43 lines
1.9 KiB
TeX
43 lines
1.9 KiB
TeX
\subsection{Regular mesh}
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At the beginning of this part we have a set of 3D splines. We sample this splines and calculate the characteristic circle associated to each sampled point. Then we match the points between successive circles and use a triangular mesh.
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\subsubsection{Characteristic circles computation}
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We take a point sampled on the spline. It gives us information about the sphere located on this point : $\overrightarrow{C}(t)$ the sphere center coordinates and r(t) its radius. We also have their derivatives : $\overrightarrow{C'}(t)$ and r'(t). The idea is to find the intersection between this sphere and the characteristic plane, which would give us the characteristic circle.
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If a point P is on the circle, then we can write the formula \ref{eq1} :
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\begin{equation}
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<\overrightarrow{C'}(t),\overrightarrow{PC}(t)> -r'(t)r(t) = 0
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\label{eq1}
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\end{equation}
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and then we calculate the center of the characteristic circle with the formula \ref{eq2}:
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\begin{equation}
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\overrightarrow{Cp}(t) = \overrightarrow{C}(t) - \frac{r'(t)r(t)}{\| \overrightarrow{C'}(t)\| ^{2}} \times \overrightarrow{C'}(t)
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\label{eq2}
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\end{equation}
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\begin{figure}[H]
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\begin{center}
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\includegraphics[scale=0.8]{img/characteristicCircle}
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\caption{\label{initialSchedule} Characteristic circle calculation}
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\end{center}
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\end{figure}
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\subsubsection{Meshing}
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After obtaining characteristic circles we sample them.
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Then we need to find matching points. We decided to make points sampling start in one direction of our mark.
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When this is done we project the mark on the new circle and sample again. Then we link each point to the point of the same index on the next circle.
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The final mesh can be seen Figure \ref{regularMesh}.
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\begin{figure}[H]
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\begin{center}
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\includegraphics[scale=0.4]{img/regularMesh}
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\caption{\label{regularMesh} The regular mesh}
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\end{center}
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\end{figure}
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%TODO
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