• Using ScalingTransform to project to the three directions.
• Only need p1 in the original code.

{{xmin, xmax}, {ymin, ymax}, {zmin, zmax}} = 
  PlotRange /. AbsoluteOptions[p1, PlotRange];
Graphics3D[{GeometricTransformation[p1[[1]], 
    ScalingTransform[10^-3, {0, 0, 1}, {0, 0, zmin}]] /. 
   Line[pts_] :> {Green, Line[pts]}, 
  GeometricTransformation[p1[[1]], 
    ScalingTransform[10^-3, {0, 1, 0}, {0, ymax, 0}]] /. 
   Line[pts_] :> {Blue, Line[pts]}, 
  GeometricTransformation[p1[[1]], 
    ScalingTransform[10^-3, {1, 0, 0}, {xmin, 0, 0}]] /. 
   Line[pts_] :> {Orange, Line[pts]}, p1[[1]]}]

• Using ScalingTransform to project to the three planes.
• Only need p1 in the original code.

{{xmin, xmax}, {ymin, ymax}, {zmin, zmax}} = 
  PlotRange /. AbsoluteOptions[p1, PlotRange];
Graphics3D[{GeometricTransformation[p1[[1]], 
    ScalingTransform[10^-3, {0, 0, 1}, {0, 0, zmin}]] /. 
   Line[pts_] :> {Green, Line[pts]}, 
  GeometricTransformation[p1[[1]], 
    ScalingTransform[10^-3, {0, 1, 0}, {0, ymax, 0}]] /. 
   Line[pts_] :> {Blue, Line[pts]}, 
  GeometricTransformation[p1[[1]], 
    ScalingTransform[10^-3, {1, 0, 0}, {xmin, 0, 0}]] /. 
   Line[pts_] :> {Orange, Line[pts]}, p1[[1]]}]