Two basic examples of the Gauss map
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The two pictures below show the Gauss map for two basic examples: a saddle-shaped surface (e.g. the graph of \(f(x,y) = x^2 - y^2\)) and a bowl-shaped surface (e.g. the graph of \(f(x,y) = -x^2 - y^2\)). Both pictures show the domain of the parametrisation on the left. At one point in the domain (initially at the origin in both pictures) the corresponding unit normal vector on the surface is shown in the 3D picture. On the left of the surface you can see the Gauss map, which represents the unit normal vector as a point in the unit sphere.
Now you can also see how the derivative of the Gauss map affects the shape of the surface. In the first example of a saddle-shaped surface, you can click and drag the point (initially at the origin) to move it around the domain of the parametrisation, and see how the Gauss map is affected. You can see that moving the point up or down in the domain also moves the Gauss map in the same direction, while moving it left or right in the domain moves the Gauss map in the opposite direction.
Now do the same thing for the bowl-shaped surface below. Moving the point to the left or right in the domain will also move the Gauss map in the same direction, and similarly moving the point up or down will also move the Gauss map in the same direction. Therefore we can see the difference between the two surfaces: moving the point by a small amount in the domain moves the Gauss map in different directions, depending on the shape of the surfaces. We will make this precise when we define the shape operator of a surface.
