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 $E(x,1/2)$. |