In Balance
A follow-up to Map , this is also the Mercator projection of circles on a rotating sphere. In this case, I’m taking concentric circles about the point (0,1,0), then rotating around the x-axis.
The key is to use ContourPlot[]: the concentric circles correspond to points on the sphere with a given dot product with (0,1,0), but what I’m actually doing is applying ContourPlot[] to the composition of the inverse Mercator projection (from the plane to the sphere) and the dot product with (0,1,0). To define the inverse Mercator map, recall that the Mercator projection is defined in terms of longitude and latitude by (\lambda, \phi) \mapsto (\lambda, \ln(\tan(\pi/4+\phi/2))). That means that a point (x,y) (with -\pi \leq x \leq \pi) is mapped to by the point on the sphere with latitude x and longitude 2( \arctan e^y-\pi/4), which to say, the point (\operatorname{sech} y \cos x, \operatorname{sech} y \sin x,\tanh y) in Cartesian coordinates:
InverseMercator[{x_, y_}] := {Sech[y] Cos[x], Sech[y] Sin[x], Tanh[y]};
To make the animation, then, it’s just a matter of choosing a color scheme (in this case "GrayTones") and then evaluating ContourPlot[]:
Manipulate[
ContourPlot[
InverseMercator[{x, y}].RotationTransform[?, {1, 0, 0}][{0, 1, 0}],
{x, -?, ?}, {y, -?, ?}, Frame -> False,
ImageSize -> 540, ContourStyle -> None,
Contours -> Range[-.95, .95, .1], PlotRangePadding -> -0.01,
ColorFunction -> "GrayTones"],
{?, 0, 2 ?}]
