227 lines
		
	
	
		
			6.1 KiB
		
	
	
	
		
			Mathematica
		
	
	
			
		
		
	
	
			227 lines
		
	
	
		
			6.1 KiB
		
	
	
	
		
			Mathematica
		
	
	
| (* Content-type: application/vnd.wolfram.mathematica *)
 | |
| 
 | |
| (*** Wolfram Notebook File ***)
 | |
| (* http://www.wolfram.com/nb *)
 | |
| 
 | |
| (* CreatedBy='Mathematica 10.0' *)
 | |
| 
 | |
| (*CacheID: 234*)
 | |
| (* Internal cache information:
 | |
| NotebookFileLineBreakTest
 | |
| NotebookFileLineBreakTest
 | |
| NotebookDataPosition[       158,          7]
 | |
| NotebookDataLength[      6004,        217]
 | |
| NotebookOptionsPosition[      5104,        179]
 | |
| NotebookOutlinePosition[      5456,        195]
 | |
| CellTagsIndexPosition[      5413,        192]
 | |
| WindowFrame->Normal*)
 | |
| 
 | |
| (* Beginning of Notebook Content *)
 | |
| Notebook[{
 | |
| Cell["\<\
 | |
| In Quaternion.h we have Logmap, but we have to be careful when qw approaches \
 | |
| -1 (from above) or 1 (from below). The Taylor expansions below are the basis \
 | |
| for the code.\
 | |
| \>", "Text",
 | |
|  CellChangeTimes->{{3.632651837171029*^9, 3.6326518973274307`*^9}}],
 | |
| 
 | |
| Cell[CellGroupData[{
 | |
| 
 | |
| Cell[BoxData[
 | |
|  RowBox[{"angle", "=", 
 | |
|   RowBox[{"2", 
 | |
|    RowBox[{"ArcCos", "[", "qw", "]"}]}]}]], "Input",
 | |
|  CellChangeTimes->{{3.6326509558588057`*^9, 3.632650976842943*^9}}],
 | |
| 
 | |
| Cell[BoxData[
 | |
|  RowBox[{"2", " ", 
 | |
|   RowBox[{"ArcCos", "[", "qw", "]"}]}]], "Output",
 | |
|  CellChangeTimes->{{3.6326509669341784`*^9, 3.6326509795921097`*^9}}]
 | |
| }, Open  ]],
 | |
| 
 | |
| Cell[CellGroupData[{
 | |
| 
 | |
| Cell[BoxData[
 | |
|  RowBox[{"s", "=", 
 | |
|   RowBox[{"Sqrt", "[", 
 | |
|    RowBox[{"1", "-", 
 | |
|     RowBox[{"qw", "*", "qw"}]}], "]"}]}]], "Input",
 | |
|  CellChangeTimes->{{3.632650983796185*^9, 3.632650994132272*^9}}],
 | |
| 
 | |
| Cell[BoxData[
 | |
|  SqrtBox[
 | |
|   RowBox[{"1", "-", 
 | |
|    SuperscriptBox["qw", "2"]}]]], "Output",
 | |
|  CellChangeTimes->{3.63265099440246*^9}]
 | |
| }, Open  ]],
 | |
| 
 | |
| Cell[CellGroupData[{
 | |
| 
 | |
| Cell[BoxData[
 | |
|  RowBox[{"factor", " ", "=", " ", 
 | |
|   RowBox[{"angle", "/", "s"}]}]], "Input",
 | |
|  CellChangeTimes->{{3.632650999925654*^9, 3.632651001339293*^9}, {
 | |
|   3.632651070297429*^9, 3.632651071527272*^9}}],
 | |
| 
 | |
| Cell[BoxData[
 | |
|  FractionBox[
 | |
|   RowBox[{"2", " ", 
 | |
|    RowBox[{"ArcCos", "[", "qw", "]"}]}], 
 | |
|   SqrtBox[
 | |
|    RowBox[{"1", "-", 
 | |
|     SuperscriptBox["qw", "2"]}]]]], "Output",
 | |
