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revision 1.80 by jvk, Thu Nov 28 16:44:45 2002 UTC revision 1.81 by jvk, Thu Nov 28 18:03:08 2002 UTC
# Line 375  Suppose $A \subset \mathbf{R}^2$ is the Line 375  Suppose $A \subset \mathbf{R}^2$ is the
375  for example, a rectangle or an 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 make the movement of $A$ look natural.  so as to make the movement of $A$ look natural.
 There should be some maximum radius $r$ of rippling, so that  
 whether $y \in B$ depends only on $A \cap B(y, r)$. Furthermore  
 $A \cap B(y, r) = \emptyset$ should imply $y \notin B$ and  
 $A \cap B(y, r) = B(y,r)$ should imply $y \in B$.  
378    
379  One way of defining such properties is to start with the requirement that:  There should be some maximum radius $r$ of distortion such that
380  \[  whether a point $y$ is in $B$ depends only on the set $A \cap B(y, r)$,
381      A = A_1 \cup A_2 \Rightarrow B = B_1 \cup B_2,  where
382  \]  $A \cap B(y, r) = \emptyset$ implies $y \notin B$ and
383  for all $A, A_1, A_2 \subset \mathbf{R}^2$ and  $A \cap B(y, r) = B(y,r)$ implies $y \in B$.
384  matching irregular shapes $B, B_1, B_2 \subset \mathbf{R}^2$.  Additionally, slow movement of $A$ should result into small
385  From this porperty, it follows that the mapping from $A$'s to $B$'s  changes in $B$ so that the rippling is not too fast.
386  is actually defined by a point relation $R \subset \mathbf{R}^2$:  XXX: continuous mapping?
387    
388    %
389    %to start with the requirement that:
390    %\[
391    %    A = A_1 \cup A_2 \Rightarrow B = B_1 \cup B_2,
392    %\]
393    %for all $A, A_1, A_2 \subset \mathbf{R}^2$ and
394    %matching irregular shapes $B, B_1, B_2 \subset \mathbf{R}^2$.
395    %From this porperty, it follows that the mapping from $A$'s to $B$'s
396    %is actually defined by a point relation $R \subset (\mathbf{R}^2)^2$:
397    %
398    One way of defining such mapping is a relation
399    $R \subset (\mathbf{R}^2)^2$ that identifies the points of $A$
400    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 $R$ is a function, i.e., $x R y \Leftrightarrow y = G(x)$,  If $d(x,y) \le r$ for all $x R y$, the maximum distortion requirement
405    is satisfied.
406    Furthermore, $R$ should be continuous to create natural rippling.
407    
408    If $R$ is a continuous function, i.e., $x R y \Leftrightarrow y = G(x)$,
409  the irregular shapes will be connected (if the original shape is connected),  the irregular shapes will be connected (if the original shape is connected),
410  but some parts of original  but some parts of original
411  shape may map to overlapping parts in the irregular shape.  shape may map to overlapping parts in the irregular shape.
412  Furthermore, the inside of $A$ may get displaced to outside of the  Furthermore, the inside of $A$ may get displaced to outside of the
413  image of the border of $A$.  image of the border of $A$.
414    
415  If $R$ is a function in the inverse direction, i.e.,  If $R$ is a continuous function in the inverse direction, i.e.,
416  $x R y \Leftrightarrow F(y) = x$,  $x R y \Leftrightarrow F(y) = x$,
417  there will be no overlapping.  there will be no overlapping.
418  Specifically, the edge of $A$ will map to the edge of $B$.  Specifically, the edge of $A$ will map to the edge of $B$.

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