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1.09.ParamPlot - 4 Growth Authors Bill Davis Horacio Porta...

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Growth Authors: Bill Davis, Horacio Porta and Jerry Uhl ©1996-2007 Publisher: Math Everywhere, Inc. Version 6.0 1.09 Parametric Plotting BASICS B.1) Parametric plots in two dimensions: Circular parameters A handy way to plot the circle x 2 + y 2 = 9 is to write x @ t D = 3 Cos @ t D and y @ t D = 3 Sin @ t D and then to plot the points 8 x @ t D , y @ t D< = 8 3 Cos @ t D , 3 Sin @ t D< as t advances from 0 to 2 p . Clear @ x, y, t D ; 8 x @ t _ D , y @ t _ D< = 8 3Cos @ t D , 3Sin @ t D< ; circle = ParametricPlot @8 x @ t D , y @ t D< , 8 t, 0, 2Pi < , PlotStyle -> 88 Blue, Thickness @ 0.015 D<< , AspectRatio -> Automatic, PlotRange -> 88 - 4, 4 < , 8 - 4, 4 <<D - 4 - 2 2 4 - 4 - 2 2 4 In this way, the original variables x and y are plotted by means of a third variable (t in this case) called a parameter. A parameter is an auxiliary variable that plays a back-room role. · B.1.a) When you plot the circle of radius 3 centered at 8 0, 0 < parametrically through the parametric formula 8 x @ t D , y @ t D< = 8 3 Cos @ t D , 3 Sin @ t D< , what is the physical meaning of the parameter t? · Answer: Look at this plot showing the circle and the point 8 x @ t D , y @ t D< you get with t = p 4 : Clear @ ray, t D ; ray @ t _ D = Graphics @ 8 Line @88 0, 0 < , 8 x @ t D , y @ t D<<D , 8 Red, PointSize @ 0.05 D , Point @8 x @ t D , y @ t D<D<<D ; Show B circle, ray B p 4 FF - 4 - 2 2 4 - 4 - 2 2 4 Now look at this: ListAnimate B Table B Show @ circle, ray @ t D , Graphics @ Text @ t, 1.4 8 x @ t D , y @ t D<DD , AspectRatio -> Automatic, Ticks -> None, PlotRange -> 88 - 5, 5 < , 8 - 5, 5 <<D , : t, 0, 2 p , p 4 >FF 5 p 4 The labels on the graphs give the values of t that make 8 x @ t D , y @ t D< plot out at the indicated point. · B.1.b) Parameters sometimes give you plotting freedom normal plotting does not allow. Some curves are best described via parametric equations and are more difficult to describe in the usual y = f @ x D terms. To see what this means, plot the spiral x = x @ t D = t Cos @ t D y = y @ t D = t Sin @ t D . · Answer: Clear @ x, y, t D ; x @ t _ D = tCos @ t D tCos @ t D y @ t _ D = tSin @ t D tSin @ t D
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ParametricPlot @8 x @ t D , y @ t D< , 8 t, 0, 8 p < , AxesLabel -> 8 "x" , "y" < , AspectRatio -> Automatic, PlotStyle -> 88 Thickness @ 0.01 D , Red <<D - 20 - 10 10 20 x - 20 - 10 10 20 y Extend the values of the parameter t to see more of the spiral: ParametricPlot @8 x @ t D , y @ t D< , 8 t, 0, 16 p < , AxesLabel -> 8 "x" , "y" < , AspectRatio -> Automatic, PlotStyle -> 88 Thickness @ 0.01 D , Red <<D - 40 - 20 20 40 x - 40 - 20 20 40 y Bull's eye. B.2) Parametric plots of curves in three dimensions When you plot in two dimensions, you use two coordinate axes: h = 5; spacer = h 10 ; twodims = Graphics @8 Line @88 - h, 0 < , 8 h, 0 <<D , Text @ "x" , 8 h + spacer, 0 <D , Line @88 0, - h < , 8 0, h <<D , Text @ "y" , 8 0, h + spacer <D<D ; Show @ twodims, PlotRange -> All D x y To plot a point, you specify an 8 x, y < coordinate: 8 x, y < = 8 2, 3 < ; Show @ twodims, Graphics @8 Red, PointSize @ 0.03 D , Point @8 x, y <D<D , PlotRange -> All D x y To plot in three dimensions, you need three coordinate axes: h = 5; spacer = h 10 ; threedims = Graphics3D @8 8 Blue, Line @88 - h, 0, 0 < , 8 h, 0, 0 <<D< , Text @ "x" , 8 h + spacer, 0, 0 <D , 8 Blue, Line @88 0, - h, 0 < , 8 0, h, 0 <<D< , Text @ "y" , 8 0, h + spacer, 0 <D , 8 Blue, Line @88 0, 0, - h < , 8 0, 0, h <<D< , Text @ "z" , 8 0, 0, h + spacer <D<D ; Show @ threedims, PlotRange -> All, Boxed -> False D
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x y z This is Mathematica 's default view. Usually Calculus& Mathematica uses the viewpoint CMView = 8 2.7, 1.6, 1.2 < .
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