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integrated last points from Efis review

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Sven Oesau 2025-11-20 14:31:06 +01:00
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6
Barycentric_coordinates_3/doc/Barycentric_coordinates_3/Barycentric_coordinates_3.txt
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@ -133,7 +133,7 @@ We adopt the simple formula below to compute tetrahedron coordinates of the quer
\f$w_i = \frac{V_i}{V}\f$ \f$w_i = \frac{V_i}{V}\f$
</center> </center>
where \f$V_i\f$ is the signed volume of the sub-tetrahedron opposite to the vertex \f$i\f$ and \f$V\f$ where \f$V_i\f$ is the signed volume of the sub-tetrahedron opposite to the vertex \f$i\f$, i.e., the tetrahedron where the vertex \f$i\f$ is replaced by the query point `q`. \f$V\f$
is the total volume of the tetrahedron, that is \f$V = V_0 + V_1 + V_2 + V_3\f$. is the total volume of the tetrahedron, that is \f$V = V_0 + V_1 + V_2 + V_3\f$.
These coordinates can be computed exactly if an exact number type is chosen, for any query point and with respect to any non-degenerate tetrahedron. No special cases are handled. These coordinates can be computed exactly if an exact number type is chosen, for any query point and with respect to any non-degenerate tetrahedron. No special cases are handled.
@ -193,8 +193,8 @@ star-shaped polyhedron.
Efficiency is crucial in this implementation. Efficiency is crucial in this implementation.
These coordinates are used in applications that require These coordinates are used in applications that require
calculations for millions of points; thus developing metrics calculations for millions of points; thus, developing metrics
to evaluate performance is absolutely necessary. In this section, to evaluate performance is necessary. In this section,
we present benchmark results for each algorithm. we present benchmark results for each algorithm.
The benchmark and runtimes are evaluated by regularly sampling The benchmark and runtimes are evaluated by regularly sampling