Error of Points in Mollweide Projection

Question

I am currently working on a project using SnIa data and using Mathematica I have obtained the following points (in degrees) through the maximum likelihood process

point1 = {72.85 \pm 60.6242, 210.26 \pm 136.593};
point2 = {-25.17 \pm 10.23, 104.70 \pm 32.47 };
point3 = {25.17 \pm 10.22, 284.70 \pm 48.32};
point4 = {18 \pm 10, 309 \pm 23};
point5 = {-18 \pm 9, 129 \pm 23};
point6 = {-15.1 \pm 11.5, 309.4 \pm 18};

where \pm=$\pm$ and corresponds to the 1$\sigma$ error of the l and b. Mapping the best fit values of the data as follows:

GeoGraphics[{AbsolutePointSize@5, RGBColor[0.09, 0.35, 0.11], 
  Point[GeoPosition@point1], AbsolutePointSize@5, Purple, 
  Point[GeoPosition@point2], AbsolutePointSize@5, Purple, 
  Point[GeoPosition@point3], AbsolutePointSize@5, Red, 
  Point[GeoPosition@point4], AbsolutePointSize@5, Red, 
  Point[GeoPosition@point5], AbsolutePointSize@5, Blue, 
  Point[GeoPosition@point6]}, GeoRange -> {{-90, 90}, {0, 360}}, 
 GeoGridLinesStyle -> Directive[Black, Dashed], 
 GeoProjection -> "Mollweide", 
 GeoGridLines -> {Automatic, {0, 30, 60, 90, 120, 150, 180, 210, 240, 
    270, 300, 330, 360}}, GeoBackground -> White, Axes -> True, 
 ImagePadding -> 25, ImageSize -> 1000, 
 Ticks -> {ticksldeg, ticksbdeg}, Background -> White, 
 AxesStyle -> Black, TicksStyle -> 15]

I have obtained the following picture So I want in the final Mollweide plot that I have uploaded, to include a circle or an ellipse (different color for each point) that corresponds to the errors of l and b, however I am not sure how to do that.

Thank you in advance

Answer 1

I would try something like this:

point1 = {Around[72.85, 60.6242], Around[210.26, 136.593]};
point2 = {Around[-25.17, 10.23], Around[104.70, 32.47]};
point3 = {Around[25.17, 10.22], Around[284.70, 48.32]};
point4 = {Around[18, 10], Around[309, 23]};
point5 = {Around[-18, 9], Around[129, 23]};
point6 = {Around[-15.1, 11.5], Around[309.4, 18]};

The Mollweide projection maps 90 degrees to flat distance Sqrt[2], so define this scale factor:

scale = Sqrt[2]/90;

This function will create an ellipse (of that scale) on the map at the point {l, b}, where both l, b are expected to be Around objects:

ellipse[{l_, b_}] := Circle[GeoPosition[{l["Value"], b["Value"]}], {b["Uncertainty"], l["Uncertainty"]} scale]

This is the GeoGraphics call, where I have removed axes and ticks, that you have under control:

GeoGraphics[{
  RGBColor[0.09, 0.35, 0.11], ellipse[point1],
  Purple, ellipse[point2],
  Purple, ellipse[point3],
  Red, ellipse[point4],
  Red, ellipse[point5], 
  Blue, ellipse[point6]
},
 GeoRange -> {{-90, 90}, {0, 360}}, 
 GeoGridLinesStyle -> Directive[Black, Dashed], 
 GeoProjection -> "Mollweide",
 GeoGridLines -> Automatic, 
 GeoBackground -> None
]

That result corresponds to ellipses on the projected map, not ellipses on the original spheroid. Motivated by a further question from the OP, I give here an alternative way of doing this using GeoCircle, which is a geometric construction on the original spheroid. Unfortunately their is no built-in notion of geo ellipses, so I'll use geo circles whose radius is the average of the semiaxes of the ellipse. To emphasize and clarify the structure of the very large case that covers the North pole, I'll use GeoDisk instead of GeoCircle. Simply replace the definition of ellipse in the previous construction by

ellipse[{l_, b_}] := GeoDisk[GeoPosition[{l["Value"], b["Value"]}], Quantity[(b["Uncertainty"] + l["Uncertainty"])/2, "AngularDegrees"]]

Then the result is