ellipse-nb.tex

   1 \documentclass{article}
   2 \usepackage{ametsoc}
   3 \usepackage{amsfonts}
   4 \usepackage{amssymb}
   5 \usepackage{graphicx}
   6 \bibliographystyle{ametsoc}
   7 \begin{document}
   8 Write the equation of an ellipse with horizontal axis $a$ and vertical axis
   9 $b$ in parametric form
  10 $$
  11 \left[
  12 \begin{array}
  13 [c]{c}
  14 x\\
  15 y
  16 \end{array}
  17 \right]  =\left[
  18 \begin{array}
  19 [c]{c}
  20 a\cos s\\
  21 b\sin s
  22 \end{array}
  23 \right] .
  24 $$
  25 Rotate by the angle $\theta\in(-\pi,\pi]$ clockwise:
  26 $$
  27 \left[
  28 \begin{array}
  29 [c]{c}
  30 x\\
  31 y
  32 \end{array}
  33 \right]  =\left[
  34 \begin{array}
  35 [c]{cc}
  36 \cos\theta & \sin\theta\\
  37 -\sin\theta & \cos\theta
  38 \end{array}
  39 \right]  \left[
  40 \begin{array}
  41 [c]{c}
  42 a\cos s\\
  43 b\sin s
  44 \end{array}
  45 \right]  .
  46 $$
  47 Multiplying out we get
  48 $$
  49 \left[
  50 \begin{array}
  51 [c]{c}
  52 x\\
  53 y
  54 \end{array}
  55 \right]  =\left[
  56 \begin{array}
  57 [c]{c}
  58 a\cos\theta\cos s+b\sin\theta\sin s\\
  59 -a\sin\theta\cos s+b\cos\theta\sin s
  60 \end{array}
  61 \right]  .
  62 $$
  63 Move the center vertically so that the point at distance $c$ from the bottom
  64 vertex on the $b$ axis is at $y=0$,
  65 $$
  66 \left[
  67 \begin{array}
  68 [c]{c}
  69 x\\
  70 y
  71 \end{array}
  72 \right]  =\left[
  73 \begin{array}
  74 [c]{c}
  75 a\cos\theta\cos s+b\sin\theta\sin s\\
  76 -a\sin\theta\cos s+b\cos\theta\sin s+(b-c)\cos\theta
  77 \end{array}
  78 \right]
  79 $$
  80 This is the equation of the ellipse from the figure. The rate of spread in the
  81 direction of the normal equivalent to the ellipse is the distance of the
  82 horizontal lines at $y=0$ and tangent to the top of the rotated shifted
  83 ellipse
  84 $$
  85 R=\max_{s}-a\sin\theta\cos s+b\cos\theta\sin s+(b-c)\cos\theta
  86 $$
  87 The find the highest point, set
  88 $$
  89 y^{\prime}\left(  s\right)  =\frac{\partial}{\partial s}\left(  -a\sin
  90 \theta\cos s+b\cos\theta\sin s+(b-c)\cos\theta\right)  =0
  91 $$
  92 which gives
  93 $$
  94 a\sin\theta\sin s+b\cos\theta\cos s=0
  95 $$
  96 We can either divide by $\sin\theta\neq0$,
  97 $$
  98 \frac{\sin s}{\cos s}+\frac{b}{a}\frac{\cos\theta}{\sin\theta}=0,
  99 $$
 100 and compute $s$ from
 101 $$
 102 s=-\arctan\left(  \frac{b\cos\theta}{a\sin\theta}\right)
 103 $$
 104 Using the arctan2 function in numpy
 105 $$
 106 s=-\mathop{arctan2}\left(  b\cos\theta,a\sin\theta\right)
 107 $$
 108 gives the correct result even for $\sin\theta=0.$ In any case, we get two solutions, $s$
 109 and $s+\pi$,  substitute in the equation of the ellipse
 110 $$
 111 y=-a\sin\theta\cos s+b\cos\theta\sin s+\left(  b-c\right)  \cos\theta
 112 $$
 113 and take the larger value:
 114 $$
 115 R=\max\left\{  u,-u\right\}  +c\cos\theta,\quad u=-a\sin\theta\cos
 116 s+b\cos\theta\sin s.
 117 $$
 118
 119
 120 \nocite{Mandel-2009-DAW}
 121 \bibliography{
 122 /Users/jmandel/daseminar/references/bigdata.bib,
 123 /Users/jmandel/daseminar/references/by_Aime.bib,
 124 /Users/jmandel/daseminar/references/epi.bib,
 125 /Users/jmandel/daseminar/references/extra.bib,
 126 /Users/jmandel/daseminar/references/geo.bib,
 127 /Users/jmandel/daseminar/references/jm.bib,
 128 /Users/jmandel/daseminar/references/ml.bib,
 129 /Users/jmandel/daseminar/references/other.bib,
 130 /Users/jmandel/daseminar/references/quad-jm.bib,
 131 /Users/jmandel/daseminar/references/slides.bib,
 132 /Users/jmandel/daseminar/references/spdes.bib
 133 }
 134
 135 \end{document}