Rev 643 | Blame | Last modification | View Log | Download
(* Content-type: application/mathematica *)(*** Wolfram Notebook File ***)(* http://www.wolfram.com/nb *)(* CreatedBy='Mathematica 7.0' *)(*CacheID: 234*)(* Internal cache information:NotebookFileLineBreakTestNotebookFileLineBreakTestNotebookDataPosition[ 145, 7]NotebookDataLength[ 15724, 569]NotebookOptionsPosition[ 14134, 508]NotebookOutlinePosition[ 14471, 523]CellTagsIndexPosition[ 14428, 520]WindowFrame->Normal*)(* Beginning of Notebook Content *)Notebook[{Cell[CellGroupData[{Cell[BoxData[RowBox[{"Eliminate", "[",RowBox[{RowBox[{"{",RowBox[{RowBox[{SuperscriptBox["a", "2"], "==",RowBox[{SuperscriptBox["y", "2"], "+",SuperscriptBox["x", "2"]}]}], ",",RowBox[{SuperscriptBox["b", "2"], "==",RowBox[{SuperscriptBox["y", "2"], "+", " ",SuperscriptBox[RowBox[{"(",RowBox[{"r", "-", "x"}], ")"}], "2"]}]}]}], "}"}], ",", "x"}],"]"}]], "Input",CellChangeTimes->{{3.48025742799592*^9, 3.480257444164321*^9}, {3.480257523613083*^9, 3.4802575457893877`*^9}, {3.4802575824866943`*^9,3.480257595868989*^9}}],Cell[BoxData[RowBox[{RowBox[{RowBox[{"-",SuperscriptBox["b", "4"]}], "+",RowBox[{"2", " ",SuperscriptBox["b", "2"], " ",SuperscriptBox["r", "2"]}], "-",SuperscriptBox["r", "4"], "-",RowBox[{"4", " ",SuperscriptBox["r", "2"], " ",SuperscriptBox["y", "2"]}]}], "\[Equal]",RowBox[{SuperscriptBox["a", "4"], "+",RowBox[{SuperscriptBox["a", "2"], " ",RowBox[{"(",RowBox[{RowBox[{RowBox[{"-", "2"}], " ",SuperscriptBox["b", "2"]}], "-",RowBox[{"2", " ",SuperscriptBox["r", "2"]}]}], ")"}]}]}]}]], "Output",CellChangeTimes->{3.480257547792871*^9, 3.4802575969300547`*^9}]}, Open ]],Cell[CellGroupData[{Cell[BoxData[RowBox[{"Simplify", "[",RowBox[{RowBox[{RowBox[{"-",SuperscriptBox["b", "4"]}], "+",RowBox[{"2", " ",SuperscriptBox["b", "2"], " ",SuperscriptBox["r", "2"]}], "-",SuperscriptBox["r", "4"], "-",RowBox[{"4", " ",SuperscriptBox["r", "2"], " ",SuperscriptBox["y", "2"]}]}], "\[Equal]",RowBox[{SuperscriptBox["a", "4"], "+",RowBox[{SuperscriptBox["a", "2"], " ",RowBox[{"(",RowBox[{RowBox[{RowBox[{"-", "2"}], " ",SuperscriptBox["b", "2"]}], "-",RowBox[{"2", " ",SuperscriptBox["r", "2"]}]}], ")"}]}]}]}], "]"}]], "Input",CellChangeTimes->{{3.480258949696538*^9, 3.4802589666213627`*^9}}],Cell[BoxData[RowBox[{RowBox[{SuperscriptBox["a", "4"], "+",SuperscriptBox["b", "4"], "+",SuperscriptBox["r", "4"], "+",RowBox[{"4", " ",SuperscriptBox["r", "2"], " ",SuperscriptBox["y", "2"]}]}], "\[Equal]",RowBox[{RowBox[{"2", " ",SuperscriptBox["b", "2"], " ",SuperscriptBox["r", "2"]}], "+",RowBox[{"2", " ",SuperscriptBox["a", "2"], " ",RowBox[{"(",RowBox[{SuperscriptBox["b", "2"], "+",SuperscriptBox["r", "2"]}], ")"}]}]}]}]], "Output",CellChangeTimes->{3.480258967385964*^9}]}, Open ]],Cell[CellGroupData[{Cell[BoxData[RowBox[{"Solve", "[",RowBox[{RowBox[{RowBox[{SuperscriptBox["a", "4"], "+",SuperscriptBox["b", "4"], "+",SuperscriptBox["r", "4"], "+",RowBox[{"4", " ",SuperscriptBox["r", "2"], " ",SuperscriptBox["y", "2"]}]}], "\[Equal]",RowBox[{RowBox[{"2", " ",SuperscriptBox["b", "2"], " ",SuperscriptBox["r", "2"]}], "+",RowBox[{"2", " ",SuperscriptBox["a", "2"], " ",RowBox[{"(",RowBox[{SuperscriptBox["b", "2"], "+",SuperscriptBox["r", "2"]}], ")"}]}]}]}], ",", "y"}], "]"}]], "Input",CellChangeTimes->{{3.4802589709519663`*^9, 3.480258994751276*^9}}],Cell[BoxData[RowBox[{"{",RowBox[{RowBox[{"{",RowBox[{"y", "\[Rule]",RowBox[{"-",FractionBox[SqrtBox[RowBox[{RowBox[{"-",SuperscriptBox["a", "4"]}], "+",RowBox[{"2", " ",SuperscriptBox["a", "2"], " ",SuperscriptBox["b", "2"]}], "-",SuperscriptBox["b", "4"], "+",RowBox[{"2", " ",SuperscriptBox["a", "2"], " ",SuperscriptBox["r", "2"]}], "+",RowBox[{"2", " ",SuperscriptBox["b", "2"], " ",SuperscriptBox["r", "2"]}], "-",SuperscriptBox["r", "4"]}]],RowBox[{"2", " ", "r"}]]}]}], "}"}], ",",RowBox[{"{",RowBox[{"y", "\[Rule]",FractionBox[SqrtBox[RowBox[{RowBox[{"-",SuperscriptBox["a", "4"]}], "+",RowBox[{"2", " ",SuperscriptBox["a", "2"], " ",SuperscriptBox["b", "2"]}], "-",SuperscriptBox["b", "4"], "+",RowBox[{"2", " ",SuperscriptBox["a", "2"], " ",SuperscriptBox["r", "2"]}], "+",RowBox[{"2", " ",SuperscriptBox["b", "2"], " ",SuperscriptBox["r", "2"]}], "-",SuperscriptBox["r", "4"]}]],RowBox[{"2", " ", "r"}]]}], "}"}]}], "}"}]], "Output",CellChangeTimes->{3.480258995585805*^9}]}, Open ]],Cell[CellGroupData[{Cell[BoxData[RowBox[{"Eliminate", "[",RowBox[{RowBox[{"{",RowBox[{RowBox[{SuperscriptBox["a", "2"], "==",RowBox[{SuperscriptBox["y", "2"], "+",SuperscriptBox["x", "2"]}]}], ",",RowBox[{SuperscriptBox["b", "2"], "==",RowBox[{SuperscriptBox["y", "2"], "+", " ",SuperscriptBox[RowBox[{"(",RowBox[{"r", "-", "x"}], ")"}], "2"]}]}]}], "}"}], ",", "y"}],"]"}]], "Input",CellChangeTimes->{{3.480259014363867*^9, 3.480259014483967*^9}}],Cell[BoxData[RowBox[{RowBox[{SuperscriptBox["b", "2"], "-",SuperscriptBox["r", "2"], "+",RowBox[{"2", " ", "r", " ", "x"}]}], "\[Equal]",SuperscriptBox["a", "2"]}]], "Output",CellChangeTimes->{3.48025901538303*^9}]}, Open ]],Cell[CellGroupData[{Cell[BoxData[RowBox[{"Simplify", "[",RowBox[{RowBox[{SuperscriptBox["b", "2"], "-",SuperscriptBox["r", "2"], "+",RowBox[{"2", " ", "r", " ", "x"}]}], "\[Equal]",SuperscriptBox["a", "2"]}], "]"}]], "Input",CellChangeTimes->{{3.480259022777849*^9, 