Euclidean Affine Space

up:: Affine Space

An affine space $\left(E,\vec{E},+\right)$ is said to be an Euclidean affine space when its underlying vector space $\vec{E}$ is a Euclidean Vector Space with a Metric Function in $E$ (induced from its Inner Product $⟨\cdot ,\cdot ⟩$).

That is, given any points $P,Q\in E$ (i.e. separated by a vector $\overrightarrow{PQ}\in \vec{E}$), their distance can be defined as

$$\rho (P,Q)=||\overrightarrow{PQ}||=\sqrt{⟨\overrightarrow{PQ},\overrightarrow{PQ}⟩}$$

This structure is meant to represent physical space ─ in particular Newtonian Reference Frames.

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References

• Euclidean space - Wikipedia