/[gzz]/gzz/Documentation/Manuscripts/Irregu/irregu.tex
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revision 1.98 by jvk, Sat Nov 30 07:59:28 2002 UTC revision 1.99 by jvk, Sat Nov 30 08:16:45 2002 UTC
# Line 540  $N(x)$ so that $E(x,y) = E(x,1/2) + (y-1 Line 540  $N(x)$ so that $E(x,y) = E(x,1/2) + (y-1
540  The envelope should not intersect itself.  The envelope should not intersect itself.
541    
542  This formulation allows us to consider the distortion only  This formulation allows us to consider the distortion only
543  in the normal ($y$) direction for each point $x$ in the edge.  in the normal direction ($y$-axis) for each point $x$ in the edge.
544  On the other hand, we will need to specifically consider the corners  On the other hand, we will need to specifically consider the corners
545  later.  later.
546    
547  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
548  displacing the edge in the normal direction by a function  displacing the edge in the normal direction by a function
549  $0\le f(\p)\le 1$ which only depends on the location $\p$ of the spine:  $f(\p)$: $C(x) = E(x, 1/2 + f(E(x,1/2)))$.
 $C(x) = E(x, f(E(x,1/2)))$.  
550    
551  The scattered case, on the other hand, can be obtained through a decision process:  The scattered case, on the other hand, can be obtained by offsetting
552  again using a function $f(\p)$ of location $\p$,  in the normal direction: using an offset function $f(\p)$ of location $\p$:
553  a given point $E(x,y)$ is {\em inside} the tear-out,  a given point $E(x,y)$ is {\em inside} the tear-out,
554  iff $f(E(x,y)) > y$.  iff $y - f(E(x,y)) < 1/2$. Here, the offset function is negated
555    for reasons of other formulations. XXX
556    
557  A suitable choice for $f$ would be  A suitable choice for $f$ would be
558  a function with noise at different frequencies, but with lower frequencies  a function with noise at different frequencies, but with lower frequencies
559  emphasized more, such as turbulence\cite{perlin-noise-intro}.  emphasized more, such as turbulence\cite{perlin-noise-intro}.
560    
561  These two algorithms correspond to one-dimensional displacement  These one-dimensional displacement and offset mappings correspond
562  and offset distortions, where the one dimension is in the normal direction.  to the following 2D forward and inverse functions:
 In the relation formulation, the corresponding forward and inverse functions  
 are  
563  \begin{eqnarray}  \begin{eqnarray}
564    G_n(\p) &=& \p + (r N) (2f(\p) - 1), \\    G_N(\p) &=& \p + (r N) f(\p), \\
565    F_n(\q) &=& \q - (r N) (2f(\q) - 1),    F_N(\q) &=& \q - (r N) f(\q),
566  \end{eqnarray}  \end{eqnarray}
567  where $N$ is the unit normal, $r$ is the distortion radius,  where $N$ is the unit normal, $r$ is the distortion radius,
568  and the undistorted shape $A$ is the region inside the spine $E(x,1/2)$.  and the undistorted shape $A$ is the region inside the spine $E(x,1/2)$.

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