(*Euler angles*)
MatrixForm[
 Rz = {{Cos[α], Sin[α], 0}, {-Sin[α],
    Cos[α], 0}, {0, 0, 1}}]
MatrixForm[
 Rxprime = {{1, 0, 0}, {0, Cos[β],
    Sin[β]}, {0, -Sin[β], Cos[β]}}]
MatrixForm[
 Rzdoubleprime = {{Cos[γ], Sin[γ], 0}, {-Sin[γ],
    Cos[γ], 0}, {0, 0, 1}}]

(*"Total Rotator"*)
    (R[α_, β_, γ_] = Rzdoubleprime.Rxprime.Rz) // MatrixForm

(*Translation*)
OffsetTrans[d_, e_, f_] := {d, e, f};

(*Parametric Equations of Arbitrarily Rotated and Translated Ellipsoid*)

X0[x0_, y0_, z0_] := {x0, y0, z0};

XPar[θ_, ϕ_, a_, b_, c_, α_, β_, γ_] =
  R[α, β, γ].X0[x0, y0, z0] /. {x0 ->
     a Cos[θ] Sin[ϕ], y0 -> b Sin[θ] Sin[ϕ],
    z0 -> c Cos[ϕ]};
XParTrans[d_, e_, f_, a_, b_, c_, α_, β_, γ_] =
  OffsetTrans[d, e, f] +
   XPar[θ, ϕ, a, b, c, α, β, γ];

(*Ellipsoidal Axes*)

(*Non rotated; no translated*)
Cross0[a_, b_, c_] := {Line[{{-a, 0, 0}, {a, 0, 0}}],
  Line[{{0, -b, 0}, {0, b, 0}}], Line[{{0, 0, -c}, {0, 0, c}}]}

(*rotated but not translated*)
CrossRot0[a_, b_,
  c_, α_, β_, γ_] := {Line[{R[α, β,
γ].{-a, 0, 0}, R[α, β, γ].{a, 0, 0}}],
  Line[{R[α, β, γ].{0, -b, 0},
    R[α, β, γ].{0, b, 0}}],
  Line[{R[α, β, γ].{0, 0, -c},
    R[α, β, γ].{0, 0, c}}]}

(*rotated and translated; final coordinate system: xyz*)
CrossRotTrans[d_, e_, f_, a_, b_,
  c_, α_, β_, γ_] := {Line[{R[α, β,
γ].{-a, 0, 0} + {d, e, f},
    R[α, β, γ].{a, 0, 0} + {d, e, f}}],
  Line[{R[α, β, γ].{0, -b, 0} + {d, e, f},
    R[α, β, γ].{0, b, 0} + {d, e, f}}],
  Line[{R[α, β, γ].{0, 0, -c} + {d, e, f},
    R[α, β, γ].{0, 0, c} + {d, e, f}}]}

(*rotated and translated; first intermediate coordinate system: x'y'z'*)
    CrossRotTransAlpha[d_, e_, f_, a_, b_,
      c_, α_, β_, γ_] := {Line[{R[α, 0, 0].{-a, 0,
           0} + {d, e, f}, R[α, 0, 0].{a, 0, 0} + {d, e, f}}],
      Line[{R[α, 0, 0].{0, -b, 0} + {d, e, f},
        R[α, 0, 0].{0, b, 0} + {d, e, f}}],
      Line[{R[α, 0, 0].{0, 0, -c} + {d, e, f},
        R[α, 0, 0].{0, 0, c} + {d, e, f}}]}

 (*rotated and translated; second intermediate  coordinate system: x''y''z''*)
    CrossRotTransBetaAlpha[d_, e_, f_, a_, b_,
      c_, α_, β_, γ_] := {Line[{R[α, β, 0].{-a,
           0, 0} + {d, e, f}, R[α, β, 0].{a, 0, 0} + {d, e, f}}],
      Line[{R[α, β, 0].{0, -b, 0} + {d, e, f},
        R[α, β, 0].{0, b, 0} + {d, e, f}}],
      Line[{R[α, β, 0].{0, 0, -c} + {d, e, f},
        R[α, β, 0].{0, 0, c} + {d, e, f}}]}


(* Cross sections*)

    CrossSectionX[d_, e_, f_, a_, b_, c_, α_, β_, γ_] :=
      ParametricPlot3D[
       OffsetTrans[d, e, f] +
        R[α, β, γ].{0, b Cos[ϕ], c Sin[ϕ]}, {ϕ,
         0, 2 Pi}, PlotStyle -> {Dashed, Gray}];

    CrossSectionY[d_, e_, f_, a_, b_, c_, α_, β_, γ_] :=
      ParametricPlot3D[
       OffsetTrans[d, e, f] +
        R[α, β, γ].{a Cos[ϕ], 0, c Sin[ϕ]}, {ϕ,
         0, 2 Pi}, PlotStyle -> {Dotted, Gray}];

