(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 5.0' Mathematica-Compatible Notebook This notebook can be used with any Mathematica-compatible application, such as Mathematica, MathReader or Publicon. 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). 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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[ 283748, 8174]*) (*NotebookOutlinePosition[ 284661, 8203]*) (* CellTagsIndexPosition[ 284617, 8199]*) (*WindowFrame->Normal*) Notebook[{ Cell[TextData[{ StyleBox["Lab 2A - Parametrizing Curves", FontSize->24, FontWeight->"Bold", FontVariations->{"Underline"->True}], "\nMath 2374 - University of Minnesota\nhttp://www.math.umn.edu/math2374\n\ Questions to: rogness@math.umn.edu" }], "Text", CellFrame->True, TextAlignment->Center, TextJustification->0, FontColor->GrayLevel[1], Background->RGBColor[0, 0, 1]], Cell[CellGroupData[{ Cell[TextData[StyleBox["Introduction", FontSize->16]], "Section"], Cell[TextData[{ "Other than a short bit about derivatives at the end of this notebook, Lab \ 2 is almost entirely devoid of calculus. Instead, it's about visualizing and \ describing different curves and surfaces. These skills will be very \ important later in this class,which is why we're spending so much time on \ them now. \n\nThis week we will consider parametric equations of curve. The \ term \"curve\" has not been defined yet, but roughly speaking,a curve (or an \ \"arc\") is a curvy line in space. If you like, you could think of a curve \ as a piece of string that's sitting in two- or three-dimensional space.\n\n\ Examples are a segment of a straight line, a circle, or a section of any \ graph of a function ", Cell[BoxData[ \(TraditionalForm\`y = f(x)\)]], ". Sometimes the curve will have two ends; sometimes the two ends of the \ curve will actually be the same point -- i.e.the two ends of the string are \ taped together. In this case we call the curve a ", StyleBox["closed", FontSlant->"Italic"], " curve. The following pictures are all examples of curves. 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431.5}, {348, 278.062}} -> {-0.66036, 2.4044, \ 0.00695926, 0.00695926}, {{75.875, 183.625}, {346.312, 279.688}} -> {-3.2932, \ 13.9975, 0.0233877, 0.0378422}, {{194.438, 302.188}, {346.312, 279.688}} -> \ {-4.83953, 7.66499, 0.0155918, 0.0252281}, {{313, 420.75}, {346.312, \ 279.688}} -> {-11.2255, 10.706, 0.0233877, 0.0311705}}] }, Closed]], Cell[CellGroupData[{ Cell[TextData[StyleBox["Curves in Two Dimensions", FontSize->16]], "Section"], Cell[TextData[{ "Why do we care about parametric equations? There are many reasons, but \ one of the most important is that using parametric equations allows us to \ graph things that we otherwise wouldn't be able to graph. For \ example,consider the unit circle ", Cell[BoxData[ \(TraditionalForm\`x\^2\)]], "+", Cell[BoxData[ \(TraditionalForm\`y\^2\)]], "=1. You know from Lab 1A that one way to graph a function y=f(x) is the \ Mathematica command ", StyleBox["Plot", FontWeight->"Bold"], ". To use this command we need to solve our equation for y, and this is \ where the trouble starts; we get two solutions, ", Cell[BoxData[ \(TraditionalForm\`y = \@\(1 - x\^2\)\)]], " and ", Cell[BoxData[ \(TraditionalForm\`y = \(-\@\(1 - x\^2\)\)\)]], ".\n\nWe can plot these separately, but they each represent only half of \ the circle:" }], "Text"], Cell[BoxData[{ \(upper = Plot[Sqrt[1 - x^2], {x, \(-1\), 1}, AspectRatio \[Rule] Automatic]\), "\[IndentingNewLine]", \(lower = Plot[\(-Sqrt[1 - x^2]\), {x, \(-1\), 1}, AspectRatio \[Rule] Automatic]\)}], "Input"], Cell[TextData[{ "(The ", StyleBox["AspectRatio\[Rule]Automatic", FontWeight->"Bold"], " option forces ", StyleBox["Mathematica", FontSlant->"Italic"], " to scale the graph so it looks like a piece of a circle; try removing the \ option to see what happens.)\n\nIf we wish to see the whole circle, we can \ show the two graphs together:" }], "Text"], Cell[BoxData[ \(Show[upper, lower]\)], "Input"], Cell[TextData[{ "So we do in fact get to see the whole circle, but we haven't really ", StyleBox["graphed", FontSlant->"Italic"], " the circle. We've just pasted together two graphs to make it look like \ one circle. Using parametric equations can solve this problem. You've \ probably seen the parametric equations for the unit circle before: ", Cell[BoxData[ \(TraditionalForm\`f(t) = \((cos\ t, \ sin\ t)\)\)]], ", where 0\[LessEqual]t\[LessEqual]2\[Pi]. We can define these equations \ in Mathematica with the following commands:" }], "Text"], Cell[BoxData[{ \(\(x[t_] = Cos[t];\)\), "\[IndentingNewLine]", \(\(y[t_] = Sin[t];\)\), "\[IndentingNewLine]", \(f[t_] = {x[t], y[t]}\)}], "Input"], Cell["\<\ (We don't actually have to define x and y separately; we could the \ whole definition in one step by typing f[t_]={Cos[t],Sin[t]}.) Once we have the parametric equation set up,we can graph it using the \ command ParametricPlot:\ \>", "Text"], Cell[BoxData[ \(ParametricPlot[f[t], {t, 0, 2 Pi}, AspectRatio \[Rule] Automatic]\)], "Input"], Cell[TextData[{ "Now we have the same picture as above, but we produced it with one \ graphing command, instead of three! Also -- and perhaps more importantly, in \ the long run -- we have a function of one variable which describes the entire \ circle, instead of the implicit function ", Cell[BoxData[ \(TraditionalForm\`x\^2 + y\^2 = 1\)]], ", which has two variables in it and is harder to work with.