Gauss Map and Curvature

The Gauss map maps the unit normal of a surface on the right to the unit sphere on the left The area surrounding the point on the surface is thus mapped to an area on the unit sphere As the radius of the loop approaches zero the ratio of these areas approaches the Gaussian curvature of the surface at the point which is also equal to the product of the principal curvatures the maximum and minimum curvatures of the normal sections through the points

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