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quantumelixir/rotation_matrix_3d.nb

Created June 17, 2017 16:42
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Rotation Matrix in 3D
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rotation_matrix_3d.nb
In[79]:= Rz[x_] := ( {
{Cos[x], -Sin[x], 0},
{Sin[x], Cos[x], 0},
{0, 0, 1}
} );
Ry[x_] := ( {
{Cos[x], 0, Sin[x]},
{0, 1, 0},
{-Sin[x], 0, Cos[x]}
} );
R[a_, b_, c_, \[Theta]_] :=
Rz[-Subscript[\[Theta], 1]].Ry[-Subscript[\[Theta],
2]].Rz[\[Theta]].Ry[Subscript[\[Theta], 2]].Rz[
Subscript[\[Theta], 1]] /. {Subscript[\[Theta], 1] ->
ArcSin[b/Sqrt[1 - c^2]],
Subscript[\[Theta], 2] -> ArcSin[Sqrt[1 - c^2]]};
Assuming[Element[Subscript[u, x], Reals] &&
Element[Subscript[u, y], Reals] && Element[Subscript[u, z], Reals],
FullSimplify[
R[Subscript[u, x], Subscript[u, y], Subscript[u,
z], \[Theta]]]] // MatrixForm
Out[82]//MatrixForm= \!\(
TagBox[
RowBox[{"(", "", GridBox[{
{
RowBox[{"1", "+",
RowBox[{
RowBox[{"(",
RowBox[{
RowBox[{"-", "1"}], "+",
RowBox[{"Cos", "[", "\[Theta]", "]"}]}], ")"}], " ",
SubsuperscriptBox["u", "y", "2"]}], "+",
RowBox[{
RowBox[{"(",
RowBox[{
RowBox[{"-", "1"}], "+",
RowBox[{"Cos", "[", "\[Theta]", "]"}]}], ")"}], " ",
SubsuperscriptBox["u", "z", "2"]}]}],
RowBox[{
RowBox[{"-", "2"}], " ",
RowBox[{"Sin", "[",
FractionBox["\[Theta]", "2"], "]"}], " ",
RowBox[{"(",
RowBox[{
RowBox[{
RowBox[{"Abs", "[",
SubscriptBox["u", "z"], "]"}], " ",
RowBox[{"Cos", "[",
FractionBox["\[Theta]", "2"], "]"}]}], "+",
RowBox[{
RowBox[{"Sin", "[",
FractionBox["\[Theta]", "2"], "]"}], " ",
SubscriptBox["u", "y"], " ",
SqrtBox[
RowBox[{"1", "-",
SubsuperscriptBox["u", "z", "2"]}]], " ",
SqrtBox[
FractionBox[
RowBox[{
RowBox[{"-", "1"}], "+",
SubsuperscriptBox["u", "y", "2"], "+",
SubsuperscriptBox["u", "z", "2"]}],
RowBox[{
RowBox[{"-", "1"}], "+",
SubsuperscriptBox["u", "z", "2"]}]]]}]}], ")"}]}],
RowBox[{"2", " ",
RowBox[{"Sin", "[",
FractionBox["\[Theta]", "2"], "]"}], " ",
RowBox[{"(",
RowBox[{
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RowBox[{"Cos", "[",
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SubscriptBox["u", "z"], "]"}], " ",
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SqrtBox[
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SubsuperscriptBox["u", "z", "2"]}]], " ",
SqrtBox[
FractionBox[
RowBox[{
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SubsuperscriptBox["u", "y", "2"], "+",
SubsuperscriptBox["u", "z", "2"]}],
RowBox[{
RowBox[{"-", "1"}], "+",
SubsuperscriptBox["u", "z", "2"]}]]]}]}], ")"}]}]},
{
RowBox[{"2", " ",
RowBox[{"Sin", "[",
FractionBox["\[Theta]", "2"], "]"}], " ",
RowBox[{"(",
RowBox[{
RowBox[{
RowBox[{"Abs", "[",
SubscriptBox["u", "z"], "]"}], " ",
RowBox[{"Cos", "[",
FractionBox["\[Theta]", "2"], "]"}]}], "-",
RowBox[{
RowBox[{"Sin", "[",
FractionBox["\[Theta]", "2"], "]"}], " ",
SubscriptBox["u", "y"], " ",
SqrtBox[
RowBox[{"1", "-",
SubsuperscriptBox["u", "z", "2"]}]], " ",
SqrtBox[
FractionBox[
RowBox[{
RowBox[{"-", "1"}], "+",
SubsuperscriptBox["u", "y", "2"], "+",
SubsuperscriptBox["u", "z", "2"]}],
RowBox[{
RowBox[{"-", "1"}], "+",
