Rate of change for no stretch/compression

Question

I am reading about cloth simulation from here.

This is what one of the part says -

Shouldn't the condition for no compression/stretching be

W_u = 0

If there is no stretch/compression along u, the rate of change along u should be 0, right ?

Why does he instead points to a condition -

||W_u|| = 1.

for same.

Answer 1

$||\textbf{w}_u || = 0$ would imply that the cloth has been compressed to a line along the $v$-direction.

Notice, that the coordinates $(u,v)$ correspond to points on the cloth, in a type of coordinate system that follows the stretching/compression. If you took a sufficiently stretchy piece of cloth and drew a grid onto it, the grid would deform when you stretch the cloth.

When $||\textbf{w}_u || = 1$, we have that the change in cloth position corresponds exactly to a change in spatial position, in the $u$-direction. That is, the grid on the cloth has not been stretched or compressed in the $u$-direction.