Transforming surface normal vectors and tangent vectors

Question

According to a book Physically Based Rendering: From Theory to Implementation. By Matt Pharr, Greg Humphreys (link, p. 86-87),
surface tangent vectors are transformed as common vectors, using transformation matrix M, but
surface normal vectors are transformed using .

I wonder why scaling does make a normal incorrect, but doesn't touch a tangent vector? Why are normals so special?

See the figure from the book.

I have read that such transformation for normal is needed to maintain orthogonality of the normal and tangent. But I'd like to get some intuitive explanation.

Answer 1

For me, the intuition would be that for rotations (and in general all transformation that can be described by an orthogonal matrix) satisfy . That means that , so for those kinds of transformations the treatment is not special at all.

A simple example with a non-orthogonal, symmetric matrix that illustrates that it is not sufficient to use to transform the normal is

Here see that you need to transform the normal using , which in the symmetric case is equal to .

Notice that this covers already quite a lot of transformations. Personally, I find that transformations that are not orthogonal and not symmetric are themselves not very intuitive, so I resort to the mathematical explanation that it is required to maintain orthogonality. Since this is the defining property of a surface normal I find this argument quite plausible. Maybe writing it out makes things a bit clearer:

So the advantage of the transformation rule from the book is that it gives you the right normal for all transformation you can think of.

Hope this helps.