\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}