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In differential geometry, the Gauss map of a surface is a function that maps each point in the surface to a unit vector that is orthogonal to the surface at that point. Namely, given a surface X in Euclidean space R3, the Gauss map is a map N: X → S2 (where S2 is the unit sphere) such that for each p in X, the function value N(p) is a unit vector orthogonal to X at p. The Gauss map is named after Carl F. Gauss.
The Gauss map can be defined (globally) if and only if the surface is orientable, in which case its degree is half the Euler characteristic. The Gauss map can always be defined locally (i.e. on a small piece of the surface). The Jacobian determinant of the Gauss map is equal to Gaussian curvature, and the differential of the Gauss map is called the shape operator.
Gauss first wrote a draft on the topic in 1825 and published in 1827.
There is also a Gauss map for a link, which computes linking number.