9.8.1  What is the problem?

People often find an annoying problem when applying normal modifier patterns to objects. It is said that one image tells more than a thousand words, and this saying also applies here. This image shows two cases where the problem appears:

Sometimes odd shadow lines appear on certain objects
Sometimes odd shadow lines appear on certain objects

As you notice, there are two clear artifacts in the image. The sphere has a straight shadow line which seems unnatural and the mesh has a non-straight shadow line when it's supposed to have a straight one.

Although the artifacts look quite different in nature, they are, in fact caused by the exact same problem.

The image one would expect
The image one would expect

What one could expect would be something like this image (don't mind about the bright part under the triangle mesh; this is explained later).