1 .. Objectives

     2 .. ----------

     3 

     4 .. A - Students and teachers from Science and engineering backgrounds

     5    B -

     6    C - 

     7    D - 

     8 

     9 .. Prerequisites

    10 .. -------------

    11 

    12 ..   1. Getting started with lists

    13      

    14 .. Author              : Nishanth Amuluru

    15    Internal Reviewer   : 

    16    External Reviewer   :

    17    Checklist OK?       : <put date stamp here, if OK> [2010-10-05]

    18 

    19 Script

    20 ------

    21 

    22 Hello friends, welcome to the tutorial on "Plotting using SAGE".

    23 

    24 {{{ Show the outline slide }}}

    25 

    26 In this tutorial we shall look at 

    27  

    28  * 2D plotting in SAGE

    29  * 3D plotting in SAGE

    30 

    31 We shall first create a symbolic variable ``x``

    32 ::

    33 

    34     x = var('x')

    35 

    36 We shall plot the function ``sin(x) - cos(x) ^ 2`` in the range (-5, 5).

    37 ::

    38 

    39     plot(sin(x) - cos(x) ^ 2, (x, -5, 5))

    40 

    41 As we can see, the plot is shown.

    42 

    43 ``plot`` command takes the symbolic function as the first argument and the

    44 range as the second argument.

    45 

    46 {{{ Pause here and try out the following exercises }}}

    47 

    48 %% 1 %% Define a variable ``y`` and plot the function ``y^2 + 5y - 7`` in the

    49         range (-3, 3)

    50 

    51 {{{ continue from paused state }}}

    52 

    53 ::

    54 

    55     y = var('y')

    56     plot(y^2 + 5*y -7, (y, -3, 3))

    57 

    58 We have seen that plot command plots the given function on a linear range.

    59 

    60 What if the x and y values are functions of another variable.

    61 For instance, lets plot the trajectory of a projectile.

    62 

    63 A projectile was thrown at 50 m/s^2 and at an angle of 45 degrees from the 

    64 ground. We shall plot the trajectory of the particle for 5 seconds.

    65 

    66 These types of plots can be drawn using the parametric_plot function.

    67 We first define the time variable.

    68 ::

    69 

    70     t = var('t')

    71 

    72 Then we define the x and y as functions of t.

    73 ::

    74 

    75     f_x = 50 * cos(pi/4)

    76     f_y = 50 * sin(pi/4) * t - 1/2 * 9.81 * t^2 )

    77 

    78 We then call the ``parametric_plot`` function as

    79 ::

    80 

    81     parametric_plot((f_x, f_y), (t, 0, 5))

    82 

    83 And we can see the trajectory of the projectile.

    84 

    85 The ``parametric_plot`` funciton takes a tuple of two functions as the first

    86 argument and the range over which the independent variable varies as the second

    87 argument.

    88 

    89 {{{ Pause here and try out the following exercises }}}

    90 

    91 %% 2 %% A particle is thrown into the air at 10 m/s^2 and at angle of 60 degrees

    92         from the top of a 100 m tower. Plot the trajectory of the particle.

    93 

    94 {{{ continue from paused state }}}

    95 

    96 ::

    97 

    98     t = var('t')

    99     f_x = 10 * cos(pi/3) * t

   100     f_y = 100 + 10 * sin(pi/3) * t - 1/2 * 9.81 * t^2

   101     parametric_plot((f_x, f_y), (t,0,5))

   102 

   103 Now we shall look at how to plot a set of points.

   104 

   105 We have the ``line`` function to acheive this.

   106 

   107 We shall plot sin(x) at few points and join them.

   108 

   109 First we need the set of points.

   110 ::

   111 

   112     points = [ (x, sin(x)) for x in srange(-2*float(pi), 2*float(pi), 0.75) ]

   113 

   114 ``srange`` takes a start, a stop and a step argument and returns a list of

   115 point. We generate list of tuples in which the first value is ``x`` and second

   116 is ``sin(x)``.

   117 

   118 ::

   119 

   120     line(points)

   121 

   122 plots the points and joins them with a line.

   123 

   124 {{{ Pause here and try out the following exercises }}}

   125 

   126 %% 3 %% Plot the cosine function using line function.

   127 

   128 {{{ continue from paused state }}}

   129 

   130 ::

   131 

   132     points = [ (x, cos(x)) for x in srange(-2*float(pi), 2*float(pi), 0.75) ]

   133     line(points)

   134 

   135 The ``line`` function behaves like the plot command in matplotlib. The

   136 difference is that ``plot`` command takes two sequences while line command

   137 expects a sequence of co-ordinates.

