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science forum beginner

Joined: 10 Jul 2006
Posts: 12

Posted: Mon Jul 10, 2006 10:29 am    Post subject: Projected small circles

Hi,

I'm having a problem with drawing small circles (in other words,
ranges) of radius r around a centre point C on a projected map (Plate
Carrée (or Equirectangular) projection).
For example, I'd like an outline of all the places in the world x
distance away from London.

Lat/long coords of the centre poiunt are simple to work out via the
map. The ideal answer would be to return more lat/long coords to draw
on the map.

I know I need to move from R^2 into R^3 then back to R^2, but any
attempts at this has just confused me! I was only ever good at pure
maths at university.

So, any help would be appreciated...
Chris Riddle
matt271829-news@yahoo.co.
science forum Guru

Joined: 11 Sep 2005
Posts: 846

Posted: Mon Jul 10, 2006 6:15 pm    Post subject: Re: Projected small circles

 Quote: Hi, I'm having a problem with drawing small circles (in other words, ranges) of radius r around a centre point C on a projected map (Plate Carrée (or Equirectangular) projection). For example, I'd like an outline of all the places in the world x distance away from London. Lat/long coords of the centre poiunt are simple to work out via the map. The ideal answer would be to return more lat/long coords to draw on the map. I know I need to move from R^2 into R^3 then back to R^2, but any attempts at this has just confused me! I was only ever good at pure maths at university.

Let r be the earth's radius, and assume the earth is a perfect sphere.
Let lat_0 and lng_0 be the latitude and longitude of the central point
C, and let d be the "radius" of the circle drawn around it, measured as
the great circle arc length from C. (Sorry, I switched your variables
names.)

Let angle t be a parameter that ranges from zero to 2*pi (radians) to
describe this circle. Set up x-y-z coordinate axes, with the centre of
earth at the origin, positive z axis through the north pole, equator
lying on the x-y plane, and positive x axis through latitude 0,
longitude 0. I get the "normalised" (x,y,z) coordinates of the point on
the circle corresponding to t as given by

x = Cos(d/r)*Cos(lat_0)*Cos(lng_0) -
Sin(d/r)*Sin(lat_0)*Cos(lng_0)*Sin(t) - Sin(d/r)*Sin(lng_0)*Cos(t)
y = Cos(d/r)*Cos(lat_0)*Sin(lng_0) -
Sin(d/r)*Sin(lat_0)*Sin(lng_0)*Sin(t) + Sin(d/r)*Cos(lng_0)*Cos(t)
z = Cos(d/r)*Sin(lat_0) + Sin(d/r)*Cos(lat_0)*Sin(t)

(Probably this could be expressed more elegantly in matrix notation!)
"Normalised" means that I have divided all the coordinates through by
r, as if the point were on the unit sphere, since the r's will just
cancel in the next step anyway. Note that the angle d/r is in radians.

And then the latitude and longitude of this point are given by

lat = arcsin(z)
lng = arctan(x,y)

where the two-parameter "quadrant sensitive" version of arctan is used.
science forum beginner

Joined: 10 Jul 2006
Posts: 12

 Posted: Tue Jul 11, 2006 8:54 am    Post subject: Re: Projected small circles Perfect... Thanks for your help! Chris Riddle

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