Why is the normal transformation a inverse transpose? (3)

The difference between normal vectors and usual vectors

Actually, normal vectors and usual vectors are different type of vectors. A usual vector itself defines direction or position. Let's recall how the normal vector is defined. A normal vector is not defined by itself. First a face is defined using usual vectors, then, the normal vector is defined according to this surface. The definition of the normal vector is a unit vector that perpendicular to the surface, in other words, it is defined as the inner product of surface tangent vector and the normal vector is 0. We should think these vectors are different vectors when they follow a transformation. As we saw in the Figure 1, when we transform a normal vector, the meaning of normal is lost. If we consider this vector is a usual vector, twice large x component is still fine, just it is not normal anymore. In this sense, normal vector has a special meaning, that should be always perpendicular to the surface.

The following three sections, we will see this problem more deeply. All three explanations are mathematically the same, but I think each has a slightly different intuition, so I will write all the three explanations.

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