How do you draw a path on a parametric sphere?

Question

Given a parameterized sphere defined by:

X[u_,v_]:=Cos[v]Sin[u];
Y[u_,v_]:=Sin[v]Sin[u];
Z[u_,v_]:=Cos[u];
R[u_,v_]:={X[u,v],Y[u,v],Z[u,v]};
ParametricPlot3D[{R[u, v]}, {u, 0, Pi}, {v, 0, 2*Pi}]

I want to draw the line from R[0,0] to R[$\pi$,$\pi$]. What is the simplest way? The end result should look something like this:

Answer 1

If you want to add a line that follows the path R[u, u] from u = 0 to u = π, you can use a single parameter ParametricPlot3D:

Show[ParametricPlot3D[R[u, v], {u, 0, Pi}, {v, 0, 2  Pi}, PlotStyle -> Opacity[.5]], 
 ParametricPlot3D[R[u, u], {u, 0, Pi}, PlotStyle -> Directive[Red, Thick]]]

Update: If you want to draw meridians from R[0,0] to R[π, π], you can use the option MeshFunctions -> {#5&} (or MeshFunctions -> {Function[{x,y,z,u,v},v]}) and specify the meridians you wish to show using the option Mesh:

ParametricPlot3D[R[u, v], {u, 0, Pi}, {v, 0, 2 Pi}, 
  PlotStyle -> Opacity[.5], 
  MeshFunctions -> {#5 &}, 
  Mesh -> {{{0, Purple}, {Pi/2, Red}, {Pi, Green}, {Pi/4, Magenta}, {5 Pi/3, Blue}}}, 
  MeshStyle -> Thick, BoundaryStyle -> None, 
  Method -> {"BoundaryOffset" -> False}, 
  PlotRangePadding -> Scaled[.05], 
  PlotLegends -> LineLegend[{Purple, Red, Green, Magenta, Blue},
     {0, Pi/2, Pi, Pi/4, 5 Pi/3}, LegendFunction -> Panel, LegendLabel -> "v"]] /. 
  Line -> (Tube[#, .025] &)