Spherical Grids

A spherical grid is a finite list of directions used to represent the sphere in a numerical calculation. Each direction in the list is a node. Such nodes are needed to integrate a spherical function, sample one, or draw it.

A sphere cannot be covered by one rectangular angular mesh without distortion or a coordinate singularity. Grid constructions balance node spacing against other useful properties. These include equal-area cells, hierarchical structure, and regular indexing in spherical angles.

Read Vectors first for the MTEX direction model. Spherical Projections explains the upper-hemisphere view. The examples construct directions on the complete sphere. The plot option 'upper' hides the lower hemisphere; it does not identify a direction with its negative. That distinction is explained in Axes.

For these figures, the plotting convention lays the reference frame out with y up and x to the right. It does not alter the constructed nodes.

plottingConvention.default('y↑→x');

Four Constructions

MTEX offers the regularS2Grid and the equispacedS2Grid. Two further choices are the HEALPixS2Grid and fibonacciS2Grid.

The regular grid is a tensor product of polar and azimuth angles. The equispaced grid changes the number of azimuth steps from one latitude to the next. HEALPix supplies the centres of hierarchical, equal-area, iso-latitude pixels. The Fibonacci grid follows a golden-angle spiral.

Each constructor accepts a target angular resolution. The target does not guarantee one exact nearest-neighbour distance. Each construction adjusts or rounds it according to its own geometry.

% a regular grid in the two spherical angles
grids{1} = regularS2Grid('resolution',7*degree);

% the MTEX equispaced grid
grids{2} = equispacedS2Grid('resolution',7*degree);

% the HEALPix grid
grids{3} = HEALPixS2Grid('resolution',7*degree);

% the Fibonacci grid
grids{4} = fibonacciS2Grid('resolution',7*degree);

gridNames = {'regular','equispaced','HEALPix','Fibonacci'};

The upper-hemisphere view makes the different node patterns visible.

plot(grids{1},'upper','layout',[1 4])
mtexTitle(gridNames{1})

for k = 2:4
  nextAxis
  plot(grids{k},'upper')
  mtexTitle(gridNames{k})
end

Notice how the latitude rows of the regular grid converge near the centre. At the pole, every azimuth represents the same direction. The equispaced and HEALPix patterns reduce the number of nodes on shorter latitude rings. The Fibonacci spiral has no latitude-ring structure.

The requested resolution also produces different numbers of nodes.

nodeCounts = cellfun(@length,grids)

nodeCounts =
        1404         812         768         827

At \(7^{\circ}\) the four constructions contain 1404, 812, 768 and 827 nodes, respectively. These counts include both hemispheres. They reflect how each constructor interprets the target resolution and are not by themselves a ranking of grid quality.

Comparison of Uniformity

Node uniformity can be measured instead of only judged from a plot. Here every node receives the same weight. Density Estimation smooths the resulting discrete measure into a function on the sphere. MTEX normalizes that function to have mean value 1. A uniform equal-weight node measure should therefore be close to the constant 1.

This diagnostic measures equal-weight node placement at the chosen halfwidth. It does not test cell areas, nearest-neighbour distances, or the accuracy of a quadrature rule.

for k = 1:4
  density(k) = calcDensity(grids{k},'halfwidth',5*degree);
end

clf
for k = 1:4
  plot(density(k),'upper','layout',[2,2]);
  mtexTitle(gridNames{k})
  if k < 4, nextAxis, end
end
setColorRange('equal')
mtexColorbar

All four panels use the same colour range. The regular grid has a strong maximum at the pole, whereas the other three panels remain close to the uniform value 1. The numerical comparison below uses the complete sphere, not only the displayed upper hemisphere.

The \(L^2\) norm gives the square root of the integrated squared deviation.

l2Deviation = norm(density-1).'

l2Deviation =
    4.0141    0.0317    0.0426    0.0201

The integrated absolute deviation is an \(L^1\) measure of the same error.

l1Deviation = sum(abs(density-1)).'

l1Deviation =
    5.7668
    0.0600
    0.0674
    0.0320

At a \(5^{\circ}\) halfwidth, the \(L^2\) deviations are 4.0141, 0.0317, 0.0426 and 0.0201. The \(L^1\) deviations are 5.7668, 0.0600, 0.0674 and 0.0320. The regular grid is about two orders of magnitude less uniform by both measures in this experiment. The Fibonacci grid has the smallest deviation.

This comparison is not a universal ranking. Changing the resolution or smoothing halfwidth changes the numbers, and a downstream algorithm may require cell areas, quadrature weights, or a particular grid structure.

Choosing a Grid

Use a regular grid when values must sit on a rectangular raster in the two spherical angles, for example for a surface or contour representation. Use an equispaced or Fibonacci grid when nearly uniform equal-weight nodes matter more than rectangular indexing. Use HEALPix when the centres of its standard equal-area pixelization are required.

A nearly uniform point set is not automatically a quadrature rule. If an integral must be exact for a specified class of functions, use the nodes and weights prescribed by that integration method.

Further Reading

Next

Grids on the rotation group are covered in Orientation Grids. Those grids can respect crystal symmetry. Choosing informative nodes for a spherical function is a different question, treated in Sampling.