/[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 1.83 by jvk, Thu Nov 28 19:02:19 2002 UTC revision 1.84 by jvk, Fri Nov 29 08:00:27 2002 UTC
# Line 378  so as to make the movement of $A$ look n Line 378  so as to make the movement of $A$ look n
378    
379  There should be some maximum radius $r$ of distortion such that  There should be some maximum radius $r$ of distortion such that
380  whether a point $y$ is in $B$ depends only on the set $A \cap B(y, r)$,  whether a point $y$ is in $B$ depends only on the set $A \cap B(y, r)$,
381  where  where $B(y, r)$ is a ball of radius $r$ centered at $y$.
382  $A \cap B(y, r) = \emptyset$ implies $y \notin B$ and  $A \cap B(y, r) = \emptyset$ should imply $y \notin B$ and
383  $A \cap B(y, r) = B(y,r)$ implies $y \in B$.  $A \cap B(y, r) = B(y,r)$ should imply $y \in B$.
384  Additionally, the mapping from $A$'s to $B$'s should be continuous  Additionally, the mapping from $A$'s to $B$'s should be continuous
385  so that slow movement of $A$ results into ``rippling'' in $B$.  so that slow movement of $A$ results into smooth ``rippling'' in $B$.
386  Furthermore, the rippling should not be too fast.  Furthermore, the rippling should not be too fast.
387    
388  %  %
# Line 401  with points in $B$: Line 401  with points in $B$:
401  \[  \[
402      B = \{\, y \mid \exists x \in A: x R y \,\}.      B = \{\, y \mid \exists x \in A: x R y \,\}.
403  \]  \]
404  If $d(x,y) \le r$ for all $x R y$, the maximum distortion requirement  If $\Vert x - y\Vert \le r$ for all $x R y$, the maximum distortion requirement
405  is satisfied.  is satisfied.
406    
407  If $R$ is a continuous function, i.e., $x R y \Leftrightarrow y = G(x)$,  If $R$ is a continuous function, i.e., $x R y \Leftrightarrow y = G(x)$,
408  the irregular shapes will be connected (if the original shape is connected),  the irregular shapes will be connected (if the original shape is connected),
409  but some parts of original  but some parts of original
410  shape may map to overlapping parts in the irregular shape.  shape may map to overlapping parts in the irregular shape.
411  Furthermore, the inside of $A$ may get displaced to outside of the  Furthermore, the inside of $A$ may get displaced to the outside of the
412  image of the border of $A$.  image of its border.
413    
414  If $R$ is a continuous function in the inverse direction, i.e.,  If $R$ is a continuous function in the inverse direction, i.e.,
415  $x R y \Leftrightarrow F(y) = x$,  $x R y \Leftrightarrow F(y) = x$,
# Line 424  The inverse function mapping is actually Line 424  The inverse function mapping is actually
424  texture shading hardware:  texture shading hardware:
425  the image of the undistorted shape can be stored in a texture and accessed  the image of the undistorted shape can be stored in a texture and accessed
426  with texture coordinates read from (or offset by) another texture storing the  with texture coordinates read from (or offset by) another texture storing the
427  inverse function (or $F(y) - y$). This is called a dependent (or offset)  inverse function (or the offset $F(y) - y$).
428  texture access.  This is called a dependent (or offset) texture access.
429    
430  The forward function cannot be efficiently implemented on fragment level,  The forward function cannot be efficiently implemented on fragment level,
431  because each fragment may depend on multiple values of the function.  because each fragment may depend on multiple values of the function.
# Line 433  The edge of the shape could be displaced Line 433  The edge of the shape could be displaced
433  but that is likely to not yield good performance if the shape has fine detail.  but that is likely to not yield good performance if the shape has fine detail.
434  Furthermore, intersections in the edge may cause additional problems.  Furthermore, intersections in the edge may cause additional problems.
435  If the function is bijection, so as to avoid any intersections, we  If the function is bijection, so as to avoid any intersections, we
436  can just as well use the invert it and use the inverse mapping.  can just as well use the inverse mapping.
437    
438  ---  ---
439    
# Line 453  Consider a half-plane shape Line 453  Consider a half-plane shape
453  \[  \[
454      A = \{\, x \mid n \cdot (x - x_0) \le 0 \,\}      A = \{\, x \mid n \cdot (x - x_0) \le 0 \,\}
455  \]  \]
456  representing one line segment, where $n$ is the normal of the line.  representing one line segment, where $n$ is the unit normal of the line.
457    
458  The offset mapping yields $y \in B$, iff  The offset mapping yields $y \in B$, iff
459  \[  \[
# Line 466  i.e., iff Line 466  i.e., iff
466  Thus, offset has only effect in the normal direction and the problem  Thus, offset has only effect in the normal direction and the problem
467  is essentially reduced to one dimension.  is essentially reduced to one dimension.
468  We can simply define the ``scattered case'' as: $F(y) = -n f(y)$,  We can simply define the ``scattered case'' as: $F(y) = -n f(y)$,
469  where the real function $f(y)$ specifies offset in the normal direction.  where the real function $f(y)$ specifies offset in the (negative)
470    normal direction.
471  This yields $y \in B$, iff  This yields $y \in B$, iff
472  \begin{equation}  \begin{equation}
473      f(y) \ge n \cdot (y - x_0),      f(y) \ge n \cdot (y - x_0),
474  \end{equation}  \end{equation}
475  The movement is still natural. XXX  The movement is still natural, because the resulting distortion
476    is continuous and local. XXX: rotation?
477    
478  An anlogy to the edge displacement case, i.e. the distorted edge defined  Consider then the edge displacement case, i.e. the distorted edge defined
479  by $y = x + G(x)$ for $x$ in the undistorted edge, by  by $y = x + G(x)$ for $x$ in the undistorted edge.
480  limiting the displacement of the shape edge to the normal direction, i.e,  Analogously to the offset function, the displacement can be limited
481  by defining $G(x) = n f(x)$, where $f(x)$ is the normal displacement.  to the normal direction by defining $G(x) = n f(x)$,
482    where $f(x)$ is the normal displacement.
483  With one-dimensional displacement, the distorted edge cannot intersect  With one-dimensional displacement, the distorted edge cannot intersect
484  itself, and we obtain $y \in B$, iff  itself, and we obtain $y \in B$, iff
485  \begin{equation}  \begin{equation}

Legend:
Removed from v.1.83  
changed lines
  Added in v.1.84

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