SubsuperscriptBox["u", "z", "2"]}]]]}]}], ")"}]}],
RowBox[{
RowBox[{"Cos", "[", "\[Theta]", "]"}], "-",
RowBox[{
RowBox[{"(",
RowBox[{
RowBox[{"-", "1"}], "+",
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SubsuperscriptBox["u", "y", "2"]}]}],
RowBox[{
RowBox[{
RowBox[{"-",
RowBox[{"Abs", "[",
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RowBox[{"(",
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RowBox[{"Cos", "[", "\[Theta]", "]"}]}], ")"}], " ",
SubscriptBox["u", "y"]}], "+",
RowBox[{
RowBox[{"Sin", "[", "\[Theta]", "]"}], " ",
SqrtBox[
RowBox[{"1", "-",
SubsuperscriptBox["u", "z", "2"]}]], " ",
SqrtBox[
FractionBox[
RowBox[{
RowBox[{"-", "1"}], "+",
SubsuperscriptBox["u", "y", "2"], "+",
SubsuperscriptBox["u", "z", "2"]}],
RowBox[{
RowBox[{"-", "1"}], "+",
SubsuperscriptBox["u", "z", "2"]}]]]}]}]},
{
RowBox[{
RowBox[{"-", "2"}], " ",
RowBox[{"Sin", "[",
FractionBox["\[Theta]", "2"], "]"}], " ",
RowBox[{"(",
RowBox[{
RowBox[{
RowBox[{"Cos", "[",
FractionBox["\[Theta]", "2"], "]"}], " ",
SubscriptBox["u", "y"]}], "+",
RowBox[{
RowBox[{"Abs", "[",
SubscriptBox["u", "z"], "]"}], " ",
RowBox[{"Sin", "[",
FractionBox["\[Theta]", "2"], "]"}], " ",
SqrtBox[
RowBox[{"1", "-",
SubsuperscriptBox["u", "z", "2"]}]], " ",
SqrtBox[
FractionBox[
RowBox[{
RowBox[{"-", "1"}], "+",
SubsuperscriptBox["u", "y", "2"], "+",
SubsuperscriptBox["u", "z", "2"]}],
RowBox[{
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SubsuperscriptBox["u", "z", "2"]}]]]}]}], ")"}]}],
RowBox[{
RowBox[{
RowBox[{"-",
RowBox[{"Abs", "[",
SubscriptBox["u", "z"], "]"}]}], " ",
RowBox[{"(",
RowBox[{
RowBox[{"-", "1"}], "+",
RowBox[{"Cos", "[", "\[Theta]", "]"}]}], ")"}], " ",
SubscriptBox["u", "y"]}], "-",
RowBox[{
RowBox[{"Sin", "[", "\[Theta]", "]"}], " ",
SqrtBox[
RowBox[{"1", "-",
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SqrtBox[
FractionBox[
RowBox[{
RowBox[{"-", "1"}], "+",
SubsuperscriptBox["u", "y", "2"], "+",
SubsuperscriptBox["u", "z", "2"]}],
RowBox[{
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SubsuperscriptBox["u", "z", "2"]}]]]}]}],
RowBox[{
RowBox[{"Cos", "[", "\[Theta]", "]"}], "-",
RowBox[{
RowBox[{"(",
RowBox[{
RowBox[{"-", "1"}], "+",
RowBox[{"Cos", "[", "\[Theta]", "]"}]}], ")"}], " ",
SubsuperscriptBox["u", "z", "2"]}]}]}
},
GridBoxAlignment->{
"Columns" -> {{Center}}, "ColumnsIndexed" -> {},
"Rows" -> {{Baseline}}, "RowsIndexed" -> {}},
GridBoxSpacings->{"Columns" -> {
Offset[0.27999999999999997`], {
Offset[0.7]},
Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> {
Offset[0.2], {
Offset[0.4]},
Offset[0.2]}, "RowsIndexed" -> {}}], "", ")"}],
Function[BoxForm`e$,
MatrixForm[BoxForm`e$]]]\)
In[66]:= Assuming[
Subscript[u, x]*Subscript[u, x] + Subscript[u, y]*Subscript[u, y] +
Subscript[u, z]*Subscript[u, z] == 1 &&
Element[Subscript[u, x], Reals] &&
Element[Subscript[u, y], Reals] &&
Element[Subscript[u, z], Reals] && Subscript[u, z] >= 0,
FullSimplify[
R[Subscript[u, x], Subscript[u, y], Subscript[u,
z], \[Theta]]]] // MatrixForm
Out[66]//MatrixForm= \!\(
TagBox[
RowBox[{"(", "", GridBox[{
{
RowBox[{"1", "+",
RowBox[{
RowBox[{"(",
RowBox[{
