Maths - Spherical Polar Coordinates

Spherical coordinates allow points to be specified using one linear distances and two angles:

• r = radius (distance from axis of sphere)
• Latitude
• Longitude

This can be used to represent points on the surface of a sphere such as the earth as explained on this page.

Cartesian to Spherical Polar Coordinates

r latitude longitude

=

sqrt(x2 + y2+ z2) arcos(z/r) atan2(y,x)

Spherical Polar Coordinates to Cartesian

We can assign an arbitrary x,y,z coordinate system in the local frame of the earth:

so from this diagram we can see that:

z = r sin(latitude)

and if we are on the Greenwich meridian then:

x = r cos(latitude)

but if we are not on the Greenwich meridian then this has to be modified depending on the latitude, so,

x = r cos(latitude) cos (longitude)

the y can be calculated from:

r² = x² + y² + z²

therefore y = r *sqrt(1 - sin(latitude) - cos(latitude) cos (longitude))

x y z

=

r cos(latitude) r *sqrt(1 - sin(latitude) - cos(latitude) cos (longitude)) r sin(latitude)

EuclideanSpace

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