# Conformal energy and first fundamental form

In my reference in mesh processing I came across the following equation

Where $$\bf{I}$$ is the first fundametal form of some surface patch $$\bf{x}$$, I don't know the exact meaning of the notation $$\bf{x}_u$$ orthogonal (I don't know how to do the symbol in the formula). I've tried to expand the expression, but I don't end up with anything.

My assumption was to write the rejection of $$\bf{x}_u$$ respect to $$\bf{x}_v$$, but it doesn't seem to lead me to anything... Any suggestion?

• It would be good if you gave the full reference from where you got this expression. Sep 27, 2018 at 7:48
• @YuriVyatkin books.google.co.uk/…, Chapter 5, section 5.4 Sep 27, 2018 at 9:15
• My guess is that $\cdot^\perp$ denotes a rotation by $\pi/2$ in the tangent plane, since we would then have $x_v -x_u^\perp = 0$ if and only if $x$ is conformal. Sep 30, 2018 at 1:27
• Regardless of what the $\perp$ notation exactly denotes, this "conformal energy" is a well-known quantity in geometry processing, and is more commonly written $\frac12 \int (\sigma_1-\sigma_2)^2\, dA$ where $\sigma_i$ are the singular values of the Jacobian of $\bf x$. Sep 30, 2018 at 1:54
• @AnthonyCarapetis and how does the equality follows? Oct 1, 2018 at 8:27