3.480259039864859*^9}}],Cell[BoxData[RowBox[{RowBox[{SuperscriptBox["a", "2"], "+",RowBox[{"r", " ",RowBox[{"(",RowBox[{"r", "-",RowBox[{"2", " ", "x"}]}], ")"}]}]}], "\[Equal]",SuperscriptBox["b", "2"]}]], "Output",CellChangeTimes->{3.480259042740219*^9}]}, Open ]],Cell[CellGroupData[{Cell[BoxData[RowBox[{"Solve", "[",RowBox[{RowBox[{RowBox[{SuperscriptBox["a", "2"], "+",RowBox[{"r", " ",RowBox[{"(",RowBox[{"r", "-",RowBox[{"2", " ", "x"}]}], ")"}]}]}], "\[Equal]",SuperscriptBox["b", "2"]}], ",", "x"}], "]"}]], "Input",CellChangeTimes->{{3.480259057756564*^9, 3.480259075038864*^9}}],Cell[BoxData[RowBox[{"{",RowBox[{"{",RowBox[{"x", "\[Rule]",FractionBox[RowBox[{SuperscriptBox["a", "2"], "-",SuperscriptBox["b", "2"], "+",SuperscriptBox["r", "2"]}],RowBox[{"2", " ", "r"}]]}], "}"}], "}"}]], "Output",CellChangeTimes->{3.4802590759762087`*^9}]}, Open ]],Cell[BoxData[RowBox[{"ParametricPlot", "[",RowBox[{RowBox[{"{",RowBox[{FractionBox[RowBox[{SuperscriptBox["a", "2"], "-",SuperscriptBox["b", "2"], "+",SuperscriptBox["0.1", "2"]}], "0.2"], ",", " ",FractionBox[SqrtBox[RowBox[{RowBox[{"-",SuperscriptBox["a", "4"]}], "+",RowBox[{"2", " ",SuperscriptBox["a", "2"], " ",SuperscriptBox["b", "2"]}], "-",SuperscriptBox["b", "4"], "+",RowBox[{"2", " ",SuperscriptBox["a", "2"], " ",SuperscriptBox["0.1", "2"]}], "+",RowBox[{"2", " ",SuperscriptBox["b", "2"], " ",SuperscriptBox["0.1", "2"]}], "-",SuperscriptBox["0.1", "4"]}]], "0.2"]}], "}"}], ",",RowBox[{"{",RowBox[{"a", ",", "0", ",", "10"}], "}"}], ",", " ",RowBox[{"{",RowBox[{"b", ",", "0", ",", "10"}], "}"}]}], "]"}]], "Input",CellChangeTimes->{{3.480260726697213*^9, 3.480260795029454*^9}, {3.4802610157613573`*^9, 3.480261151606037*^9}, {3.480261184002676*^9,3.480261316392874*^9}, {3.48026136441684*^9, 3.48026148678514*^9}}],Cell[CellGroupData[{Cell[BoxData[RowBox[{"\[IndentingNewLine]",RowBox[{"Eliminate", "[",RowBox[{RowBox[{"{",RowBox[{RowBox[{SuperscriptBox["a", "2"], "==",RowBox[{SuperscriptBox["y", "2"], "+",SuperscriptBox["x", "2"]}]}], ",",RowBox[{SuperscriptBox["b", "2"], "==",RowBox[{SuperscriptBox["y", "2"], "+", " ",SuperscriptBox[RowBox[{"(",RowBox[{"r", "-", "x"}], ")"}], "2"]}]}]}], "}"}], ",", "a"}],"]"}]}]], "Input",CellChangeTimes->{{3.480263655197631*^9, 3.480263663686249*^9}}],Cell[BoxData[RowBox[{RowBox[{SuperscriptBox["r", "2"], "-",RowBox[{"2", " ", "r", " ", "x"}], "+",SuperscriptBox["x", "2"], "+",SuperscriptBox["y", "2"]}], "\[Equal]",SuperscriptBox["b", "2"]}]], "Output",CellChangeTimes->{{3.480263658468478*^9, 3.480263664271557*^9}}]}, Open ]],Cell[CellGroupData[{Cell[BoxData[RowBox[{"Simplify", "[",RowBox[{RowBox[{SuperscriptBox["r", "2"], "-",RowBox[{"2", " ", "r", " ", "x"}], "+",SuperscriptBox["x", "2"], "+",SuperscriptBox["y", "2"]}], "\[Equal]",SuperscriptBox["b", "2"]}], "]"}]], "Input",CellChangeTimes->{{3.4802636687522917`*^9, 3.48026368255702*^9}}],Cell[BoxData[RowBox[{RowBox[{SuperscriptBox["b", "2"], "+",RowBox[{"2", " ", "r", " ", "x"}]}], "\[Equal]",RowBox[{SuperscriptBox["r", "2"], "+",SuperscriptBox["x", "2"], "+",SuperscriptBox["y", "2"]}]}]], "Output",CellChangeTimes->{3.480263685526928*^9}]}, Open ]],Cell[CellGroupData[{Cell[BoxData[RowBox[{"Solve", "[",RowBox[{RowBox[{RowBox[{SuperscriptBox["b", "2"], "+",RowBox[{"2", " ", "r", " ", "x"}]}], "\[Equal]",RowBox[{SuperscriptBox["r", "2"], "+",SuperscriptBox["x", "2"], "+",SuperscriptBox["y", "2"]}]}], ",", "b"}], "]"}]], "Input",CellChangeTimes->{{3.480263691906734*^9, 3.480263707606902*^9}}],Cell[BoxData[RowBox[{"{",RowBox[{RowBox[{"{",RowBox[{"b", "\[Rule]",RowBox[{"-",SqrtBox[RowBox[{SuperscriptBox["r", "2"], "-",RowBox[{"2", " ", "r", " ", "x"}], "+",SuperscriptBox["x", "2"], "+",SuperscriptBox["y", "2"]}]]}]}], "}"}], ",",RowBox[{"{",RowBox[{"b", "\[Rule]",SqrtBox[RowBox[{SuperscriptBox["r", "2"], "-",RowBox[{"2", " ", "r", " ", "x"}], "+",SuperscriptBox["x", "2"], "+",SuperscriptBox["y", "2"]}]]}], "}"}]}], "}"}]], "Output",CellChangeTimes->{3.480263708377795*^9}]}, Open ]],Cell[CellGroupData[{Cell[BoxData[RowBox[{"Eliminate", "[",RowBox[{RowBox[{"{",RowBox[{RowBox[{SuperscriptBox["a", "2"], "==",RowBox[{SuperscriptBox["y", "2"], "+",SuperscriptBox["x", "2"]}]}], ",",RowBox[{SuperscriptBox["b", "2"], "==",RowBox[{SuperscriptBox["y", "2"], "+", " ",SuperscriptBox[RowBox[{"(",RowBox[{"r", "-", "x"}], ")"}], "2"]}]}]}], "}"}], ",", "b"}],"]"}]], "Input",CellChangeTimes->{{3.4802637422228947`*^9, 3.480263742630278*^9}}],Cell[BoxData[RowBox[{RowBox[{SuperscriptBox["x", "2"], "+",SuperscriptBox["y", "2"]}], "\[Equal]",SuperscriptBox["a", "2"]}]], "Output",CellChangeTimes->{3.48026374413827*^9}]}, Open ]],Cell[CellGroupData[{Cell[BoxData[RowBox[{"Solve", "[",RowBox[{RowBox[{RowBox[{SuperscriptBox["x", "2"], "+",SuperscriptBox["y", "2"]}], "\[Equal]",SuperscriptBox["a", "2"]}], ",", "a"}], "]"}]], "Input",CellChangeTimes->{{3.480263751875348*^9, 3.480263763613769*^9}}],Cell[BoxData[RowBox[{"{",RowBox[{RowBox[{"{",RowBox[{"a", "\[Rule]",RowBox[{"-",SqrtBox[RowBox[{SuperscriptBox["x", "2"], "+",SuperscriptBox["y", "2"]}]]}]}], "}"}], ",",RowBox[{"{",RowBox[{"a", "\[Rule]",SqrtBox[RowBox[{SuperscriptBox["x", "2"], "+",SuperscriptBox["y", "2"]}]]}], "}"}]}], "}"}]], "Output",CellChangeTimes->{3.480263765039091*^9}]}, Open ]],Cell[CellGroupData[{Cell[BoxData[RowBox[{"FullSimplify", "[",FractionBox[SqrtBox[RowBox[{RowBox[{"-",SuperscriptBox["a", "4"]}], "+",RowBox[{"2", " ",SuperscriptBox["a", "2"], " ",SuperscriptBox["b", "2"]}], "-",SuperscriptBox["b", "4"], "+",RowBox[{"2", " ",SuperscriptBox["a", "2"], " ",SuperscriptBox["r", "2"]}], "+",RowBox[{"2", " ",SuperscriptBox["b", "2"], " ",SuperscriptBox["r", "2"]}], "-",SuperscriptBox["r", "4"]}]],RowBox[{"2", " ", "r"}]], "]"}]], "Input",CellChangeTimes->{{3.480334626025627*^9, 3.480334632855236*^9}, {3.4803347434308167`*^9, 3.480334744290421*^9}}],Cell[BoxData[FractionBox[SqrtBox[RowBox[{RowBox[{"-",RowBox[{"(",RowBox[{"a", "-", "b", "-", "r"}], ")"}]}], " ",RowBox[{"(",RowBox[{"a", "+", "b", "-", "r"}], ")"}], " ",RowBox[{"(",RowBox[{"a", "-", "b", "+", "r"}], ")"}], " ",RowBox[{"(",RowBox[{"a", "+", "b", "+", "r"}], ")"}]}]],RowBox[{"2", " ", "r"}]]], "Output",CellChangeTimes->{3.480334634541192*^9, 3.4803347455989037`*^9}]}, Open ]]},WindowSize->{1434, 824},WindowMargins->{{0, Automatic}, {Automatic, 25}},FrontEndVersion->"7.0 for Linux x86 (64-bit) (February 25, 2009)",StyleDefinitions->"Default.nb"](* End of Notebook Content *)(* Internal cache information *)(*CellTagsOutlineCellTagsIndex->{}*)(*CellTagsIndexCellTagsIndex->{}*)(*NotebookFileOutlineNotebook[{Cell[CellGroupData[{Cell[567, 22, 648, 20, 32, "Input"],Cell[1218, 44, 692, 23, 33, "Output"]}, Open ]],Cell[CellGroupData[{Cell[1947, 72, 749, 24, 32, "Input"],Cell[2699, 98, 579, 19, 33, "Output"]}, Open ]],Cell[CellGroupData[{Cell[3315, 122, 694, 21, 32, "Input"],Cell[4012, 145, 1369, 42, 54, "Output"]}, Open ]],Cell[CellGroupData[{Cell[5418, 192, 547, 18, 32, "Input"],Cell[5968, 212, 239, 7, 33, "Output"]}, Open ]],Cell[CellGroupData[{Cell[6244, 224, 303, 8, 32, "Input"],Cell[6550, 234, 273, 9, 33, "Output"]}, Open ]],Cell[CellGroupData[{Cell[6860, 248, 366, 11, 32, "Input"],Cell[7229, 261, 316, 10, 50, "Output"]}, Open ]],Cell[7560, 274, 1175, 32, 66, "Input"],Cell[CellGroupData[{Cell[8760, 310, 599, 19, 55, "Input"],Cell[9362, 331, 299, 8, 33, "Output"]}, Open ]],Cell[CellGroupData[{Cell[9698, 344, 340, 9, 32, "Input"],Cell[10041, 355, 289, 9, 33, "Output"]}, Open ]],Cell[CellGroupData[{Cell[10367, 369, 382, 11, 32, "Input"],Cell[10752, 382, 626, 20, 46, "Output"]}, Open ]],Cell[CellGroupData[{Cell[11415, 407, 549, 18, 32, "Input"],Cell[11967, 427, 195, 6, 33, "Output"]}, Open ]],Cell[CellGroupData[{Cell[12199, 438, 283, 8, 32, "Input"],Cell[12485, 448, 450, 16, 46, "Output"]}, Open ]],Cell[CellGroupData[{Cell[12972, 469, 684, 20, 66, "Input"],Cell[13659, 491, 459, 14, 51, "Output"]}, Open ]]}]*)(* End of internal cache information *)