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. |
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There should be some maximum radius $r$ of rippling, so that |
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whether $y \in B$ depends only on $A \cap B(y, r)$. Furthermore |
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$A \cap B(y, r) = \emptyset$ should imply $y \notin B$ and |
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$A \cap B(y, r) = B(y,r)$ should imply $y \in B$. |
|
378 |
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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 |
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388 |
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% |
389 |
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%to start with the requirement that: |
390 |
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%\[ |
391 |
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% A = A_1 \cup A_2 \Rightarrow B = B_1 \cup B_2, |
392 |
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%\] |
393 |
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%for all $A, A_1, A_2 \subset \mathbf{R}^2$ and |
394 |
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%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 |
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% |
398 |
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One way of defining such mapping is a relation |
399 |
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$R \subset (\mathbf{R}^2)^2$ that identifies the points of $A$ |
400 |
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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 |
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is satisfied. |
406 |
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Furthermore, $R$ should be continuous to create natural rippling. |
407 |
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408 |
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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 |
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|
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 $B$. |