.MCAD 308000000 \  docDocumentMmcObject[.. d2_graph_format graphData% axisFormat)L)Ltrace2D&&&&&&&&& & & & & &&& dim_formatTmasslengthtimecharge temperature luminosity substanceNumericalFormatQdii shpRectVhmmcDocumentObjectState\ mcPageModelK????mcHeaderFooterI@I CHeaderFooterJ@J@J@JMbP?MbP? TextState? TextStyle>@ ArialSerial_ParPropDefaultW?Normalfont_style_listO font_styleP  VariablesTimes New Roman@P  ConstantsTimes New Roman@P TextArial@P Greek VariablesSymbol@P User 1Arial@P User 2 Courier New@P User 3System@P User 4Script@P User 5Roman@P User 6Modern@P User 7Times New Roman@P SymbolsSymbol@P Current Selection FontArial@P Undefined Font@P HeaderArial@P FooterArial@P Rotated Math FontTimes New Roman TextRegion* docRegionGshpBoxUhjh CharacterMap-RangeMap;F Investigating Conics ... Eccentricity 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, x2/a2 + y2/b2 = 1, b2 = a2 - c2, e = c/a. The Mathematica code below generates an animation that illustrates the shape of the conic as the eccentricity varies. When e = 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. This 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. The code below generates an animation that displays the foci and the ellipse as the eccentricity varies. To display the animation, View->Animate. Select the region to be animated. Set the FRAME variable to vary from 0 to at least 30; the playback speed can be changed. (Suggestion: one frame per second is a nice playback speed.) Alternatively, you can view the avi file by clicking on the button below.  ChrPropMap7% RangeElem<  ChrPropData8 RangeData= <% 8nArial255,0,0 < 8 uz0%se7)ù헋23'<-.")"/&/$" -$ȓن.+}6/)-1.:1y80|( a s|~~}~~y{ wdZ xk {}~}|}}~|| y `!%5,+)3%++ %/,~ %!+')-)$&  #D@2$/Ó!*#*1"'#2. 2.$.   - 1( +0+,' !   (! &( a s|~~}~~|}}qckZ wp}}}}~~|| y `! !'   #   us"  ػ5$$  琁,%!'!ϑ"    &/     ## %* b`$   |z%;0( a s|~~}~~|}} |X!"fQz z~}}}~|| y `! "$ ϣ  &"%(Ҵ  '% $0y  ǣ, $ % &+ҕ ) #       .Զ 㓕 Ǝ ! 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