Here is a static solution to the problem. It shows a mesh on the sphere that represents the normal lat-long coordinate system.
A function representing the equator.
equator[θ_] := {Cos[θ], Sin[θ], 0}
A function and a plot representing the inclined circle. Note that the inclination is accomplished by a rotation of the equator about the x-axis.
inclinedCircle[θ_, i_] :=
RotationTransform[i, {1, 0, 0}][equator[θ]];
circlePlot[i_] :=
ParametricPlot3D[
Evaluate[inclinedCircle[θ, i °]], {θ, 0, 2 π},
PlotStyle -> {Thickness[0.005]}]
A function and a plot representing the sphere.
sphere[u_, v_] :=
{Cos[u] Cos[v], Sin[u] Cos[v], Sin[v]};
spherePlot =
ParametricPlot3D[
Evaluate[sphere[u, v]], {u, 0, 2 π}, {v, -π, π}];
A great circle inclined at 60°.
Show[{spherePlot, circlePlot[60]},
Axes -> False, Boxed -> False,
PlotRange -> All, SphericalRegion -> True,
ViewPoint -> {5, 0, 0},
ViewAngle -> 12 °]

Update 1
With a little modification to inclinedCircle
and circlePlot
to handle nodal point shift, it is easy to construct an interactive version. A rotation about the z-axis has been added to accomplish the nodal shift.
inclinedCircle[θ_, i_, offset_] :=
RotationTransform[offset, {0, 0, 1}]
[RotationTransform[i, {1, 0, 0}][equator[θ]]];
circlePlot[i_, offset_: 0] :=
ParametricPlot3D[
Evaluate[inclinedCircle[θ, i °, offset °]], {θ, 0, 2 π},
PlotStyle -> {Thickness[0.005]}]
The parameter ι is the inclination and the parameter α is the equatorial ascending node.
Manipulate[
Show[{spherePlot, circlePlot[ι, α]},
Axes -> False,
Boxed -> False,
PlotRange -> All,
SphericalRegion -> True,
ViewPoint -> {5, 0, 0},
ViewAngle -> 12 °],
{{ι, 60.}, 0., 90., 5., Appearance -> "Labeled"},
{{α, 0.}, 0., 355., 5., Appearance -> "Labeled"},
SaveDefinitions -> True]

Update 2
Rather than describing a great circle as the composition of rotations applied to the equator, those rotations can be used to derive a more conventional description, a parametic function built from elementary trigonometric functions. To do this, first apply inclinedCircle
to some value-free symbols.
Clear[θ, ι, φ]; inclinedCircle[θ, ι, φ]
{Cos[θ] Cos[φ] - Cos[ι] Sin[θ] Sin[φ],
Cos[ι] Cos[φ] Sin[θ] + Cos[θ] Sin[φ],
Sin[θ] Sin[ι]}
From the output expression, write a function generator, inclinedCircle
, that when given the inclination ι and the ascending node φ, returns a pure function producing the points on the desired great circle.
inclinedCircleF[ι_, φ_] =
{Cos[#] Cos[φ] - Cos[ι] Sin[#] Sin[φ],
Cos[ι] Cos[φ] Sin[#] + Cos[#] Sin[φ],
Sin[#] Sin[ι]}&;
Next write a plotting function that will use the function generator rather than the rotations.
circlePlot2[ι_, φ_: 0.] :=
ParametricPlot3D[
inclinedCircleF[ι °, φ °][θ], {θ, 0., 2. π},
PlotStyle -> {Thickness[0.005]}]
Finally, substitute the new plotting function for the original one in the Manipulate
.
Manipulate[
Show[{spherePlot, circlePlot2[ι, φ]},
Axes -> False,
Boxed -> False,
PlotRange -> All,
SphericalRegion -> True,
ViewPoint -> {5, 0, 0},
ViewAngle -> 12 °],
{{ι, 60.}, 0., 90., 5., Appearance -> "Labeled"},
{{φ, 0.}, 0., 355., 5., Appearance -> "Labeled"},
SaveDefinitions -> True]
The output from evaluating this Manipulate
expression is exactly the same as with the one using rotations, so I won't repeat it here.
Manipulate
? The angle of inclination? The equatorial nodal points? $\endgroup$