1 \documentclass{article
}
6 \bibliographystyle{ametsoc
}
8 Write the equation of an ellipse with horizontal axis $a$ and vertical axis
25 Rotate by the angle $
\theta\in(-
\pi,
\pi]$ clockwise:
36 \cos\theta &
\sin\theta\\
37 -
\sin\theta &
\cos\theta
47 Multiplying out we get
58 a
\cos\theta\cos s+b
\sin\theta\sin s\\
59 -a
\sin\theta\cos s+b
\cos\theta\sin s
63 Move the center vertically so that the point at distance $c$ from the bottom
64 vertex on the $b$ axis is at $y=
0$,
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
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
85 R=
\max_{s
}-a
\sin\theta\cos s+b
\cos\theta\sin s+(b-c)
\cos\theta
87 The find the highest point, set
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
94 a
\sin\theta\sin s+b
\cos\theta\cos s=
0
96 We can either divide by $
\sin\theta\neq0$,
98 \frac{\sin s
}{\cos s
}+
\frac{b
}{a
}\frac{\cos\theta}{\sin\theta}=
0,
102 s=-
\arctan\left(
\frac{b
\cos\theta}{a
\sin\theta}\right)
104 Using the arctan2 function in numpy
106 s=-
\mathop{arctan2
}\left( b
\cos\theta,a
\sin\theta\right)
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
111 y=-a
\sin\theta\cos s+b
\cos\theta\sin s+
\left( b-c
\right)
\cos\theta
113 and take the larger value:
115 R=
\max\left\
{ u,-u
\right\
} +c
\cos\theta,
\quad u=-a
\sin\theta\cos
120 \nocite{Mandel-
2009-DAW
}
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