I want to copy this square here {{-12, -1, 4}, {-9, 14, 8}, {0, 9, 20}, {-3, -6, 16}}. The square has vertices that all lie on the sphere (x-2)^2 + (y-4)^2 + (z-6)^2 = 15^2. I am trying to draw this square like this 
I don't know how to start. How can I draw it?
The GIF image is here
We can use @CarlWoll's ResourceFunction["SplineCircle"] to do all the heavy lifting for us:
ct = {2, 4, 6}; pts = {{-12, -1, 4}, {-9, 14, 8}, {0, 9, 20}, {-3, -6, 16}}; sphereLine[ct_][{pt1_, pt2_}] := ResourceFunction["SplineCircle"][ ct, Norm[pt1 - ct], {pt1 - ct, pt2 - ct}, {0, VectorAngle[pt1 - ct, pt2 - ct]} ] Graphics3D[{ Sphere[ct, Norm[First@pts - ct]], Black, Thick, sphereLine[ct] /@ Partition[pts, 2, 1, {1, 1}] }] 
To get the dashed lines, do the following:
DynamicModule[{vp = {1, 1, 1}, vv = {0, 0, 1}, vc = {0.5, 0.5, 0.5}, va = 60 °}, Overlay[{Graphics3D[{ Sphere[ct, Norm[First@pts - ct]], Black, Thick, sphereLine[ct] /@ Partition[pts, 2, 1, {1, 1}] }, ViewPoint -> Dynamic@vp, ViewVertical -> Dynamic@vv, ViewCenter -> Dynamic@vc, ViewAngle -> Dynamic@va], Graphics3D[{ {Transparent, Sphere[ct, Norm[First@pts - ct]]}, Black, Thick, Dashed, sphereLine[ct] /@ Partition[pts, 2, 1, {1, 1}] }, ViewPoint -> Dynamic@vp, ViewVertical -> Dynamic@vv, ViewCenter -> Dynamic@vc, ViewAngle -> Dynamic@va, Boxed -> False ] }, All, 1] ] 
This is achieved by synchronizing the view for two Graphics3D expressions that are stacked on top of each other: The lower one has the original graphics, while the upper one only includes the dashed lines. You can see an illustration of this below, where I show the two individual graphics side by side with the resulting one:
Clear[pts,center,r]; pts = {{-12, -1, 4}, {-9, 14, 8}, {0, 9, 20}, {-3, -6, 16}}; center = {2, 4, 6}; r = 15; Show[ParametricPlot3D[ Threaded@center + r*Normalize /@ ({1 - s, s} . # & /@ Partition[pts - Threaded@center, 2, 1, 1]), {s, 0, 1}], Graphics3D[Ball[center, r]], Boxed -> False, Axes -> False, PlotRange -> All] 
pts = {{-12, -1, 4}, {-9, 14, 8}, {0, 9, 20}, {-3, -6, 16}}; center = {2, 4, 6}; r = 15; axis = {1, -1, 1} - {-1, 1, -1}; figs = Table[ Block[{rot = RotationMatrix[t, axis], v = RotationMatrix[t, axis] . {1, 0, 0}, g1, g2, ball}, g1 = ParametricPlot3D[ Threaded@center + r*Normalize /@ ({1 - s, s} . # & /@ Partition[pts - Threaded@center, 2, 1, 1]) // Evaluate, {s, 0, 1}, PlotStyle -> Directive@{LightYellow, AbsoluteThickness[5]}, RegionFunction -> Function[{x, y, z}, ({x, y, z} - center) . v > 0], ViewProjection -> "Orthographic"]; g2 = ParametricPlot3D[ Threaded@center + r*Normalize /@ ({1 - s, s} . # & /@ Partition[pts - Threaded@center, 2, 1, 1]) // Evaluate, {s, 0, 1}, PlotStyle -> Directive@{Darker@Yellow, AbsoluteDashing[{1, 10}, 0, "Round"], AbsoluteThickness[4]}, RegionFunction -> Function[{x, y, z}, ({x, y, z} - center) . v < 0], ViewProjection -> "Orthographic"]; ball = Graphics[{RadialGradientFilling[{LightGray, Black}, {0.4, 0.7}], Disk[Rest[center . rot], r]}]; g = Show[g1, g2, ViewProjection -> "Orthographic"]; Graphics[{ball[[1]], g[[1]] /. {x_Real, y_Real, z_Real} -> Rest[{x, y, z} . rot]}, PlotRange -> All]], {t, 0, 2 π, .1}]; Export["test.gif", figs] // SystemOpen 
Do[Export["test" <> ToString[i] <> ".pdf", figs[[i]]], {i, 1, Length@figs}]; \documentclass{article} \usepackage{animate} \usepackage{graphicx} \begin{document} \begin{center} \animategraphics[controls,loop,width=4in]{5}{test}{1}{63} \end{center} \end{document} 