Spherical Coordinates

Nestharus · 05-22-2011, 02:51 AM

The spherical coordinate system maps points relative to an origin given two angles and a distance.

What makes this different from the Cartesian coordinate system is that rather than defining a point by its x,y,z distance from the origin, it defines a point by the magnitude of an x,y,z point and two angles (magnitude being distance).

The magnitude is defined as ||p|| or ||r|| = SquareRoot(x*x+y*y+z*z)

Notice how it has 3 dimensions. The z just needs to be added in order to get that third dimension.

The two angles define direction.

In the polar coordinate system, one angle and a distance is needed. This angle is theta, which is how far around a circle a given point goes.

Theta-

Code:

x = r*cos(theta)
y = r*sin(theta)
r = ||p|| = SquareRoot(x*x + y*y)
tan(theta) = y/x

Note that because z is 0, it doesn't do anything in the polar coordinate system.

By adding a z, the magnitude expands and a second angle is needed, this angle being how far the point is from the northern z-axis (easier this way).

The range of phi goes from 0 to pi, 0 being north and pi being south.

In polar coordinates, phi is always pi/2.

Equations are as follows

Code:

x = p*cos(theta)*sin(phi)
y = p*sin(theta)*sin(phi)
z = p*cos(phi)

p = SquareRoot(x*x + y*y + z*z)
r = SquareRoot(x*x + y*y)
tan(theta) = y/x
tan(phi) = r/z

Angle Between Points (x0, y0, z0) and (x, y, z).
(x0, y0, z0) would be the first point

Code:

where
----xd = x - x0
----yd = y - y0
----zd = z - z0
----pd = SquareRoot(xd*xd + yd*yd + zd*zd)
----rd = SquareRoot(xd*xd + yd*yd)

tan(theta) = yd/xd
tan(phi) = rd/zd

Distance between two 3D points

Code:

where
----xd = x - x0
----yd = y - y0
----zd = z - z0
----pd = SquareRoot(xd*xd + yd*yd + zd*zd)

distance = pd

Jass:

thetaBetween = Atan2((y - y0), (x - x0))

phiBetween = Atan2(SquareRoot((x - x0)*(x - x0) + (y - y0)*(y - y0)), z - z0)

distanceBetween = SquareRoot((x - x0)*(x - x0) + (y - y0)*(y - y0) + (z - z0)*(z - z0))

More information can be found at these sites

Magtheridon96 · 05-22-2011, 12:54 PM

I put them in a library for you ^^

Jass:

library CoordUtils

    // All the math done by Nestharus

    // Compiled to one library by Magtheridon96

    // I based it on locations, but it can be easily changed to work with coordinates :)

    function GetDistance takes location v, location w returns real

        local real xd=GetLocationX(w)-GetLocationX(v)

        local real yd=GetLocationY(w)-GetLocationY(v)

        local real zd=GetLocationZ(w)-GetLocationZ(v)

        return SquareRoot(xd*xd+yd*yd+zd*zd)

    endfunction

    // Returns angle in radians

    function GetPHI takes location v, location w returns real

        return Atan2(GetDistance(v,w),GetLocationZ(w)-GetLocationZ(v))

    endfunction

    // Returns the theta in radians

    function GetTHETA takes location v, location w returns real

        return Atan2(GetLocationY(w)-GetLocationY(v),GetLocationX(w)-GetLocationX(v))

    endfunction

endlibrary