|  CellChangeTimes->{3.632651001671771*^9, 3.632651072007021*^9}]
 | |
| }, Open  ]],
 | |
| 
 | |
| Cell[CellGroupData[{
 | |
| 
 | |
| Cell[BoxData[
 | |
|  RowBox[{"Expand", "[", 
 | |
|   RowBox[{"Series", "[", 
 | |
|    RowBox[{
 | |
|     FractionBox[
 | |
|      RowBox[{"2", " ", 
 | |
|       RowBox[{"ArcCos", "[", "qw", "]"}]}], 
 | |
|      SqrtBox[
 | |
|       RowBox[{"1", "-", 
 | |
|        SuperscriptBox["qw", "2"]}]]], ",", 
 | |
|     RowBox[{"{", 
 | |
|      RowBox[{"qw", ",", "1", ",", "1"}], "}"}], ",", 
 | |
|     RowBox[{"Assumptions", "->", 
 | |
|      RowBox[{"(", 
 | |
|       RowBox[{"qw", "<", "1"}], ")"}]}]}], "]"}], "]"}]], "Input",
 | |
|  CellChangeTimes->{{3.6326510739355927`*^9, 3.632651117949705*^9}, {
 | |
|   3.6326511716876993`*^9, 3.632651189491748*^9}, {3.632651248821335*^9, 
 | |
|   3.632651267905816*^9}}],
 | |
| 
 | |
| Cell[BoxData[
 | |
|  InterpretationBox[
 | |
|   RowBox[{"2", "-", 
 | |
|    FractionBox[
 | |
|     RowBox[{"2", " ", 
 | |
|      RowBox[{"(", 
 | |
|       RowBox[{"qw", "-", "1"}], ")"}]}], "3"], "+", 
 | |
|    InterpretationBox[
 | |
|     SuperscriptBox[
 | |
|      RowBox[{"O", "[", 
 | |
|       RowBox[{"qw", "-", "1"}], "]"}], 
 | |
|      RowBox[{"3", "/", "2"}]],
 | |
|     SeriesData[$CellContext`qw, 1, {}, 0, 3, 2],
 | |
|     Editable->False]}],
 | |
|   SeriesData[$CellContext`qw, 1, {2, 0, 
 | |
|     Rational[-2, 3]}, 0, 3, 2],
 | |
|   Editable->False]], "Output",
 | |
|  CellChangeTimes->{{3.632651102947558*^9, 3.632651118218814*^9}, {
 | |
|   3.632651179610784*^9, 3.6326511898522263`*^9}, {3.632651249719887*^9, 
 | |
|   3.632651268312502*^9}}]
 | |
| }, Open  ]],
 | |
| 
 | |
| Cell[CellGroupData[{
 | |
| 
 | |
| Cell[BoxData[
 | |
|  RowBox[{"ArcCos", "[", 
 | |
|   RowBox[{"-", "1"}], "]"}]], "Input",
 | |
|  CellChangeTimes->{{3.632651352754121*^9, 3.63265135286866*^9}}],
 | |
| 
 | |
| Cell[BoxData["\[Pi]"], "Output",
 | |
|  CellChangeTimes->{3.632651353300222*^9}]
 | |
| }, Open  ]],
 | |
| 
 | |
| Cell[CellGroupData[{
 | |
| 
 | |
| Cell[BoxData[
 | |
|  RowBox[{"Expand", "[", 
 | |
|   RowBox[{"Series", "[", 
 | |
|    RowBox[{
 | |
|     FractionBox[
 | |
|      RowBox[{
 | |
|       RowBox[{"-", "2"}], 
 | |
|       RowBox[{"ArcCos", "[", 
 | |
|        RowBox[{"-", "qw"}], "]"}]}], 
 | |
|      SqrtBox[
 | |
|       RowBox[{"1", "-", 
 | |
|        SuperscriptBox["qw", "2"]}]]], ",", 
 | |
|     RowBox[{"{", 
 | |
|      RowBox[{"qw", ",", 
 | |
|       RowBox[{"-", "1"}], ",", "1"}], "}"}], ",", 
 | |
|     RowBox[{"Assumptions", "->", 
 | |
|      RowBox[{"(", 
 | |
|       RowBox[{"qw", ">", 
 | |
|        RowBox[{"-", "1"}]}], ")"}]}]}], "]"}], "]"}]], "Input",
 | |
|  CellChangeTimes->{{3.6326510739355927`*^9, 3.632651117949705*^9}, {
 | |
|   3.6326511716876993`*^9, 3.6326512088422937`*^9}, {3.632651301817163*^9, 
 | |
|   3.6326513406015453`*^9}, {3.63265150259446*^9, 3.632651505055284*^9}, {
 | |
|   3.632651744223112*^9, 3.632651772717318*^9}}],
 | |
| 
 | |
| Cell[BoxData[
 | |
|  InterpretationBox[
 | |
|   RowBox[{
 | |
|    RowBox[{"-", "2"}], "-", 
 | |
|    FractionBox[
 | |
|     RowBox[{"2", " ", 
 | |
|      RowBox[{"(", 
 | |
|       RowBox[{"qw", "+", "1"}], ")"}]}], "3"], "+", 
 | |
|    InterpretationBox[
 | |
|     SuperscriptBox[
 | |
|      RowBox[{"O", "[", 
 | |
|       RowBox[{"qw", "+", "1"}], "]"}], 
 | |
|      RowBox[{"3", "/", "2"}]],
 | |
|     SeriesData[$CellContext`qw, -1, {}, 0, 3, 2],
 | |
|     Editable->False]}],
 | |
|   SeriesData[$CellContext`qw, -1, {-2, 0, 
 | |
|     Rational[-2, 3]}, 0, 3, 2],
 | |
|   Editable->False]], "Output",
 | |
|  CellChangeTimes->{
 | |
|   3.632651209181905*^9, 3.6326513025091133`*^9, {3.632651332608609*^9, 
 | |
|    3.632651341031022*^9}, 3.632651506516138*^9, {3.632651746679185*^9, 
 | |
|    3.632651773032124*^9}}]
 | |
| }, Open  ]]
 | |
| },
 | |
| WindowSize->{808, 751},
 | |
| WindowMargins->{{4, Automatic}, {Automatic, 4}},
 | |
| FrontEndVersion->"10.0 for Mac OS X x86 (32-bit, 64-bit Kernel) (June 27, \
 | |
| 2014)",
 | |
| StyleDefinitions->"Default.nb"
 | |
| ]
 | |
| (* End of Notebook Content *)
 | |
| 
 | |
| (* Internal cache information *)
 | |
| (*CellTagsOutline
 | |
| CellTagsIndex->{}
 | |
| *)
 | |
| (*CellTagsIndex
 | |
| CellTagsIndex->{}
 | |
| *)
 | |
| (*NotebookFileOutline
 | |
| Notebook[{
 | |
| Cell[558, 20, 263, 5, 49, "Text"],
 | |
| Cell[CellGroupData[{
 | |
| Cell[846, 29, 174, 4, 28, "Input"],
 | |
| Cell[1023, 35, 154, 3, 28, "Output"]
 | |
| }, Open  ]],
 | |
| Cell[CellGroupData[{
 | |
| Cell[1214, 43, 197, 5, 28, "Input"],
 | |
| Cell[1414, 50, 129, 4, 40, "Output"]
 | |
| }, Open  ]],
 | |
| Cell[CellGroupData[{
 | |
| Cell[1580, 59, 206, 4, 28, "Input"],
 | |
| Cell[1789, 65, 233, 7, 59, "Output"]
 | |
| }, Open  ]],
 | |
| Cell[CellGroupData[{
 | |
| Cell[2059, 77, 605, 17, 61, "Input"],
 | |
| Cell[2667, 96, 645, 19, 48, "Output"]
 | |
| }, Open  ]],
 | |
| Cell[CellGroupData[{
 | |
| Cell[3349, 120, 142, 3, 28, "Input"],
 | |
| Cell[3494, 125, 74, 1, 28, "Output"]
 | |
| }, Open  ]],
 | |
| Cell[CellGroupData[{
 | |
| Cell[3605, 131, 788, 22, 61, "Input"],
 | |
| Cell[4396, 155, 692, 21, 48, "Output"]
 | |
| }, Open  ]]
 | |
| }
 | |
| ]
 | |
| *)
 | |
| 
 | |
| (* End of internal cache information *)
 |