(*********************************************************************** Mathematica-Compatible Notebook This notebook can be used on any computer system with Mathematica 4.0, MathReader 4.0, or any compatible application. The data for the notebook starts with the line containing stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). NOTE: If you modify the data for this notebook not in a Mathematica- compatible application, you must delete the line below containing the word CacheID, otherwise Mathematica-compatible applications may try to use invalid cache data. For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. ***********************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 744380, 24362]*) (*NotebookOutlinePosition[ 745280, 24392]*) (* CellTagsIndexPosition[ 745236, 24388]*) (*WindowFrame->Normal*) Notebook[{ Cell[BoxData[ RowBox[{"\[IndentingNewLine]", StyleBox[\(Investigating\ Conics ... \[IndentingNewLine] Eccentricity\), "Title", FontColor->RGBColor[1, 0, 0]]}]], "Input", TextAlignment->Center, FontSize->14], Cell["\<\ This demo demonstrates the geometric effects of varying the eccentricity for \ conics. We start with an ellipse with fixed vertices at x = -1 and x = 1 \ with eccentricity e = 0. As the eccentricity varies, the foci and shape of \ the conic change, making a transition from ellipse to hyperbola. For an ellipse centered at the origin, x^2/a^2 + y^2/b^2 = 1, b^2 = a^2 - c^2, e = c/a.\ \>", "Text", FontSize->14], Cell[TextData[StyleBox["The Mathematica code below generates an animation \ that illustrates the shape of the conic as the eccentricity varies. When \ne \ = 0, the equations for the ellipse show that c = 0 and a = b so the foci are \ coincident at the origin. The \"ellipse\" is a circle. As e increases, the \ foci separate and b decreases. As e approaches 1, the ellipse becomes \ flatter and b approaches 0. the equation for the ellipse does not apply at \ this point, since that would require division by zero. In this case, the \ definition of ellipse requires the graph to be a line segment connecting the \ foci. As e increases from1, the foci continue to separate and the resulting \ conic is a hyperbola. Thus, we see that for 0 < e < 1, the conic is an \ ellipse; for e > 1, the conic is a hyperbola. \n\nThis can lead to a further \ discussion of whether there is a conic for which the eccentricity is exactly \ equal to 1. The answer is yes; that conic is a parabola. \n\nTo display the \ animation,select the closed cell below and from the toolbar,select \ Cell\[Rule]Animate Selected Graphics.You can control speed of the playback \ using the playback tools.", FontSize->14]], "Text"], Cell[BoxData[ \(<< Graphics`Animation`\)], "Input"], Cell[BoxData[{ \(Clear[f, g, a, b, c, e]\), "\[IndentingNewLine]", \(\(a = 4;\)\)}], "Input"], Cell[BoxData[{ \(b[e_] := Sqrt[a^2 \((e^2 - 1)\)] /; e \[GreaterEqual] 1\), "\[IndentingNewLine]", \(b[e_] := \(-Sqrt[a^2 \((1 - e^2)\)]\) /; e < 1\)}], "Input"], Cell[BoxData[{ \(f[x_, e_] := Sqrt[\((1 - x^2/a^2)\)*b[e]^2] /; b[e] \[LessEqual] 0\), "\[IndentingNewLine]", \(f[x_, e_] := Sqrt[\((1 - x^2/a^2)\)*\(-b[e]^2\)] /; b[e] > 0\), "\[IndentingNewLine]", \(g[x_, e_] := \(-1\)*Sqrt[\((1 - x^2/a^2)\)*b[e]^2] /; b[e] \[LessEqual] 0\), "\[IndentingNewLine]", \(g[x_, e_] := \(-1\)*Sqrt[\((1 - x^2/a^2)\)*\(-b[e]^2\)] /; b[e] > 0\), "\[IndentingNewLine]", \(\(c[e_] = a*e;\)\)}], "Input"], Cell[BoxData[ \(plotconic[ e_] := {Plot[{f[x, e], g[x, e]}, {x, \(-a\), a}, PlotRange \[Rule] {{\(-2.5\)*a, 2.5*a}, {\(-2.5\)*a, 2.5*a}}, AspectRatio \[Rule] 1, PlotLabel \[Rule] StyleForm[StringJoin["\", ToString[N[e, 2]]], FontSize \[Rule] 12], PlotStyle \[Rule] {{RGBColor[0, 0, 1], Thickness[ .01]}, {RGBColor[0, 0, 1], Thickness[ .01]}}, PlotPoints \[Rule] 150], Graphics[{PointSize[ .025], RGBColor[1, 0, 0], Point[{c[e], 0}]}], \ Graphics[{PointSize[ .025], RGBColor[1, 0, 0], Point[{\(-c[e]\), 0}]}]} /; e \[LessEqual] 1\)], "Input"], Cell[BoxData[ \(plotconic[ e_] := {Plot[{f[x, e], g[x, e]}, {x, \(-2.5\)*a, \(-a\)}, PlotRange \[Rule] {{\(-2.5\)*a, 2.5*a}, {\(-2.5\)*a, 2.5*a}}, AspectRatio \[Rule] 1, PlotLabel \[Rule] StyleForm[StringJoin["\", ToString[N[e, 2]]], FontSize \[Rule] 12], PlotStyle \[Rule] {{RGBColor[0, 0, 1], Thickness[ .01]}, {RGBColor[0, 0, 1], Thickness[ .01]}}, PlotPoints \[Rule] 150], Graphics[{PointSize[ .025], RGBColor[1, 0, 0], Point[{c[e], 0}]}], \ 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