# Recommend a space to analyze the bearing of the vector between any two points

In Euclidean space, given any two points, the vector connecting them can be characterized by length (distance) and direction (bearing). Now I am only interested in the bearing part. And I found it is inconvenient to analyze the bearing property because in Euclidean space distance is always involved in the analysis.

Is there a space I can map the Euclidean space to so that I can analyze the bearing between any two points better? Say some kind of projection space or something, I don't know. Any suggestions?

Here is an example. Consider a triangle in Euclidean space. What I am interested is the shape, i.e., interior angles of the triangle. I don't care the scale, i.e., length of the sides. How can I map this triangle into another space where I need not care the distance?

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## 1 Answer

You should become familiar with the notion of a configuration space or moduli space in mathematics (unfortunately these both have more precise technical meanings that at this point will distract from the underlying idea). If you are interested in a family of objects, you can often construct a space whose points describe those objects in some way. Moreover this space should have a topology so that you can make sense of a "continuous family" of objects.

In your first example, the space of all directions in $\mathbb{R}^n$ is the sphere $S^{n-1}$. There is also a configuration space $F_2(\mathbb{R}^n)$ describing ordered pairs of distinct points in $\mathbb{R}^n$ and consequently a natural map $F_2(\mathbb{R}^n) \to S^{n-1}$ sending a pair of vectors $u, v$ to $\frac{u-v}{|u-v|}$.

In your second example, the space of all triangles in $\mathbb{R}^n$ can be identified with the configuration space $C_3(\mathbb{R}^n)$ describing unordered triples of distinct points in $\mathbb{R}^n$ (namely the vertices). There is a natural map which translates the triangle until its vertices $a, b, c$ satisfy $a + b + c = 0$ (presumably you don't care about translation either) and another one which scales $a, b, c$ so that the triangle has unit area.

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