Merge remote-tracking branch 'cgal/master' into gsoc2022-isosurface
This commit is contained in:
@@ -996,15 +996,13 @@ to derive new closed-form equations for the corrected curvature measures. These
|
||||
curvature measures are the first step for computing the curvatures. For a triangle \f$ \tau_{ijk} \f$,
|
||||
with vertices \a i, \a j, \a k:
|
||||
|
||||
\f[
|
||||
\begin{align*}
|
||||
\f{align*}{
|
||||
\mu^{(0)}(\tau_{ijk}) = &\frac{1}{2} \langle \bar{\mathbf{u}} \mid (\mathbf{x}_j - \mathbf{x}_i) \times (\mathbf{x}_k - \mathbf{x}_i) \rangle, \\
|
||||
\mu^{(1)}(\tau_{ijk}) = &\frac{1}{2} \langle \bar{\mathbf{u}} \mid (\mathbf{u}_k - \mathbf{u}_j) \times \mathbf{x}_i + (\mathbf{u}_i - \mathbf{u}_k) \times \mathbf{x}_j + (\mathbf{u}_j - \mathbf{u}_i) \times \mathbf{x}_k \rangle, \\
|
||||
\mu^{(2)}(\tau_{ijk}) = &\frac{1}{2} \langle \mathbf{u}_i \mid \mathbf{u}_j \times \mathbf{u}_k \rangle, \\
|
||||
\mu^{\mathbf{X},\mathbf{Y}}(\tau_{ijk}) = & \frac{1}{2} \big\langle \bar{\mathbf{u}} \big| \langle \mathbf{Y} | \mathbf{u}_k -\mathbf{u}_i \rangle \mathbf{X} \times (\mathbf{x}_j - \mathbf{x}_i) \big\rangle
|
||||
-\frac{1}{2} \big\langle \bar{\mathbf{u}} \big| \langle \mathbf{Y} | \mathbf{u}_j -\mathbf{u}_i \rangle \mathbf{X} \times (\mathbf{x}_k - \mathbf{x}_i) \big\rangle,
|
||||
\end{align*}
|
||||
\f]
|
||||
\f}
|
||||
where \f$ \langle \cdot \mid \cdot \rangle \f$ denotes the usual scalar product,
|
||||
\f$ \bar{\mathbf{u}}=\frac{1}{3}( \mathbf{u}_i + \mathbf{u}_j + \mathbf{u}_k )\f$.
|
||||
|
||||
|
||||
Reference in New Issue
Block a user