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Fix typos in the User Manual
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afabri committed Oct 2, 2023
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Expand Up @@ -121,7 +121,7 @@ In order to approximate complex boundaries well, more anchors are generated by r
\f[ d = d / input\_mesh\_average\_edge\_length. \f]
Optionally, \f$ d \f$ can be measured as the ratio of the chord length:
\f[ d = d / \Vert(\mathbf{a}, \mathbf{b})\Vert. \f]
Also, we can add a dihedral angle weight \f$ sin(\mathbf{N}_i,\mathbf{N}_j) \f$ to the distance measurement, where \f$ \mathbf{N}_i,\mathbf{N}_j \f$ are the normals of the proxies separated by the chord \f$ (\mathbf{a}, \mathbf{b}) \f$. If the angle between proxy \f$ P_i \f$ and \f$ P_j \f$ is rather small, then a coarse approximation will do as it does not add geometric information on the shape. Trivial chords (less than 4 edges) are not subdivided if they are non-circular. In case of circular chords, additional anchors maybe added to maintain the topology even if they are trivial, as detailed in Section \ref sma_anchors_additional.
Also, we can add a dihedral angle weight \f$ sin(\mathbf{N}_i,\mathbf{N}_j) \f$ to the distance measurement, where \f$ \mathbf{N}_i,\mathbf{N}_j \f$ are the normals of the proxies separated by the chord \f$ (\mathbf{a}, \mathbf{b}) \f$. If the angle between proxy \f$ P_i \f$ and \f$ P_j \f$ is rather small, then a coarse approximation will do as it does not add geometric information on the shape. Trivial chords (made of a single edge) are not subdivided. In case of circular chords, additional anchors may be added to maintain the topology, as detailed in Section \ref sma_anchors_additional.

\cgalFigureBegin{chord, chord.jpg}
Varying the chord error. From left to right: clustering partition, and meshing with decreasing absolute chord error 5, 3 and 1 without dihedral angle weight. The boundaries of the partition (red lines) are approximated with increasing accuracy.
Expand All @@ -132,7 +132,7 @@ Varying the chord error. From left to right: clustering partition, and meshing w
For a boundary cycle without any anchor such as the hole depicted Figure \cgalFigureRef{operations}, we first add a starting anchor to the boundary. We then subdivide this circular chord to ensure that every boundary cycle has at least 2 anchors (i.e., every chord is connecting 2 different anchors, Figure \cgalFigureRef{anchors}). Finally, we add additional anchors to ensure that at least three anchor vertices are generated on every boundary cycle.

\cgalFigureBegin{anchors, anchors.jpg}
Adding anchors. From left to right: starting from a partition (grey) with a hole (while) and two encircled regions (green and blue), we add a starting anchor (orange disk) to the boundary cycle (red dash line) without any anchor (2nd), subdivide the circular chord (3rd, the number indicates the level of recursion) and add anchors to the boundary cycle with less than 2 anchors (4th, red dash lines).
Adding anchors. From left to right: starting from a partition (grey) with a hole (white) and two encircled regions (green and blue), we add a starting anchor (orange disk) to the boundary cycle (red dash line) without any anchor (2nd), subdivide the circular chord (3rd, the number indicates the level of recursion) and add anchors to the boundary cycle with less than 2 anchors (4th, red dash lines).
\cgalFigureEnd

\subsubsection sma_triangulation Discrete Triangulation
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