Type I pairs with the variable that runs vertically in the usual representation of the coordinate system. The remaining types are paired with the rest of the variables in ascending order.
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I’ve always had trouble remembering the definitions of type I, II, and III regions. In particular, it’s tricky to remember which type (I, II, III) is paired with which axis ($x,$ $y,$ $z$).
In $2$-dimensional space, type I and II regions are paired with the $x$ and $y$ axes, respectively:
- A region is of type I if it can be decomposed into vertical segments parallel to the $y$-axis, with each $x$-value having at most $1$ segment and the segment endpoints tracing out continuous functions.
- A region is of type II if it can be decomposed into horizontal segments parallel to the $x$-axis, with each $y$-value having at most $1$ segment and the segment endpoints tracing out continuous functions.
But in $3$-dimensional space, type I, II, and III regions are paired with the $z,$ $x,$ and $y$ axes, respectively:
- A region is of type I if it can be decomposed into vertical segments parallel to the $z$-axis, with each $xy$-value having at most $1$ segment and the segment endpoints tracing out continuous functions.
- A region is of type II if it can be decomposed into horizontal segments parallel to the $x$-axis, with each $yz$-value having at most $1$ segment and the segment endpoints tracing out continuous functions.
- A region is of type III if it can be decomposed into horizontal segments parallel to the $y$-axis, with each $xz$-value having at most $1$ segment and the segment endpoints tracing out continuous functions.
It seems as though these definitions aren’t even consistent with each other! Type I pairs with $y$ in $2$-dimensional space, but with $z$ in $3$-dimensional space.
Further complicating the matter, they aren’t even consistent in being inconsistent! Type II pairs with $x$ in $2$-dimensional space, and again with $x$ in $3$-dimensional space.
However, I recently noticed a trend that seems to generalize well:
- Type I pairs with the variable that runs vertically in the usual representation of the coordinate system.
- In $2$-dimensional space, that’s $y,$ and in $3$-dimensional space, that’s $z.$
- The remaining types are paired with the rest of the variables in ascending order.
- In $2$-dimensional space, we’ve already assigned $y$ to type I, and the only remaining variable is $x,$ so we assign it to type II.
- In $3$-dimensional space, we’ve already assigned $z$ to type I, and the remaining variables in alphbetical order are $x,y,$ so we assign $x$ to type II and $y$ to type III.
- In $2$-dimensional space, we’ve already assigned $y$ to type I, and the only remaining variable is $x,$ so we assign it to type II.
Generalizing to $N$-dimensional space with coordinates $(x_1, x_2, x_3, \ldots, x_N),$ we have the following pairings:
- Type I is paired with $x_N$
- Type II is paired with $x_1$
- Type III is paired with $x_2$
- Type IV is paired with $x_3$
- (and so on)
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