{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

1.09.ParamPlot

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

This preview shows pages 1–4. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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
x y z This is Mathematica 's default view. Usually Calculus& Mathematica uses the viewpoint CMView = 8 2.7, 1.6, 1.2 < .

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern