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

enter image description here

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?

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  • $\begingroup$ It would be good if you gave the full reference from where you got this expression. $\endgroup$ – Yuri Vyatkin Sep 27 '18 at 7:48
  • $\begingroup$ @YuriVyatkin books.google.co.uk/…, Chapter 5, section 5.4 $\endgroup$ – user8469759 Sep 27 '18 at 9:15
  • $\begingroup$ 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. $\endgroup$ – Anthony Carapetis Sep 30 '18 at 1:27
  • $\begingroup$ 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 $\bf x$. $\endgroup$ – Anthony Carapetis Sep 30 '18 at 1:54
  • $\begingroup$ @AnthonyCarapetis and how does the equality follows? $\endgroup$ – user8469759 Oct 1 '18 at 8:27

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