diff -r c507e9c413c6 -r 75fd106303dc plotting_using_sage/script.rst --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/plotting_using_sage/script.rst Fri Oct 08 11:40:21 2010 +0530 @@ -0,0 +1,257 @@ +.. Objectives +.. ---------- + +.. A - Students and teachers from Science and engineering backgrounds + B - + C - + D - + +.. Prerequisites +.. ------------- + +.. 1. Getting started with lists + +.. Author : Nishanth Amuluru + Internal Reviewer : + External Reviewer : + Checklist OK? : [2010-10-05] + +Script +------ + +Hello friends, welcome to the tutorial on "Plotting using SAGE". + +{{{ Show the outline slide }}} + +In this tutorial we shall look at + + * 2D plotting in SAGE + * 3D plotting in SAGE + +We shall first create a symbolic variable ``x`` +:: + + x = var('x') + +We shall plot the function ``sin(x) - cos(x) ^ 2`` in the range (-5, 5). +:: + + plot(sin(x) - cos(x) ^ 2, (x, -5, 5)) + +As we can see, the plot is shown. + +``plot`` command takes the symbolic function as the first argument and the +range as the second argument. + +{{{ Pause here and try out the following exercises }}} + +%% 1 %% Define a variable ``y`` and plot the function ``y^2 + 5y - 7`` in the + range (-3, 3) + +{{{ continue from paused state }}} + +:: + + y = var('y') + plot(y^2 + 5*y -7, (y, -3, 3)) + +We have seen that plot command plots the given function on a linear range. + +What if the x and y values are functions of another variable. +For instance, lets plot the trajectory of a projectile. + +A projectile was thrown at 50 m/s^2 and at an angle of 45 degrees from the +ground. We shall plot the trajectory of the particle for 5 seconds. + +These types of plots can be drawn using the parametric_plot function. +We first define the time variable. +:: + + t = var('t') + +Then we define the x and y as functions of t. +:: + + f_x = 50 * cos(pi/4) + f_y = 50 * sin(pi/4) * t - 1/2 * 9.81 * t^2 ) + +We then call the ``parametric_plot`` function as +:: + + parametric_plot((f_x, f_y), (t, 0, 5)) + +And we can see the trajectory of the projectile. + +The ``parametric_plot`` funciton takes a tuple of two functions as the first +argument and the range over which the independent variable varies as the second +argument. + +{{{ Pause here and try out the following exercises }}} + +%% 2 %% A particle is thrown into the air at 10 m/s^2 and at angle of 60 degrees + from the top of a 100 m tower. Plot the trajectory of the particle. + +{{{ continue from paused state }}} + +:: + + t = var('t') + f_x = 10 * cos(pi/3) * t + f_y = 100 + 10 * sin(pi/3) * t - 1/2 * 9.81 * t^2 + parametric_plot((f_x, f_y), (t,0,5)) + +Now we shall look at how to plot a set of points. + +We have the ``line`` function to acheive this. + +We shall plot sin(x) at few points and join them. + +First we need the set of points. +:: + + points = [ (x, sin(x)) for x in srange(-2*float(pi), 2*float(pi), 0.75) ] + +``srange`` takes a start, a stop and a step argument and returns a list of +point. We generate list of tuples in which the first value is ``x`` and second +is ``sin(x)``. + +:: + + line(points) + +plots the points and joins them with a line. + +{{{ Pause here and try out the following exercises }}} + +%% 3 %% Plot the cosine function using line function. + +{{{ continue from paused state }}} + +:: + + points = [ (x, cos(x)) for x in srange(-2*float(pi), 2*float(pi), 0.75) ] + line(points) + +The ``line`` function behaves like the plot command in matplotlib. The +difference is that ``plot`` command takes two sequences while line command +expects a sequence of co-ordinates. + +As we can see, the axes limits are set by SAGE. Often we would want to set them +ourselves. Moreover, the plot is shown here since the last command that is +executed produces a plot. + +Let us try this example +:: + + plot(cos(x), (x,0,2*pi)) + # Does the plot show up?? + +As we can see here, the plot is not shown since the last command does not +produce a plot. + +The actual way of showing a plot is to use the ``show`` command. + +:: + + p1 = plot(cos(x), (x,0,2*pi)) + show(p1) + # What happens now?? + +As we can see the plot is shown since we used it with ``show`` command. + +``show`` command is also used set the axes limits. + +:: + + p1 = plot(cos(x), (x,0,2*pi)) + show(p1, xmin=0, xmax=2*pi, ymin=-1.2, ymax=1.2) + +As we can see, we just have to pass the right keyword arguments to the ``show`` +command to set the axes limits. + +{{{ Pause here and try out the following exercises }}} + +%% 4 %% Plot the cosine function in the range (-2pi, 2pi) and set the x-axis + limits to (-5, 5) and y-axis limits to (-2, 2) respectively. + +{{{ continue from paused state }}} + +:: + + p1 = plot(cos(x), (x, 0, 2*pi)) + show(p1, xmin=-5, xmax=5, ymin=-2, ymax=2) + +The ``show`` command can also be used to show multiple plots. +:: + + p1 = plot(cos(x), (x, 0, 2*pi)) + p2 = plot(sin(x), (x, 0, 2*pi)) + show(p1+p2) + +As we can see, we can add the plots and use them in the ``show`` command. + +{{{ Pause here and try out the following exercises }}} + +%% 5 %% Plot sin(x) and sin(2*x) in the range (0, 2pi) + +{{{ continue from paused state }}} + +:: + + p1 = plot(sin(x), (x, 0, 2*pi)) + p2 = plot(sin(2*x), (x, 0, 2*pi)) + show(p1+p2) + +Now we shall look at 3D plotting in SAGE. + +We have the ``plot3d`` function that takes a function in terms of two +independent variables and the range over which they vary. + +:: + + x, y = var('x y') + plot3d(x^2 + y^2, (x, 0, 2), (y, 0, 2)) + +We get a 3D plot which can be rotated and zoomed using the mouse. + +{{{ Pause here and try out the following exercises }}} + +%% 6 %% Plot the function sin(x)^2 + cos(y)^2 for x in range (0,2) and y in + range (-2, 2) + +{{{ continue from paused state }}} + +:: + + x, y = var("x y") + plot3d( sin(x)^2 + cos(y)^2, (x, 0, 2), (y, -2, 2)) + +``parametric_plot3d`` function plots the surface in which x, y and z are +functions of another variable. + +:: + + u, v = var("u v") + f_x = u + f_y = v + f_z = u^2 + v^2 + parametric_plot3d((f_x, f_y, f_z), (u, 0, 2), (v, 0, 2)) + +{{{ Show summary slide }}} + +This brings us to the end of the tutorial. +we have learnt + + * How to draw 2D plots using plot comand + * How to use the parametric_plot and line functions + * How to use show command for multiple plots and setting axes limits + * How to draw 3D plots + +{{{ Show the "sponsored by FOSSEE" slide }}} + +#[Nishanth]: Will add this line after all of us fix on one. +This tutorial was created as a part of FOSSEE project, NME ICT, MHRD India + +Hope you have enjoyed and found it useful. +Thankyou +