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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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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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