   138 

   139 As we can see, the axes limits are set by SAGE. Often we would want to set them

   140 ourselves. Moreover, the plot is shown here since the last command that is

   141 executed produces a plot. 

   142 

   143 Let us try this example

   144 ::

   145 

   146     plot(cos(x), (x,0,2*pi))

   147     # Does the plot show up??

   148 

   149 As we can see here, the plot is not shown since the last command does not

   150 produce a plot.

   151 

   152 The actual way of showing a plot is to use the ``show`` command.

   153 

   154 ::

   155 

   156     p1 = plot(cos(x), (x,0,2*pi))

   157     show(p1)

   158     # What happens now??

   159 

   160 As we can see the plot is shown since we used it with ``show`` command.

   161 

   162 ``show`` command is also used set the axes limits.

   163 

   164 ::

   165 

   166     p1 = plot(cos(x), (x,0,2*pi))

   167     show(p1, xmin=0, xmax=2*pi, ymin=-1.2, ymax=1.2)

   168 

   169 As we can see, we just have to pass the right keyword arguments to the ``show``

   170 command to set the axes limits.

   171 

   172 {{{ Pause here and try out the following exercises }}}

   173 

   174 %% 4 %% Plot the cosine function in the range (-2pi, 2pi) and set the x-axis

   175         limits to (-5, 5) and y-axis limits to (-2, 2) respectively.

   176 

   177 {{{ continue from paused state }}}

   178 

   179 ::

   180 

   181     p1 = plot(cos(x), (x, 0, 2*pi))

   182     show(p1, xmin=-5, xmax=5, ymin=-2, ymax=2)

   183 

   184 The ``show`` command can also be used to show multiple plots.

   185 ::

   186 

   187     p1 = plot(cos(x), (x, 0, 2*pi))

   188     p2 = plot(sin(x), (x, 0, 2*pi))

   189     show(p1+p2)

   190 

   191 As we can see, we can add the plots and use them in the ``show`` command.

   192 

   193 {{{ Pause here and try out the following exercises }}}

   194 

   195 %% 5 %% Plot sin(x) and sin(2*x) in the range (0, 2pi)

   196 

   197 {{{ continue from paused state }}}

   198 

   199 ::

   200 

   201     p1 = plot(sin(x), (x, 0, 2*pi))

   202     p2 = plot(sin(2*x), (x, 0, 2*pi))

   203     show(p1+p2)

   204 

   205 Now we shall look at 3D plotting in SAGE.

   206 

   207 We have the ``plot3d`` function that takes a function in terms of two 

   208 independent variables and the range over which they vary.

   209 

   210 ::

   211 

   212     x, y = var('x y')

   213     plot3d(x^2 + y^2, (x, 0, 2), (y, 0, 2))

   214 

   215 We get a 3D plot which can be rotated and zoomed using the mouse.

   216 

   217 {{{ Pause here and try out the following exercises }}}

   218 

   219 %% 6 %% Plot the function sin(x)^2 + cos(y)^2 for x in range (0,2) and y in

   220         range (-2, 2)

   221 

   222 {{{ continue from paused state }}}

   223 

   224 ::

   225 

   226     x, y = var("x y")

   227     plot3d( sin(x)^2 + cos(y)^2, (x, 0, 2), (y, -2, 2))

   228 

   229 ``parametric_plot3d`` function plots the surface in which x, y and z are

   230 functions of another variable.

   231 

   232 ::

   233 

   234    u, v = var("u v")

   235    f_x = u

   236    f_y = v

   237    f_z = u^2 + v^2

   238    parametric_plot3d((f_x, f_y, f_z), (u, 0, 2), (v, 0, 2))

   239 

   240 {{{ Show summary slide }}}

   241 

   242 This brings us to the end of the tutorial.

   243 we have learnt

   244 

   245  * How to draw 2D plots using plot comand

   246  * How to use the parametric_plot and line functions

   247  * How to use show command for multiple plots and setting axes limits

   248  * How to draw 3D plots

   249 

   250 {{{ Show the "sponsored by FOSSEE" slide }}}

   251 

   252 #[Nishanth]: Will add this line after all of us fix on one.

   253 This tutorial was created as a part of FOSSEE project, NME ICT, MHRD India

   254 

   255 Hope you have enjoyed and found it useful.

   256 Thankyou

   257