectively, and the ripples are contained between these two
504    smooth curves.
505  An envelope can be defined with a \emph{spine} $E(x,1/2)$ and a normal vector  An envelope can be defined with a \emph{spine} $E(x,1/2)$ and a normal vector
506  $N(x)$ so that $E(x,y) = E(x,1/2) + (y-1/2) N(x)$.  $N(x)$ so that $E(x,y) = E(x,1/2) + (y-1/2) N(x)$.
507  The envelope should not intersect itself.  The envelope should not intersect itself.
508    
509    This formulation allows us to consider the distortion only
510    in the normal ($y$) direction for each point $x$ in the edge.
511    On the other hand, we will need to specifically consider the corners
512    later.
513    
514  The edge curve $C(x)$ of a conneccted shape can be obtained by simply  The edge curve $C(x)$ of a conneccted shape can be obtained by simply
515  displacing the edge in the normal direction of the envelope by a function  displacing the edge in the normal direction by a function
516  $0\le f(\p)\le 1$ which only depends on the location $\p$ of the spine:  $0\le f(\p)\le 1$ which only depends on the location $\p$ of the spine:
517  $C(x) = E(x, f(E(x,1/2)))$.  $C(x) = E(x, f(E(x,1/2)))$.
518    
# Line 521  emphasized more, such as turbulence\cite Line 527  emphasized more, such as turbulence\cite
527    
528  These two algorithms correspond to one-dimensional displacement  These two algorithms correspond to one-dimensional displacement
529  and offset distortions, where the one dimension is in the normal direction.  and offset distortions, where the one dimension is in the normal direction.
530  The corresponding function and inverse function are  In the relation formulation, the corresponding forward and inverse functions
531    are
532  \begin{eqnarray}  \begin{eqnarray}
533    G_n(\p) &=& \p + (r N) (2f(\p) - 1), \\    G_n(\p) &=& \p + (r N) (2f(\p) - 1), \\
534    F_n(\q) &=& \q - (r N) (2f(\q) - 1),    F_n(\q) &=& \q - (r N) (2f(\q) - 1),

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