In Euclidean geometry two polygons are said to be similar if, by rotation and scaling, one can be transformed into the other, and vice versa. If we consider a general metric space, does this notion exist? I can think of how this works by scaling: you consider $d(a_i , a_{I+1} )$, where $a_i$ are the vertices of the polygon, then two figures are similar if each pair can be scaled with the same constant to yield the distances of the second figure. My problem is more with rotation and angle. The notion if homeomorphic is too general because all (non-self intersecting) polygons are homeomorphic.
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1Not just rotation and scaling, also reflection and translation. – Eric Wofsey Aug 14 '16 at 21:16
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1Hm... So all the symmetries in the plane? Would it help to consider the actions of the symmetry group (is this defined for all metric spaces?) on the polygon? – Juan Sebastian Lozano Aug 14 '16 at 21:21
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In a metric space $(X,d)$ a similitude can be defined as a function $f:X\to X$ such that $ d(f(x),f(y))=rd(x,y) $ for $r>0$. If $X$ is a vector space and the metric is derived from an inner product (so that the notion of angle can be defined) than similarities preserve the angles.
Emilio Novati
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Why does one need an inner product on a vector space in order to be able to define angles? Is the concept of angle only defined for inner product spaces? – Chill2Macht Apr 02 '17 at 09:37
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1In a vector space, without some notion of orthogonality, I don't see haw we can define the angle between two vectors. – Emilio Novati Apr 02 '17 at 09:53
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There are two different ways you can set this up in the language of metric spaces. If $X$ and $Y$ are metric spaces, say that a bijection $f:X\to Y$ is a similarity if there exists a constant $r>0$ such that $d(f(x),f(y))=rd(x,y)$ for all $x,y\in X$. There are then two definitions we could make:
- Let $X$ and $Y$ be metric spaces. Then $X$ and $Y$ are similar if there exists a similarity $f:X\to Y$.
- Let $Z$ be a metric space and $X,Y\subseteq Z$ be subsets. Then $X$ and $Y$ are similar in $Z$ if there exists a similarity $f:Z\to Z$ such that $f(X)=Y$.
Clearly, if $X$ and $Y$ are similar in $Z$, then they are similar by the first definition, since you can just restrict $f:Z\to Z$ to a map $X\to Y$ to get a similarity from $X$ to $Y$. The converse is not true in general, though: there might exist a similarity $X\to Y$ that can't be extended to a similarity from all of $Z$ to itself. However, when $Z=\mathbb{R}^2$, it turns out that any similarity between subsets of $Z$ extends to a similarity from $Z$ to itself. So for subsets of the plane, these two definitions coincide, and are the usual definition of similarity.
What does this have to do with rotations (and reflections and translations)? Well, (compositions of) those are exactly the isometries of the plane. And it turns out that every similarity from $\mathbb{R}^2$ to itself can be written as a composition of a scaling followed by an isometry. In general, though, the right definition of "similarity" is any map which multiplies distances by a fixed nonzero constant.
Eric Wofsey
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