The Sphere Manifold

The surface of a sphere requires at least two charts to be represented as a Manifold. One way to do this is to use two planes, one attached to the north pole, the other to the south pole. The points in these planes are projected onto the surface of the sphere by drawing a line from the opposite pole to the point on the plane, and taking the projection to be the point at which this line intersects the sphere.

Imagine a sphere with two circles, one mapped to each pole

For a point in the north pole plane, (r,s,1), the opposite pole is (0,0,-1), and the line between them is

                    x=t*r
                    y=t*s
                    z=-1+2*t

The south pole is t=0, and the point in the north pole plane is t=1. The intersection with the sphere satisfies x^2+y^2+z^2=1, which determines t. This gives a quadratic, which must have one solution of t=0 (since the south pole is on the sphere). The quadratic is

              t^2 (r^2+s^2+4) -4 t = 0

So t=-4/(r^2+s^2+4), and the north pole projection is

              x(r,s)=4r/(r^2+s^2+4)
              y(r,s)=4s/(r^2+s^2+4)
              z(r,s)=1-8/(r^2+s^2+4)

The south pole is similar, and it turns out to be of the same form, with a z coordinate of the opposite sign.

Imagine a sphere with chart mappings labeled

Now that we have the chart mappings, we must decide how much of the base spaces we need to include to make sure that the charts overlap. The equator (z=0) is r^2+s^2=4 (for either chart). So as long as we keep a little more of the base space than a circle of radius 2 the charts will overlap. How much overlap to allow is pretty much arbitrary. I chose to extend the north pole chart down to z=-1/2 and the south pole up to z=1/2. This gives circles of radius 2 sqrt(3).

Finally, since there are two charts and they overlap, we must compute the mapping between the charts. This is simply a matter of writing the equation. If the point in chart 0 (the north pole mapping) is (r,s), and in chart 1 (the south pole mapping) is (u,v), then for these to represent the same point on the sphere, we have

      x(r,s) = 4r/(r^2+s^2+4)  = 4u/(u^2+v^2+4)  = x(u,v)
      y(r,s) = 4s/(r^2+s^2+4)  = 4v/(u^2+v^2+4)  = y(u,v)
      z(r,s) = 1-8/(r^2+s^2+4) = 8/(u^2+v^2+4)-1 = z(u,v)

From the z equation we have that

     r^2+s^2+4= 4(u^2+v^2+4)/(u^2+v^2)
     u^2+v^2+4= 4(r^2+s^2+4)/(r^2+s^2)

So that

      r = u(r^2+s^2+4)/(u^2+v^2+4) = 4u/(u^2+v^2)
      s = v(r^2+s^2+4)/(u^2+v^2+4) = 4v/(u^2+v^2)
and
      u = r(u^2+v^2+4)/(r^2+s^2+4) = 4r/(r^2+s^2)
      v = s(u^2+v^2+4)/(r^2+s^2+4) = 4s/(r^2+s^2)

So, for the sphere manifold we have

Base Space Dimension = 2.
numberOfChartsInAtlas=2.
isitInChart((r,s),chart 0)=
isitInChart((r,s),chart 1)=
Target Space Dimension = 3.
Target Space = Euclidean 3-space.
getPoint((r,s),chart 0)=
getPoint((r,s),chart 1)=
mapBetweenCharts((r,s),chart 0, chart 1)=
```
(4r/(r*r+s*s),4s/(r*r+s*s))
```
mapBetweenCharts((r,s),chart 1, chart 0)=
```
(4r/(r*r+s*s),4s/(r*r+s*s))
```
getChartsThatOverlap(chart 0, list )=1, and on return list={1}
getChartsThatOverlap(chart 1, list )=1, and on return list={0}
isitDifferentiable=TRUE.
isitOpen=TRUE.
isitDiscrete=FALSE.

For more details, see the reference for the NASphere class.

[Comments | NAO Home Page]

[Research home page]

[ IBM home page | Order | Privacy | ContactIBM | Legal ]