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circles

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emiliie 2015-03-12 19:00:06 +01:00
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report/subsections/circles.tex
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\paragraph{Characteristic circles computation}
We take a point sampled on the spline. It gives us information about the sphere located on this point : C(t) the sphere center coordinates and r(t) its radius. We also have their derivatives : 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.
If a point P is on the circle, then we can write this formula :
$<C'(t),PC(t)> - r'(t)r(t) = 0$
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.
If a point P is on the circle, then we can write this formula : \\
$<\overrightarrow{C'}(t),\overrightarrow{PC}(t)> -r'(t)r(t) = 0$
and then we calculate the center of the characteristic circle with this formula :
$Cp(t) = C(t) – r'(t)r(t)/norm(C'(t))² * C'(t)$
$\overrightarrow{Cp}(t) = \overrightarrow{C}(t) - r'(t)r(t)/ /| (\overrightarrow{C'}(t)/| ^{2} * \overrightarrow{C'}(t)$
\begin{figure}[H]
\begin{center}