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\input preamble.tex
\def\fr{\displaystyle\frac}
% ---------------------------------------------------------------------------
\begin{document}
\unitlength1.125cm
\begin{center}
{\Huge \bf{Ellipse Parameters}}
\bigskip
\begin{lapdf}(16,16)(-8,-11)
\Lingrid(10)(1,1)(-8,8)(-11,5)
\Setwidth(0.01)
\Dash(1)
\Polygon(-7.2,-3.2)(0,4)(7.2,-3.2)(-7.2,-3.2)(0,-10.4)(7.2,-3.2) \Stroke
\Polygon(5.33,-1.33)(-5.33,-1.33)(0,-6.67)(5.33,-1.33) \Stroke
\Polygon(4.5,-0.5)(-4.5,-0.5)(0,-5)(4.5,-0.5) \Stroke
\Dash(0)
\Setwidth(0.02)
\Red
\Rcurve(128)(-4,0,3)(0,4,2)(4,0,3) \Stroke
\Rcurve(128)(-4,0,3)(0,4,-2)(4,0,3) \Stroke
\Green
\Rcurve(96)(-4,0,2)(0,4,1)(4,0,2) \Stroke
\Rcurve(96)(-4,0,2)(0,4,-1)(4,0,2) \Stroke
\Blue
\Rcurve(64)(-4,0,3)(0,4,1)(4,0,3) \Stroke
\Rcurve(64)(-4,0,3)(0,4,-1)(4,0,3) \Stroke
\Point(1)(-4,0)
\Point(1)(0,4)
\Point(1)(4,0)
\Point(1)(7.2,-3.2)
\Point(1)(4,-6.4)
\Point(1)(0,-10.4)
\Point(1)(-4,-6.4)
\Point(1)(-7.2,-3.2)
\Point(1)(-5.33,-1.33)
\Point(1)(-4,-2.67)
\Point(1)(0,-6.67)
\Point(1)(4,-2.67)
\Point(1)(5.33,-1.33)
\Point(1)(-4.5,-0.5)
\Point(1)(-4,-1)
\Point(1)(0,-5)
\Point(1)(4,-1)
\Point(1)(4.5,-0.5)
\Point(0)(0,-3.2)
\Point(0)(0,-1.33)
\Point(0)(0,-0.5)
\end{lapdf}
{\large $w=2/3$, $w=1/2$, $w=1/3$}
\end{center}
\parskip0.2cm
We know the center $M=(x_m,y_m)$ and the values of $a$ and $b$. We want
to calculate the curve points $P_0$, $P_1$ and $P_2$ and the weight
$w$ to draw the ellipse. With $r=\sqrt{a^2+b^2}$ we get:
\begin{equation}
P_0={{x_m-\fr{a^2}{r}}\choose{y_m+\fr{b^2}{r}}} \quad
P_1={{xm}\choose{ym+r}} \quad
P_2={{x_m+\fr{a^2}{r}}\choose{y_m+\fr{b^2}{r}}} \quad
w_0=1 \quad
w_1=\pm\fr{b}{r} \quad
\end{equation}
With these weights we can draw the ellipse with two segments. One segment
uses the positive and the other the negative weight $w_1$.
\end{document}
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