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The two triangles $xyz$ and $x^{\prime}y^{\prime}z^{\prime}$, shown below, have opposite orientations. A closed curve $abcd$ is embedded in the first triangle ($abcd$). The vertices of the corresponding closed discrete curve $a^{\prime}b^{\prime}c^{\prime}d^{\prime}$ are computed by the method of barycentric coordinates. As shown below $a^{\prime}b^{\prime}c^{\prime}d^{\prime}$ will have opposite orientation as $abcd$.

I'd like to know:

  1. why the difference in orientation of the embedding triangles leads to a difference in the orientations of the embedded discrete curves.

  2. if reversing the orientation of the triangle $xyz$ to match that of $x^{\prime}y^{\prime}z^{\prime}$ is sufficient to is sufficient to ensure that $abcd$ and $a^{\prime}b^{\prime}c^{\prime}d^{\prime}$ have the same orientations.

two triangles with opposite orientations

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