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I'm studying elementary differential geometry and I'm trying to understand how the Gauss map is a map into the unit sphere. The definition I'm working with is

$\textbf{Def}$: Let $S$ be a regular surface. A Gauss map of $S$ is a continous map $N:S\to\mathbb{R}^3$ such that

$1)$ for any $p$, if the base of the vector $N(p)$ is placed at $p$, then $N(p)$ is perpendicular to the tangent plane.

$2)$ $N(p)$ is a unit vector.

Question: what is the mathematical definition of "base" here? And is there an explicit correspodence between the Gauss map and the unit sphere?

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2 Answers 2

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The base is understood as the origin point of the vector.

The Gauss map is embedded in the unit sphere (it is the set of the endpoints of the normal vectors when you move their origin to the center).

enter image description here

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  • $\begingroup$ Those images are very helpful! Thanks $\endgroup$
    – user119264
    Commented Aug 7, 2018 at 11:10
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An euclidean vector can be defined using two points, a base (sometimes called origin or initial point) and an end (or terminal point). For example, take the points $P=(0,1,0)$ and $Q=(1,0,2)$; the vector with base $P$ and end $Q$ is $\vec{PQ}=P-Q=(-1,1,-2)$. This is just the vector that goes from $P$ to $Q$.

You can send every vector to the origin by a traslation of the base and the end. Therefore, unit vectors are those whose ends lie in the unit sphere after this translation.

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