RowBox[{"-", "1"}], "+",
RowBox[{"Cos", "[", "\[Theta]", "]"}]}], ")"}], " ",
SubsuperscriptBox["u", "y", "2"]}], "+",
RowBox[{
RowBox[{"(",
RowBox[{
RowBox[{"-", "1"}], "+",
RowBox[{"Cos", "[", "\[Theta]", "]"}]}], ")"}], " ",
SubsuperscriptBox["u", "z", "2"]}]}],
RowBox[{
RowBox[{
RowBox[{"-",
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SubscriptBox["u", "z"]}], "+",
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SubscriptBox["u", "y"], " ",
SqrtBox[
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SubsuperscriptBox["u", "y", "2"], "-",
SubsuperscriptBox["u", "z", "2"]}]]}]}],
RowBox[{
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RowBox[{"(",
RowBox[{
RowBox[{"-", "1"}], "+",
RowBox[{"Cos", "[", "\[Theta]", "]"}]}], ")"}], " ",
SubscriptBox["u", "z"], " ",
SqrtBox[
RowBox[{"1", "-",
SubsuperscriptBox["u", "y", "2"], "-",
SubsuperscriptBox["u", "z", "2"]}]]}]}]},
{
RowBox[{
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RowBox[{"Sin", "[", "\[Theta]", "]"}], " ",
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RowBox[{
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RowBox[{"Cos", "[", "\[Theta]", "]"}]}], ")"}], " ",
SubscriptBox["u", "y"], " ",
SqrtBox[
RowBox[{"1", "-",
SubsuperscriptBox["u", "y", "2"], "-",
SubsuperscriptBox["u", "z", "2"]}]]}]}],
RowBox[{
RowBox[{"Cos", "[", "\[Theta]", "]"}], "+",
RowBox[{
RowBox[{"(",
RowBox[{"1", "-",
RowBox[{"Cos", "[", "\[Theta]", "]"}]}], ")"}], " ",
SubsuperscriptBox["u", "y", "2"]}]}],
RowBox[{
RowBox[{
RowBox[{"(",
RowBox[{"1", "-",
RowBox[{"Cos", "[", "\[Theta]", "]"}]}], ")"}], " ",
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RowBox[{
RowBox[{"Sin", "[", "\[Theta]", "]"}], " ",
SqrtBox[
RowBox[{"1", "-",
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{
RowBox[{
RowBox[{
RowBox[{"-",
RowBox[{"Sin", "[", "\[Theta]", "]"}]}], " ",
SubscriptBox["u", "y"]}], "+",
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RowBox[{
RowBox[{"-", "1"}], "+",
RowBox[{"Cos", "[", "\[Theta]", "]"}]}], ")"}], " ",
SubscriptBox["u", "z"], " ",
SqrtBox[
RowBox[{"1", "-",
SubsuperscriptBox["u", "y", "2"], "-",
SubsuperscriptBox["u", "z", "2"]}]]}]}],
RowBox[{
RowBox[{
RowBox[{"(",
RowBox[{"1", "-",
RowBox[{"Cos", "[", "\[Theta]", "]"}]}], ")"}], " ",
SubscriptBox["u", "y"], " ",
SubscriptBox["u", "z"]}], "-",
RowBox[{
RowBox[{"Sin", "[", "\[Theta]", "]"}], " ",
SqrtBox[
RowBox[{"1", "-",
SubsuperscriptBox["u", "y", "2"], "-",
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RowBox[{
RowBox[{"Cos", "[", "\[Theta]", "]"}], "+",
RowBox[{
RowBox[{"(",
RowBox[{"1", "-",
RowBox[{"Cos", "[", "\[Theta]", "]"}]}], ")"}], " ",
SubsuperscriptBox["u", "z", "2"]}]}]}
},
GridBoxAlignment->{
"Columns" -> {{Center}}, "ColumnsIndexed" -> {},
"Rows" -> {{Baseline}}, "RowsIndexed" -> {}},
GridBoxSpacings->{"Columns" -> {
Offset[0.27999999999999997`], {
Offset[0.7]},
Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> {
Offset[0.2], {
Offset[0.4]},
Offset[0.2]}, "RowsIndexed" -> {}}], "", ")"}],
Function[BoxForm`e$,
MatrixForm[BoxForm`e$]]]\)
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