[PLUG-TALK] Re: Ellipsoid uniform random distribution
spike
spike at pcez.com
Wed Feb 21 19:37:11 UTC 2007
Russell Senior wrote:
>
>spike> This doesn't look uniform-on-a-sphere to me:
>
>
>Try it.
>
>According to my 1968 Schaum's Outline _Mathematical Handbook of
>Formulas and Tables_, 4.41 gived a formula for the area of a spherical
>cap of radius r and height h (from the pole). The formula is:
>
> Surface area = 2 * pi * r * h
>
>That is, the surface area is linear with respect to the height of the
>cap. In other words, the distribution of area on a sphere is linearly
>with respect to the z axis.
>
>Also, recall trigonometry. The z-dimension halfway to the pole is
>*not* at the 45th parallel. ASIN(0.5) = 30 degrees latitude. SIN(45
>degrees) = 0.707...
>
>
>
>
Thanks for setting me straight.
Yesterday I was looking at the same formula (S = 2*pi*R*h) in my
CRC Standard Mathematical Tables but the linearity didn't register.
My main error was confusing 45 degrees of colatitude (phi) with
z=1/2. I expect I'll remember the difference for several months.
spike
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