    CrossSectionZ[d_, e_, f_, a_, b_, c_, α_, β_, γ_] :=
      ParametricPlot3D[
       OffsetTrans[d, e, f] +
        R[α, β, γ].{a Cos[ϕ], b Sin[ϕ], 0}, {ϕ,
         0, 2 Pi}, PlotStyle -> {DotDashed, Gray}];


    (*XYZ system*)
    arrowAxesXYZ[arrowLength_] :=
     Map[Arrow[Tube[{{0, 0, 0}, #}]] &, arrowLength IdentityMatrix[3]]
    AxesXYZ[pos_] := {Text[Style["X", 20, Italic], {pos + 0.2, 0, 0}],
      Text[Style["Y", 20, Italic], {0, pos + 0.2, 0}],
      Text[Style["Z", 20, Italic], {0, 0, pos + 0.2}]}

Clear[g1, g2, g3, g4, g5, g6, g7, g9, EllipsoidalFunMod]
EllipsoidalFunMod[d_, e_, f_, a_, b_,
  c_, α_, β_, γ_, {opts___}] :=
 Module[{g1, g2, g3, g4, g5, g6, g7, g8},
  g2 = CrossRotTrans[d, e, f, a, b, c, α, β, γ];
  g3 = CrossSectionX[d, e, f, a, b, c, α, β, γ];
  g4 = CrossSectionY[d, e, f, a, b, c, α, β, γ];
  g5 = CrossSectionZ[d, e, f, a, b, c, α, β, γ];
  g6 = CrossRotTransAlpha[d, e, f, a, b,
    c, α, β, γ];
  g7 = CrossRotTransBetaAlpha[d, e, f, a, b,
    c, α, β, γ];
  g1 = ParametricPlot3D[
    XParTrans[d, e, f, a, b,
     c, α, β, γ], {θ, 0, 2 Pi}, {ϕ, 0,
      Pi}, PlotStyle ->
     Directive[Orange, Specularity[White, 40], Opacity[0.5]],
    Mesh -> None];
  g8 = Arrow[Tube[{{0, 0, 0}, {d, e, f}}, 0.05]];
  Legended[
   Show[{g1,
     Graphics3D[{{Black, g8}, {Red, Thick, g6}, {Green, Thick,
        g7}, {Blue, Thick, g2}, {Gray, arrowAxesXYZ[30]},
       AxesXYZ[31]}], g3, g4, g5}, Axes -> False,
    AxesOrigin -> {0, 0, 0}, PlotRange -> All, ImageSize -> Large,
    PlotLabel ->
     Style[Framed[
       "Arbitrarily Oriented & Translated EllipsoidnEllipsoidal
Semi-Axes: ta=" <> ToString[a] <> "tb=" <> ToString[b] <> "tc=" <>
        ToString[c]], Blue, 20, Background -> Lighter[Yellow]], opts],
    LineLegend[{Directive[Thick, Black], Directive[Thick, Red],
     Directive[Thick, Green],
     Directive[Thick, Blue]}, {Style[
      "Translation by " <>
       ToString[TraditionalForm[!(TraditionalForm`*
StyleBox["d", "TI"] *
OverscriptBox[
StyleBox["X", "TI"], "^"] + *
StyleBox["e", "TI"] *
OverscriptBox[
StyleBox["Y", "TI"], "^"] + *
StyleBox["f", "TI"] *
OverscriptBox[
StyleBox["Z", "TI"], "^"])]], FontSize -> 18, Black],
     Style["rotation through " <> ToString[TraditionalForm@α] <>
        " about Z–axisnNew Coordinate System x'y'z'; z'
≡Z", 18, Red],
     Style["rotation through " <> ToString[TraditionalForm@β] <>
        " about x'–axisnNew Coordinate System x''y''z''; x
''≡x'", 18, Green],
     Style["rotation through " <> ToString[TraditionalForm@γ] <>
        " about z''–axisnFinal Coordinate System xyz; z
≡z''", 18, Blue]}, LegendFunction -> "Frame",
    LegendLayout -> "Column"]]]

EllipsoidalFunMod[12, 21, 30, 3, 5, 19, Pi/4, Pi/6, Pi/3, {}]

Block[{$PerformanceGoal = "Speed"},
 Manipulate[
  EllipsoidalFunMod[d, e, f, a, b,
   c, α, β, γ, {}], {{d, 0}, -10, 10,
   1}, {{e, 0}, -10, 10, 1}, {{f, 0}, -10, 10, 1}, {{a, 1}, 1, 10,
   1}, {{b, 1}, 1, 10, 1}, {{c, 1}, 1, 100, 1}, {{α, 0}, 0,
   2 Pi, Pi/10}, {{β, 0}, 0, 2 Pi, Pi/10}, {{γ, 0}, 0,
   2 Pi, Pi/10}]]