From dcf6a64b7910a0c12a170aa50ba502b748798725 Mon Sep 17 00:00:00 2001 From: emiliie Date: Thu, 12 Mar 2015 19:00:06 +0100 Subject: [PATCH 1/2] circles --- report/subsections/circles.tex | 10 ++++++---- 1 file changed, 6 insertions(+), 4 deletions(-) diff --git a/report/subsections/circles.tex b/report/subsections/circles.tex index b9365f8..4b30567 100644 --- a/report/subsections/circles.tex +++ b/report/subsections/circles.tex @@ -4,11 +4,13 @@ \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 : - $ - 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} From f9377a14ae3f37e603a6521da14062ce17f54ba3 Mon Sep 17 00:00:00 2001 From: emiliie Date: Thu, 12 Mar 2015 19:09:01 +0100 Subject: [PATCH 2/2] modification circles --- report/subsections/circles.tex | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/report/subsections/circles.tex b/report/subsections/circles.tex index 4b30567..c0430e9 100644 --- a/report/subsections/circles.tex +++ b/report/subsections/circles.tex @@ -10,7 +10,7 @@ and then we calculate the center of the characteristic circle with this formula : - $\overrightarrow{Cp}(t) = \overrightarrow{C}(t) - r'(t)r(t)/ /| (\overrightarrow{C'}(t)/| ^{2} * \overrightarrow{C'}(t)$ + $\overrightarrow{Cp}(t) = \overrightarrow{C}(t) - r'(t)r(t)/ /| \overrightarrow{C'}(t)/| ^{2} * \overrightarrow{C'}(t)$ \begin{figure}[H] \begin{center}