\n\nWe call \ f[t] a ", StyleBox["parametrization", FontSlant->"Italic"], " of the circle. This is an important term, and you need to learn it. \ Later this semester (and even later in this lab) we will ask you to \"find a \ parametrization for...\" This means you need to find a parametric equation \ whose graph matches a certain picture. If you are asked to find a \ parametrization for the unit circle, you now know that the answer would be:\n\ \nf(t) = (cos t, sin t), 0 \[LessEqual] t \[LessEqual] 2\[Pi].\n" }], "Text"], Cell[TextData[{ "One quick comment about using ", StyleBox["ParametricPlot[]", FontWeight->"Bold"], ". With ", StyleBox["Mathematica", FontSlant->"Italic"], " v4.x, if you define a function f[t] and then plot it by typing ", StyleBox["ParametricPlot[ f[t], ......]", FontWeight->"Bold"], " as above, you'll see this blue error message:\n\n", Cell[BoxData[ \(ParametricPlot::"ppcom" \(\(:\)\(\ \)\) "Function \!\(f[t]\) cannot be compiled; plotting will proceed with \ the uncompiled function."\)], "Message", FontColor->RGBColor[0, 0, 1]], "\n\nThis isn't a problem and you shouldn't worry about it. If it really \ annoys you, ask your TA about the different ways to get rid of it. The \ simplest is probably to type and evaluate the command ", StyleBox["Off[ParametricPlot::ppcom]", FontWeight->"Bold"], ".\n\nWith ", StyleBox["Mathematica", FontSlant->"Italic"], " v5.0 this problem no longer exists." }], "Text", CellFrame->True, Background->GrayLevel[0.900008]], Cell[TextData[{ "Here are some other common examples of parametric equations:\n\n", StyleBox["Example 1", FontSize->14, FontWeight->"Bold"], StyleBox[". A circle of radius r\n", FontSize->14], "\nWe simply take our parametrization of the unit circle and multiply each \ component by r to get ", Cell[BoxData[ \(TraditionalForm\`f(t) = \((r*cos\ t, \ r*sin\ t)\)\)]], ", 0\[LessEqual]t\[LessEqual]2\[Pi]. Before you go any further, you should \ use ", StyleBox["ParametricPlot", FontWeight->"Bold"], " to graph two different circles where r=3 and r=5. Remember to include \ the ", StyleBox["AspectRatio\[RightArrow]Automatic", FontWeight->"Bold"], " option if you want the result to look like a circle." }], "Text", PageWidth->WindowWidth], Cell[TextData[{ StyleBox["Example 2", FontSize->14, FontWeight->"Bold"], StyleBox[". An ellipse ", FontSize->14], Cell[BoxData[ \(TraditionalForm\`x\^2\/a\^2\)], FontSize->14], StyleBox[" + ", FontSize->14], Cell[BoxData[ \(TraditionalForm\`y\^2\/b\^2\)], FontSize->14], StyleBox[" = 1", FontSize->14], "\n\nThis is an ellipse centered at the origin. To graph it by hand, you \ need to know that an ellipse written in this form goes through the points (\ \[PlusMinus]a, 0) and (0, \[PlusMinus] b). You would plot these four points \ and then try to draw a smooth ellipse through them. Mathematica can draw \ this precisely using parametric equations. (If you tried to use the ", StyleBox["Plot", FontWeight->"Bold"], " command you'd have the same problem as with the circle; when you solve \ for y you get " }], "Text", TextAlignment->Left], Cell[TextData[{ StyleBox["y = ", FontSize->16], Cell[BoxData[ \(TraditionalForm\`\(\[PlusMinus]\@\(\(b\^2\)(1 - x\^2\/a\^2)\)\)\)], FontSize->15] }], "Text", TextAlignment->Center], Cell[TextData[{ "and you'd have to graph it in two pieces.)\n\nTo parametrize the ellipse, \ we start with the parametrization for the unit circle, ", Cell[BoxData[ \(TraditionalForm\`f(t) = \((cos\ t, \ sint)\)\)]], ". You can think of the variables a and b as instructions for how far to \ stretch/compress this circle in the x and y directions to make our ellipse. \ In mathematical terms, this means we should multiply the x-component of f(t) \ by a, and the y-component of f(t) by b. So the parametrization of the \ ellipse is ", Cell[BoxData[ \(TraditionalForm\`\(\(f(t) = \((a*cos\ t, \ b*sin\ t)\)\)\(,\)\)\)]], " 0\[LessEqual]t\[LessEqual]2\[Pi]. You can see this is similar to the \ parametrization for a circle of radius r, and in fact if a=b then the result \ is a circle!