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If I were standing in a triangle - How do I calculate the inverse of my position? Can it be done? It's easy inside a rectangle, but I can't think of how you would do it inside of a triangle.

I'm working on a color theory in case it matters.

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3  
Inverse of position? – rschwieb Oct 19 '12 at 19:04
    
Could you elaborate with an example using the rectangle? – Jeremy Oct 19 '12 at 19:08
    
well if i'm in one corner of a rectangle i know the inverse of my position is the opposite corner. but with a triangle it is different. But i suspect there must be a way to calculate an inverse. or maybe just the farthest point from any given point. – Marshall House Oct 19 '12 at 19:14
    
equilateral by the way^ – Marshall House Oct 19 '12 at 19:15
    
Ah, you could use some transformation, like rotating $120^\circ$ around the center.. Is it a regular triangle at all? Or you can use reflections to a midline.. what would you prefer? – Berci Oct 19 '12 at 19:15

This question is quite hard to answer, because we aren't provided with a satisfying definition of "inverse of a position". So in order for my answer to make sense I wil first have to introduce a definition. I will do this for a very general case, namely: for the boundary $\partial S$ of an $n$-dimenstional convex set $S$ in the Eucledian space $\mathbb R^n$. In the OP for example, $\partial S$ is a triangle.

Consider a point $x\in \partial S$ and any point $P\in\overset{~\circ}S$. The inverse point of $x$ with respect to $P$ is now defined as the intesection between $\partial S$ and the line through $x$ and $P$, (which is not $x$ itself). Notation: $\mathfrak I_P(x)$.

The map $\mathfrak I_P:\partial S\to \partial S$ is now well-defined due to the convexness. It can even be extended to the whole of $\bar S$ in a very obvious way.

So loosely speaking, we are reflecting in the point $P$. Any interior point $P$ will do in fact.

In the OP's case we are considering an equilateral triangle. Now an equilateral triangle has one particular nice point (opinion based), which is of course its incentre. So a very natural choise would be to choose this point for $P$.


This somewhat answers the question, however as with any definition, the OP is free to make another one that suites his needs better. The most important thing for the OP to know is that the "inverse of a position" has no conventional definion in general. So in any such case one must make their own definition, check that it is well-defined and that is usefull.

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