\n\nAs an example, here is the graph of the ellipse ", StyleBox[" ", FontSize->14], Cell[BoxData[ \(TraditionalForm\`x\^2\/4\^2\)], FontSize->14], " + ", Cell[BoxData[ \(TraditionalForm\`y\^2\/3\^2\)], FontSize->14], " = 1" }], "Text", TextAlignment->Left], Cell[BoxData[ \(ParametricPlot[{4 Cos[t], 3 Sin[t]}, {t, 0, 2 Pi}, AspectRatio \[Rule] Automatic]\)], "Input"], Cell[TextData[{ StyleBox["Example 3", FontSize->14, FontWeight->"Bold"], StyleBox[". Circles and Ellipses centered away from the origin.", FontSize->14], StyleBox["\n", FontSize->14], "\nIt's actually surprisingly easy to parametrize a circle or an ellipse \ which is centered at a point other than the origin. For example, consider \ the circle of radius 3 centered at the point (1,-2). If the circle were \ centered at the origin, its parametrization would be g(t) = (3cos t, 3sin t). \ All we need to do is move all of our points 1 unit to the right and 2 units \ down to get the circle we want, so we add these numbers to the components of \ g(t):" }], "Text"], Cell[BoxData[{ \(f[t_] = {1 + 3 Cos[t], \(-2\) + 3 Sin[t]}\), "\[IndentingNewLine]", \(ParametricPlot[f[t], {t, 0, 2 Pi}, AspectRatio \[Rule] Automatic]\)}], "Input"], Cell["\<\ In general the parametrization for a circle of radius r, centered \ at (h,k), is f(t) = (h + r*cos t, k + r*sin t), where of course \ 0\[LessEqual]t\[LessEqual]2\[Pi]. You won't be surprised to find that this \ works for an ellipse, as well; to shift the ellipse from above so that it's \ centered at (1,-2), simply add 1 and -2 to the x and y components:\ \>", \ "Text"], Cell[BoxData[ \(ParametricPlot[{1 + 4 Cos[t], \(-2\) + 3 Sin[t]}, {t, 0, 2 Pi}, AspectRatio \[Rule] Automatic]\)], "Input"], Cell[TextData[{ StyleBox["Example 4", FontSize->14, FontWeight->"Bold"], StyleBox[". Spirals", FontSize->14], "\n\nConsider the following function. Try to think about what its graph \ will look like, and why." }], "Text"], Cell[BoxData[ \(f[t_] = {t*Cos[2 Pi*t], t*Sin[2 Pi*t]}\)], "Input"], Cell["\<\ It looks kind of like a circle of radius t, but because t is \ increasing, it means that instead of tracing out a circle, the graph of this \ function will spin out from the origin:\ \>", "Text"], Cell[BoxData[ \(ParametricPlot[f[t], {t, 0, 5}, AspectRatio \[Rule] Automatic]\)], "Input"], Cell[TextData[{ StyleBox["Example 5", FontSize->14, FontWeight->"Bold"], StyleBox[". Straight Line Segment", FontSize->14], StyleBox["\n", FontSize->13], "\nOne important example of a parametric equation is a straight line \ segment. Instead of simply quoting a formula, let's look at an example and \ try to parametrize such a segment.\n\nSuppose we have two points, P=(-1,2) \ and Q=(2,1). Intuitively, what we want to do is start at the point P and \ travel towards the point Q. The following picture represents this; we go \ from the origin to P, which is accomplished with the black arrow. The \ colored arrows represent our progress towards Q; the numbers above the arrows \ represent how far we've gone, percentage-wise. 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vector notation, the black arrow is the vector ", Cell[BoxData[ \(TraditionalForm\`OP\&\[RightVector]\)]], ", the vector that goes from the origin to the point P. The colored arrows \ are all multiples of the vector ", Cell[BoxData[ \(TraditionalForm\`PQ\&\[RightVector]\)]], ". (Note that most of each colored arrow is covered up by the previous \ arrows; all of the colored arrows actually start at P.) So once we're at the \ point P, we want to add on everything from 0% of ", Cell[BoxData[ \(TraditionalForm\`PQ\&\[RightVector]\)]], " to 100% of ", Cell[BoxData[ \(TraditionalForm\`PQ\&\[RightVector]\)]], ". Since percentages are really just numbers between 0 and 1, our \ parametrization for the line segment between the points P and Q is:\n\nf(t) = \ ", Cell[BoxData[ \(TraditionalForm\`OP\&\[RightVector]\)]], " + t\[CenterDot]", Cell[BoxData[ \(TraditionalForm\`PQ\&\[RightVector]\)]], ", where 0 \[LessEqual] t \[LessEqual] 1.\n\nFor our specific example, we \ have:" }], "Text"], Cell[BoxData[{ \(f[t_] = {\(-1\), 2}\ + \ t*\(({2, 1} - {\(-1\), 2})\)\), "\n", \(ParametricPlot[f[t], {t, 0, 1}]\)}], "Input"], Cell[TextData[{ "To make sure you understand this process, you should parametrize the line \ segments from (-1,-1) to (5,1) and from (4,2) to (4,-3). Plot them and \ verify that they are correct.\n\n", StyleBox["Example 6", FontSize->14, FontWeight->"Bold"], StyleBox[". A Line", FontSize->14], "\n\nIf we take the parametrization f(t) = ", Cell[BoxData[ \(TraditionalForm\`OP\&\[RightVector]\)]], " + t\[CenterDot]", Cell[BoxData[ \(TraditionalForm\`PQ\&\[RightVector]\)]], " and let t range from -\[Infinity] to +\[Infinity], instead of just 0 to \ 1, we get the entire line containing P and Q instead of the line segment \ between them. This is the parametric equation for a line.\n\n", StyleBox["Example 7", FontSize->14, FontWeight->"Bold"], StyleBox[". The Graph of a Function y=f(x)", FontSize->14], "\n\nAs we noted above, a piece of the graph of any function is a curve. \ For example, the following commands all produce examples of curves." }], "Text"], Cell[BoxData[ \(Plot[Sin[x^3], {x, 0, Pi}]\)], "Input"], Cell[BoxData[ \(Plot[x^2, {x, \(-1\), 1}]\)], "Input"], Cell[BoxData[ \(Plot[Exp[x], \ {x, \(-5\), 1}]\)], "Input"], Cell["\<\ There will be times when we want to represent these curves with \ parametric equations. There's an easy, almost silly, way to do this. If we \ want to parametrize the graph of y=g(x), we can let x be the parameter, and \ our parametrization is simply f(x) = (x, g(x)). Sometimes we wish to keep t \ as our parameter instead of switching to x. All we need to do is change the \ name of our variable, so we have f(t) = (t, g(t)). For example, the \ parametrization for the first plot above goes like this:\ \>", "Text"], Cell[BoxData[{ \(f[t_] = {t, Sin[t^3]}\), "\[IndentingNewLine]", \(ParametricPlot[f[t], {t, 0, Pi}]\)}], "Input"], Cell["\<\ The resulting graph should look the same as the first graph in this \ example. This leads to the obvious question, \"Why go to all this trouble if \ you end up with the same thing?\" There are various answers to this \ question, but perhaps the simplest is this: later on we'll learn many \ wonderful things you can do with parametrizations, and we want to be sure \ that we can do these things to plain old regular graphs of functions..\ \>", \ "Text"], Cell[TextData[{ StyleBox["Exercise 1", FontSize->16, FontWeight->"Bold"], "\n\nDuplicate the following picture as closely as possible by \ parametrizing each line segment and displaying them together. 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everything that we've graphed has been on the xy-plane. ", StyleBox["Mathematica", FontSlant->"Italic"], " can also graph parametric equations in three dimensional space. To do \ this, we use a variant of ", StyleBox["ParametricPlot", FontWeight->"Bold"], " which is named, not surprisingly, ", StyleBox["ParametricPlot3D", FontWeight->"Bold"], ". " }], "Text"], Cell[TextData[{ "Note the capitalization in ", StyleBox["ParametricPlot3D", FontWeight->"Bold"], ". If the two P's and the D are not capitalized, you will get an error \ message! Also, if you want to turn off the blue error messages in ", StyleBox["Mathematica", FontSlant->"Italic"], " 4.x about compiled functions, you can type and evaluate the command ", StyleBox["Off[ParametricPlot3D::ppcom]", FontWeight->"Bold"], " \[LongDash] essentially the same as before, but with \"3D\" in it. (Once \ again, you don't have to worry about this when using ", StyleBox["Mathematica", FontSlant->"Italic"], " 5.0.)\n\nAlso, as you might expect, we've Live-ned up the ", StyleBox["ParametricPlot3D", FontWeight->"Bold"], " command with an interactive version called ", StyleBox["ParametricPlot3DLive", FontWeight->"Bold"], ". This is in addition to the ", StyleBox["Live", FontWeight->"Bold"], " commands you learned about in Lab 1B." }], "Text", CellFrame->True, Background->GrayLevel[0.900008]], Cell[TextData[{ StyleBox["ParametricPlot3D", FontWeight->"Bold"], " expects a parametric equation with three components instead of two. We \ can actually graph any of the equations above by simply letting z=0. For \ example, the following commands will plot the unit circle:" }], "Text"], Cell[BoxData[{ \(f[t_] = {Cos[t], Sin[t], 0}\), "\[IndentingNewLine]", \(circle0 = ParametricPlot3D[f[t], {t, 0, 2 Pi}]\)}], "Input"], Cell[TextData[{ "As you can see, the difference is that now ", StyleBox["Mathematica", FontSlant->"Italic"], " treats the circle as if it lives in three dimensions instead of two. We \ can plot the unit circle for other values of z, as well:" }], "Text"], Cell[BoxData[{ \(circle1 = ParametricPlot3D[{Cos[t], Sin[t], 1}, {t, 0, 2 Pi}]\), "\[IndentingNewLine]", \(circle2 = ParametricPlot3D[{Cos[t], Sin[t], 2}, {t, 0, 2 Pi}]\)}], "Input"], Cell["\<\ We can even plot all three circles together, so you can really get \ a feel for the added dimension:\ \>", "Text"], Cell[BoxData[ \(Show[circle0, circle1, circle2]\)], "Input"], Cell[TextData[{ "The third dimension is even easier to see if you replace the ", StyleBox["Show", FontWeight->"Bold"], " command with ", StyleBox["ShowLive", FontWeight->"Bold"], " and rotate the picture a bit." }], "Text"], Cell[BoxData[ \(ShowLive[circle0, circle1, circle2]\)], "Input"], Cell[TextData[{ "Of course, the z component of the function does not have to stay constant, \ and things get a lot more interesting if we let it change. Here's a circle \ of radius 1 which is not in the xy plane. (Which plane ", StyleBox["is", FontSlant->"Italic"], " it in? The output can sometimes be a little deceiving!)" }], "Text"], Cell[BoxData[ \(circle = ParametricPlot3D[{Cos[t], 0, Sin[t]}, {t, 0, 2 Pi}, AxesLabel \[Rule] {"\", "\", "\"}]\)], "Input"], Cell[TextData[{ "(To see if you really understand what plane the circle is in, you could \ try adding the option ", StyleBox["ViewPoint\[Rule]{2,0,0}", FontWeight->"Bold"], " to the previous command and see if you can explain the output. \ Alternaticvely, you could replace ", StyleBox["ParametricPlot3D", FontWeight->"Bold"], " with ", StyleBox["ParametricPlot3DLive", FontWeight->"Bold"], " and rotate the picture to figure out what plane it's in.)" }], "Text"], Cell[TextData[{ "Let's look at a few examples of curves in three-dimensional space.\n\n", StyleBox["Example 8", FontSize->14, FontWeight->"Bold"], StyleBox[". Helix", FontSize->14], "\n\nTo introduce a helix, let's look at the following function first." }], "Text"], Cell[BoxData[ \(f[t_] = {Cos[t], Sin[t], t/2}\)], "Input"], Cell["\<\ As t moves from 0 to 2\[Pi], you can see that the x- and \ y-coordinates will go around the unit circle. But our z-coordinate starts at \ zero and increases, so instead of a circle, we get a helix twisting \ upwards.\ \>", "Text"], Cell[BoxData[ \(ParametricPlot3D[f[t], {t, 0, 2 Pi}]\)], "Input"], Cell["\<\ This helix only makes one revolution; to go around more times we \ can let t go from 0 to 4\[Pi], or 6\[Pi], etc. (We want to choose a multiple \ of 2\[Pi] because that represents a full trip around the circle.)\ \>", "Text"], Cell[BoxData[ \(ParametricPlot3D[f[t], {t, 0, 6 Pi}]\)], "Input"], Cell["\<\ Often we write the parametrization for a helix in a slightly \ different form, like this:\ \>", "Text"], Cell[BoxData[ \(f[t_] = {Cos[2 Pi*t], Sin[2 Pi*t], t}\)], "Input"], Cell["\<\ Now we make a full revolution around the circle when t increases \ from 0 to 1, instead of 2\[Pi]. Similarly, if we let t go from 0 to 2, we \ would make 2 revolutions around the helix. In general, to make n \ revolutions, we let t go from 0 to n. For example, to graph a helix which \ makes three revolutions, using the new parametrization, evaluate the \ following command:\ \>", "Text"], Cell[BoxData[ \(ParametricPlot3D[f[t], {t, 0, 3}]\)], "Input"], Cell[TextData[{ "Note one crucial difference: in this last example, the z values range from \ 0 to 3. In the previous example, the z values ranges from 0 to more than 9. \ Why is this?\n\n", StyleBox["Example 9", FontSize->14, FontWeight->"Bold"], StyleBox[". A Tornado\n", FontSize->14], "\nIf we \"combine\" the spiral from above with the helix we can make an \ interesting effect that looks like a tornado:" }], "Text"], Cell[BoxData[{ \(\(f[t_] = {t\ Cos[2 Pi\ t], \ t\ Sin[2 Pi\ t], \ 2 t};\)\), "\[IndentingNewLine]", \(\(ParametricPlot3D[f[t], {t, 0, 10}, ViewPoint \[Rule] {0, \(-10\), 1}];\)\)}], "Input"], Cell[TextData[{ "This is a little choppy, because ", StyleBox["Mathematica", FontSlant->"Italic"], " is letting the parameter t increase too much as it moves from point to \ point on the graph. To fix this we can use the option ", StyleBox["PlotPoints", FontWeight->"Bold"], " in the command, which specifies how many points to use when graphing the \ function. With a curve, the default is 75; if we increase this we can make \ the picture smoother:" }], "Text", CellFrame->True, Background->GrayLevel[0.900008]], Cell[BoxData[{ \(\(f[t_] = {t*Cos[2*Pi*t], t*Sin[2*Pi*t], 2*t};\)\), "\[IndentingNewLine]", \(ParametricPlot3D[f[t], {t, 0, 10}, ViewPoint \[Rule] {0, \(-10\), 1}, \[IndentingNewLine]\t PlotPoints \[Rule] 400]\)}], "Input"], Cell[TextData[{ "If you really want to be able to see a difference, try letting t range up \ to 40 in both commands and compare your output. We'll use this ", StyleBox["PlotPoints", FontWeight->"Bold"], " option for all of the graphs in the next example." }], "Text", CellFrame->True, Background->GrayLevel[0.900008]], Cell[TextData[{ StyleBox["Example 10", FontSize->14, FontWeight->"Bold"], StyleBox[". A Randomly Chosen Graph\n", FontSize->14], "\nThis example, unlike many of the others, will probably never be used in \ class. The point is just to show you that all kinds of silly things are \ possible -- and to continue demonstrating that we can make pretty pictures \ using parametric equations.\n\nConsider the graph of this function:" }], "Text"], Cell[BoxData[{ \(g[x_] = Sin[6 Pi\ x]\), "\[IndentingNewLine]", \(Plot[g[x], {x, \(-2\), 2}]\)}], "Input"], Cell[TextData[{ "For our silly example, let's create functions whose x- and y- values will \ follow this graph above, but whose z-values will vary with time t, instead of \ just being 0. If we ", StyleBox["do", FontSlant->"Italic"], " let z=0, the we just get the same graph again, but now \"living\" in \ three-dimensions:" }], "Text"], Cell[BoxData[{ \(f[t_] = {t, Sin[6 Pi\ t], \ 0}\), "\[IndentingNewLine]", \(ParametricPlot3D[f[t], {t, \(-2\), 2}, PlotPoints \[Rule] 300]\)}], "Input"], Cell[TextData[{ "Now let's play around with the z values. Try to predict what each graph \ will look like before evaluating the commands. You may want to use the ", StyleBox["ShowLive", FontWeight->"Bold"], " or ", StyleBox["ParametricPlot3DLive", FontWeight->"Bold"], " ", "functions to get a better look at some of these." }], "Text"], Cell[BoxData[{ \(f[t_] = {t, Sin[6 Pi\ t], \ t}\), "\[IndentingNewLine]", \(ParametricPlot3D[f[t], {t, \(-2\), 2}, PlotPoints \[Rule] 300]\)}], "Input"], Cell[BoxData[{ \(f[t_] = {t, Sin[6 Pi\ t], \ 4 - t}\), "\[IndentingNewLine]", \(ParametricPlot3D[f[t], {t, \(-2\), 2}, PlotPoints \[Rule] 300]\)}], "Input"], Cell[BoxData[{ \(f[t_] = {t, Sin[6 Pi\ t], \ 4 - Abs[t]}\), "\[IndentingNewLine]", \(ParametricPlot3D[f[t], {t, \(-2\), 2}, PlotPoints \[Rule] 300]\)}], "Input"], Cell[BoxData[{ \(f[t_] = {t, Sin[6 Pi\ t], \ t^2}\), "\[IndentingNewLine]", \(ParametricPlot3D[f[t], {t, \(-2\), 2}, PlotPoints \[Rule] 300]\)}], "Input"], Cell[BoxData[{ \(f[t_] = {t, Sin[6 Pi\ t], \ t^3}\), "\[IndentingNewLine]", \(ParametricPlot3D[f[t], {t, \(-2\), 2}]\)}], "Input"], Cell[TextData[{ StyleBox["Exercise 3\n", FontSize->14, FontWeight->"Bold"], "\nParametrize the intersection of the cylinder ", Cell[BoxData[ \(TraditionalForm\`x\^2\)]], " + ", Cell[BoxData[ \(TraditionalForm\`y\^2\)]], " = 9 with the plane z = 4-x. First you should plot the cylinder with the \ command ", StyleBox["ContourPlot3D", FontWeight->"Bold"], ", which was introduced in Lab 1B. You can plot the plane using ", StyleBox["Plot3D", FontWeight->"Bold"], ". Then use ", StyleBox["Show", FontWeight->"Bold"], " to plot them together so you can see the intersection and get an idea of \ what it should look like. (Suggested ranges of x and y for plotting both the \ cylinder and the plane are {x,-4,4}, {y,-4,4}. For ", StyleBox["ContourPlot3D", FontWeight->"Bold"], " you'll also need to specify a z range, such as {z,0,8}.)\n\nTo \ parametrize the curve which is the intersection, first note that the entire \ curve lies above the circle ", Cell[BoxData[ \(TraditionalForm\`x\^2\)]], " + ", Cell[BoxData[ \(TraditionalForm\`y\^2\)]], " = 9. You know how to choose x(t) and y(t) so that the x- and y-values \ are on this circle. All that's left is to determine what z(t) is.\n\n", StyleBox["In exercises 4 and 5, follow the directions for exercise three, \ changing the plotting ranges as needed.", FontSlant->"Italic"], "\n\n", StyleBox["Exercise 4\n", FontSize->14, FontWeight->"Bold"], "\nParametrize the intersection of the cylinder ", Cell[BoxData[ \(TraditionalForm\`x\^2\)]], " + ", Cell[BoxData[ \(TraditionalForm\`y\^2\)]], " = 4 with the surface z = ", Cell[BoxData[ \(TraditionalForm\`x\^2\)]], " - ", Cell[BoxData[ \(TraditionalForm\`y\^2\)]], ".\n\n", StyleBox["Exercise 5\n", FontSize->14, FontWeight->"Bold"], "\nParametrize the intersection of the cylinder ", Cell[BoxData[ \(TraditionalForm\`x\^2\)]], " + ", Cell[BoxData[ \(TraditionalForm\`y\^2\)]], " = 16 with the surface z = ", Cell[BoxData[ FormBox[ FractionBox[ FormBox[\(x\^2\), "TraditionalForm"], "4"], TraditionalForm]]], " - ", Cell[BoxData[ FormBox[ FractionBox[ FormBox[\(y\^2\), "TraditionalForm"], "9"], TraditionalForm]]], ".\n\n", StyleBox["Exercise 6", FontSize->14, FontWeight->"Bold"], "\n\nImagine you're in a Geography professor's office and there is a globe \ in front of you. Suppose you put a pen at the North Pole, start spinning the \ globe, and then you slowly move the pen from the North Pole to the South \ Pole. 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You should turn \ in your parametrization, with the corresponding graph, and an explanation of \ how you arrived at your answer. Simply stumbling across the correct answer \ will result in little credit. You may use the following hints:\n\n\ \[FilledSmallSquare] Every point on this curve is on the unit sphere ", Cell[BoxData[ \(TraditionalForm\`x\^2\)]], "+ ", Cell[BoxData[ \(TraditionalForm\`y\^2\)]], " + ", Cell[BoxData[ \(TraditionalForm\`z\^2\)]], " = 1, so when you think you have your final parametrization, you should \ make sure that if you square x(t), y(t), and z(t), and add them all together \ you'll get 1.\n\n\[FilledSmallSquare] There are many possible \ parametrizations, but in the one originally used to create this picture, \ z(t)=Cos[\[Pi]*t] and t goes from 0 to 1.\n\n\[FilledSmallSquare] Seen from \ the top down, the path makes twenty revolutions around the sphere. So you \ could start by parametrizing a helix which makes 20 revolutions and has the \ z(t) component given above. Then decide how to change x(t) and y(t). (Extra \ hint: you multiply them by the same thing, something which goes from 0 to 1 \ and back to 0 again as t goes from 0 to 1 itself.)\n\n\[FilledSmallSquare] \ You will probably want to used the ", StyleBox["PlotPoints", FontWeight->"Bold"], " option for ParametricPlot3D, which was used in examples 9 and 10. This \ picture was created using the option ", StyleBox["PlotPoints\[RightArrow]400", FontWeight->"Bold"], ".\n\n", StyleBox["Exercise 7", FontSize->14, FontWeight->"Bold"], "\n\nParametrize the following curve. 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00<000000?ooo`3oool05@3oool5000006/0oooo001:0?ooo`040000003oool0oooo000000/0oooo 00@000000?ooo`3oool0oooo0P0000050?ooo`030000003oool0oooo0140oooo1000001`0?ooo`00 BP3oool010000000oooo0?ooo`00000;0?ooo`030000003oool0oooo0080oooo0P0000040?ooo`03 0000003oool0oooo00d0oooo1000001d0?ooo`00BP3oool010000000oooo0?ooo`0000080?ooo`<0 00001`3oool01@000000oooo0?ooo`3oool0000000X0oooo1@00001h0?ooo`00BP3oool010000000 oooo0?ooo`0000080?ooo`030000003oool0oooo00P0oooo00@000000?ooo`3oool000001P3oool4 000007d0oooo001:0?ooo`040000003oool0oooo000000P0oooo00<000000?ooo`3oool0203oool4 00000080oooo100000210?ooo`00B`3oool2000000T0oooo1000000:0?ooo`<00000Q@3oool00>`0 oooo003/0?ooo`00k03oool00001\ \>"], ImageRangeCache->{{{0, 235.375}, {255.75, 0}} -> {0, 0, 0, 0}}], "\n\nYou can use the following hints:\n\n\[FilledSmallSquare] Every point \ on this curve is on the double cone ", Cell[BoxData[ \(TraditionalForm\`z\^2 = x\^2 + y\^2\)]], ", so when you think you have your final parametrization, you should make \ sure that if you square x(t), and y(t) and add them together, you get ", Cell[BoxData[ \(TraditionalForm\`\((z(t))\)\^2\)]], ".\n\n\[FilledSmallSquare] There are many possible parametrizations, but in \ the one originally used to create this picture, z(t)=t, and t goes from -1 \ to 1.\n\n\[FilledSmallSquare] Seen from the top down, the path makes twenty \ revolutions around the z-axis. So you could start by parametrizing a helix \ which makes 20 revolutions and has the z(t) component given above. Then \ decide how to change x(t) and y(t)." }], "Text", CellFrame->True, Background->RGBColor[0.996109, 0.597665, 0.597665]] }, Closed]], Cell[CellGroupData[{ Cell[TextData[StyleBox["Derivatives of Parametrizations", FontSize->16]], "Section"], Cell[TextData[{ "Note: depending on your instructor, you may have already seen this \ material as a part of an assigned reading. In that case, you can very \ quickly skim through the material, but you should read about how to use the \ ", StyleBox["PathAnimate", FontWeight->"Bold"], " and ", StyleBox["PathTangentAnimate", FontWeight->"Bold"], " commands. You'll be asked to use them in later labs.\n\nFor simplicity, \ we're only going to deal with two dimensions here, but the same ideas apply \ in three dimensions as well." }], "Text", CellFrame->True, Background->GrayLevel[0.833326]], Cell["\<\ Suppose we have a parametrization f(t) = (x(t), y(t)), where t \ ranges from 0 to 10. If we say that t represents time in seconds, then we \ can say f(t) represents the position of an object at time t. For example, \ consider the following function:\ \>", "Text"], Cell[BoxData[ \(\(f[t_] = {Cos[t], Sin[t]};\)\)], "Input"], Cell[TextData[{ "You should know by now that this is a parametrization for the unit circle. \ As you learned above you can plot this function using ", StyleBox["ParametricPlot", FontWeight->"Bold"], ":" }], "Text"], Cell[BoxData[ \(ParametricPlot[f[t], {t, 0, 2 Pi}, AspectRatio \[Rule] Automatic]\)], "Input"], Cell["\<\ This doesn't give you a feel for what I'm telling you, however: \ that f(t) can be viewed as the location function for a particle moving around \ the unit circle, where the particle finishes its trip when t=2\[Pi] seconds \ (or minutes, or hours, or whatever our unit is). Try looking at the output of the following command. It will produce an \ animation of the particle moving around, which is actually just a number of \ different pictures in rapid succession. Your TA will demonstrate how to view \ the animation.\ \>", "Text"], Cell[BoxData[ \(PathAnimate[f[t], {t, 0, 2 Pi}, 40]\)], "Input"], Cell[TextData[{ "Now you can see the particle moving around the curve. It looks as if the \ speed of the particle is constant; let's see if we can prove that somehow.\n\n\ Recall from section 1.10 in your textbook that if f(t) represents the \ location of a particle at time t, then v(t)=", Cell[BoxData[ \(TraditionalForm\`f\^\[Prime]\)]], "(t), the derivative of f, represents the ", StyleBox["velocity", FontSlant->"Italic"], " of the particle at time t. For example, in our case we have:" }], "Text"], Cell[BoxData[ \(v[t_] = D[f[t], t]\)], "Input"], Cell[TextData[{ "Velocity is a little different than speed. Speed is just a number \ representing how fast a particle is moving. Velocity is a vector whose ", StyleBox["length", FontSlant->"Italic"], " represents speed, and whose ", StyleBox["direction", FontSlant->"Italic"], " represents the direction the particle is moving at that specific instance \ of time.\n\nTo see an animation of the particle moving around the circle \ along with its tangent vectors, run this command:" }], "Text"], Cell[BoxData[ \(PathTangentAnimate[f[t], {t, 0, 2 Pi}, 40]\)], "Input"], Cell[TextData[{ "It certainly looks as if the length of the velocity vector is constant, \ which would confirm that the speed of the particle is constant. In fact, we \ can show this algebraically. You should find the length of the tangent \ vector v(t) on paper and show that it is constant -- it does not depend on t.\ \n\nLet's look at a different parametrization of the unit circle,\n\ng(t) = \ (", Cell[BoxData[ \(TraditionalForm\`\(\(Cos(t\^2)\)\(,\)\(\ \)\(Sin(t\^2)\)\(\ \)\)\)]], "),\t0\[LessEqual]t\[LessEqual]", Cell[BoxData[ \(TraditionalForm\`\@\(2 \[Pi]\)\)]], ".\n\nThis is a parametrization of the unit circle because as t ranges from \ 0 to ", Cell[BoxData[ \(TraditionalForm\`\@\(2 \[Pi]\)\)]], ", ", Cell[BoxData[ \(TraditionalForm\`t\^2\)]], " ranges from 0 to 2\[Pi]:" }], "Text"], Cell[BoxData[{ \(\(g[t_] = {Cos[t^2], Sin[t^2]};\)\), "\[IndentingNewLine]", \(ParametricPlot[g[t], {t, 0, Sqrt[2 Pi]}, AspectRatio \[Rule] Automatic]\)}], "Input"], Cell[TextData[{ "Now let's watch the particle whose motion is represented by ", Cell[BoxData[ \(TraditionalForm\`g(t)\)]], ":" }], "Text"], Cell[BoxData[ \(PathTangentAnimate[g[t], {t, 0, Sqrt[2 Pi]}, 40]\)], "Input"], Cell["\<\ As you can see, the particle starts out very slowly and picks up \ speed as time goes on. You can see this either by watching the particle \ itself, or by watching the length of the tangent vector grow. (Remember, the \ length of the tangent vector represents the speed of the particle!) Find the velocity of this particle by computing the derivative of g[t]. Can \ you see why the particle is speeding up? You'll be asked to investigate this \ parametrization of the circle a little more in Lab 4A.\ \>", "Text"], Cell[TextData[{ "If you'd like to play around with these two animation commands, their \ syntax is:\n\n", StyleBox["PathAnimate[f[t],{t,tmin,tmax},n]\n\ PathTangentAnimate[f[t],{t,tmin,tmax},n]", FontFamily->"Courier", FontWeight->"Bold"], StyleBox["\n", FontWeight->"Bold"], "\nwhere n represents the number of frames to draw. (So the bigger n is, \ the longer it takes to create the animation, but the smoother it looks. 40 \ seems to be a reasonable number for most parametrizations. Here are a couple \ of parametrizations to try; you can copy these and paste them into commands:\n\ \n", StyleBox["f[t_]={Cos[t],Sin[t]^3}, {t, 0, 2Pi}\t\t(at least 40 frames)\n\ f[t_]={Cos[2t],Sin[3t]}, {t, 0, 2Pi}\t\t(at least 50 frames)\n\ f[t_]={Cos[2t],Sin[4t]}, {t, 0, 2Pi}\t\t(at least 50 frames)\n\ f[t_]={Cos[5t],Sin[3t]}, {t, 0, 2Pi}\t\t(at least 100 frames)\nf[t_]={t,t^2}, \ {t, -1,1}\t\t\t\t\t(at least 30 frames)\nf[t_]={t^3,t^2}, {t, -1,1}\t\t\t\t\t\ (at least 30 frames)", FontFamily->"Courier"] }], "Text", CellFrame->True, Background->GrayLevel[0.849989]], Cell[CellGroupData[{ Cell["Credits", "Subsection"], Cell["\<\ This lab was written from scratch in December 2001/January 2002. I \ made some minor updates and added a few exercises in January 2004. The \ biggest change was the inclusion of the Animation commands, which used to \ live in Lab 4A (Arclength). Parametrizations and their derivatives are \ covered in lecture during the second week of class, so this was a more \ natural location for this material to appear in a lab. I did not think of the curve in exercise 7 on my own. I can't remember who \ first showed it to me, but presumably it was Dr. Emil Knapp at Augustana \ College, who was my advisor and my Calc III professor. I've noticed that \ Harvard's multivariable calculus class also used this curve as an example in \ one of their labs. Their course has apparently been revamped, because I \ tried to find a URL for their lab to include here, but it no longer exists. This lab is copyright 2002, 2004 by Jonathan Rogness (rogness@math.umn.edu) \ and is protected by the Creative Commons Attribution-NonCommercial-ShareAlike \ License. You can find more information on this license at \ http://creativecommons.org/licenses/by-nc-sa/1.0/ Although it's not specifically required by the license, I'd appreciate it if \ you let me know if you use parts of our labs, just so I can keep track of it. \ Please send me any questions or comments!\ \>", "Text"], Cell[BoxData[""], "Input"] }, Closed]] }, Closed]] }, FrontEndVersion->"5.0 for X", ScreenRectangle->{{0, 1024}, {0, 768}}, WindowSize->{598, 691}, WindowMargins->{{10, Automatic}, {0, Automatic}}, PrintingPageRange->{Automatic, Automatic}, PrintingOptions->{"PaperSize"->{612, 792}, "PaperOrientation"->"Portrait", "PostScriptOutputFile":>FrontEnd`FileName[{"user002", "rogness"}, \ "Newlab.nb.ps", CharacterEncoding -> "iso8859-1"], "Magnification"->1}, CellLabelAutoDelete->True ] (******************************************************************* Cached data follows. 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