.. Automatically generated reST file from Doconce source (http://code.google.com/p/doconce/) Using Pysketcher to Create Principal Sketches of Physics Problems ================================================================= :Author: Hans Petter Langtangen :Date: Apr 1, 2012 *Abstract.* Pysketcher is a Python package which allows principal sketches of physics and mechanics problems to be realized through short programs instead of interactive (and potentially tedious and inaccurate) drawing. Elements of the sketch, such as lines, circles, angles, forces, coordinate systems, etc., are realized as objects and collected in hierarchical structures. Parts of the hierarchical structures can easily change line styles and colors, or be copied, scaled, translated, and rotated. These features make it straightforward to move parts of the sketch to create animation, usually in accordance with the physics of the underlying problem. Exact dimensioning of the elements in the sketch is trivial to obtain since distances are specified in computer code. Pysketcher is easy to learn from a number of examples. Beyond essential Python programming and a knowledge about mechanics problems, no further background is required. .. Task (can be questions): make sketches of physical problems, see fig .. through user-friendly composition of basic shapes .. Desired knowledge: plotting curves, basic OO (ch. X.Y, ...) .. Required knowledge? .. Learning Goals: these targets the inner workings of pysketcher, .. which is just a part of this document... A First Glimpse of Pysketcher ============================= Formulation of physical problems makes heavy use of *principal sketches* such as the one in Figure :ref:`sketcher:fig:inclinedplane`. This particular sketch illustrates the classical mechanics problem of a rolling wheel on an inclined plane. The figure is made up many individual elements: a rectangle filled with a pattern (the inclined plane), a hollow circle with color (the wheel), arrows with label (the :math:`N` and :math:`Mg` forces, and the :math:`x` axis), an angle with symbol :math:`\theta`, and a dashed line indicating the starting location of the wheel. Drawing software and plotting programs can produce such figures quite easily in principle, but the amount of details the user needs to control with the mouse can be substantial. Software more tailored to producing sketches of this type would work with more convenient abstractions, such as circle, wall, angle, force arrow, axis, and so forth. And as soon we start *programming* to construct the figure we get a range of other powerful tools at disposal. For example, we can easily translate and rotate parts of the figure and make an animation that illustrates the physics of the problem. .. _sketcher:fig:inclinedplane: .. figure:: figs-sketcher/wheel_on_inclined_plane.png :width: 600 *Sketch of a physics problem* Basic Construction of Sketches ------------------------------ Before attacking real-life sketches as in Figure :ref:`sketcher:fig1` we focus on the significantly simpler drawing shown in in Figure :ref:`sketcher:fig:vehicle0`. This toy sketch consists of several elements: two circles, two rectangles, and a "ground" element. .. _sketcher:fig:vehicle0: .. figure:: figs-sketcher/vehicle0_dim.png :width: 600 *Sketch of a simple figure* Basic Drawing ~~~~~~~~~~~~~ A typical program creating these five elements is shown next. After importing the ``pysketcher`` package, the first task is always to define a coordinate system. Some graphics operations are done with a helper object called ``drawing_tool`` (imported from ``pysketcher``). With the drawing area in place we can make the first ``Circle`` object in an intuitive fashion: .. code-block:: python from pysketcher import * R = 1 # radius of wheel L = 4 # distance between wheels H = 2 # height of vehicle body w_1 = 5 # position of front wheel drawing_tool.set_coordinate_system(xmin=0, xmax=w_1 + 2*L + 3*R, ymin=-1, ymax=2*R + 3*H) wheel1 = Circle(center=(w_1, R), radius=R) By using symbols for characteristic lengths in the drawing, rather than absolute lengths, it is easier to change dimensions later. To translate the geometric information about the ``wheel1`` object to instructions for the plotting engine (in this case Matplotlib), one calls the ``wheel1.draw()``. To display all drawn objects, one issues ``drawing_tool.display()``. The typical steps are hence: .. code-block:: python wheel1 = Circle(center=(w_1, R), radius=R) wheel1.draw() # Define other objects and call their draw() methods drawing_tool.display() drawing_tool.savefig('tmp.png') # store picture The next wheel can be made by taking a copy of ``wheel1`` and translating the object a distance (to the right) described by the vector :math:`(4,0)`: .. code-block:: python wheel2 = wheel1.copy() wheel2.translate((L,0)) The two rectangles are made in an intuitive way: .. code-block:: python under = Rectangle(lower_left_corner=(w_1-2*R, 2*R), width=2*R + L + 2*R, height=H) over = Rectangle(lower_left_corner=(w_1, 2*R + H), width=2.5*R, height=1.25*H) Groups of Objects ~~~~~~~~~~~~~~~~~ Instead of calling the ``draw`` method of every object, we can group objects and call ``draw``, or perform other operations, for the whole group. For example, we may collect the two wheels in a ``wheels`` group and the ``over`` and ``under`` rectangles in a ``body`` group. The whole vehicle is a composition of its ``wheels`` and ``body`` groups. The codes goes like .. code-block:: python wheels = Composition({'wheel1': wheel1, 'wheel2': wheel2}) body = Composition({'under': under, 'over': over}) vehicle = Composition({'wheels': wheels, 'body': body}) The ground is illustrated by an object of type ``Wall``, mostly used to indicate walls in sketches of mechanical systems. A ``Wall`` takes the ``x`` and ``y`` coordinates of some curve, and a ``thickness`` parameter, and creates a "thick" curve filled with a simple pattern. In this case the curve is just a flat line so the construction is made of two points on the ground line (:math:`(w_1-L,0)` and :math:`(w_1+3L,0)`): .. code-block:: python ground = Wall(x=[w_1 - L, w_1 + 3*L], y=[0, 0], thickness=-0.3*R) The negative thickness makes the pattern-filled rectangle appear below the defined line, otherwise it appears above. We may now collect all the objects in a "top" object that contains the whole figure: .. code-block:: python fig = Composition({'vehicle': vehicle, 'ground': ground}) fig.draw() # send all figures to plotting backend drawing_tool.display() drawing_tool.savefig('tmp.png') The ``fig.draw()`` call will visit all subgroups, their subgroups, and so in the hierarchical tree structure that we have collected, and call ``draw`` for every object. Changing Line Styles and Colors ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Controlling the line style, line color, and line width is fundamental when designing figures. The ``pysketcher`` package allows the user to control such properties in single objects, but also set global properties that are used if the object has no particular specification of the properties. Setting the global properties are done like .. code-block:: python drawing_tool.set_linestyle('dashed') drawing_tool.set_linecolor('black') drawing_tool.set_linewidth(4) At the object level the properties are specified in a similar way: .. code-block:: python wheel1.set_linestyle('solid') wheel1.set_linecolor('red') and so on. Geometric figures can be specified as *filled*, either with a color or with a special visual pattern: .. code-block:: py # Set filling of all curves drawing_tool.set_filled_curves(color='blue', pattern='/') # Turn off filling of all curves drawing_tool.set_filled_curves(False) # Fill the wheel with red color wheel1.set_filled_curves('red') .. http://packages.python.org/ete2/ for visualizing tree structures! The Figure Composition as an Object Hierarchy ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ The composition of objects is hierarchical, as in a family, where each object has a parent and a number of children. Do a ``print fig`` to display the relations: .. code-block:: python ground wall vehicle body over rectangle under rectangle wheels wheel1 arc wheel2 arc The indentation reflects how deep down in the hierarchy (family) we are. This output is to be interpreted as follows: * ``fig`` contains two objects, ``ground`` and ``vehicle`` * ``ground`` contains an object ``wall`` * ``vehicle`` contains two objects, ``body`` and ``wheels`` * ``body`` contains two objects, ``over`` and ``under`` * ``wheels`` contains two objects, ``wheel1`` and ``wheel2`` More detailed information can be printed by .. code-block:: python print fig.show_hierarchy('std') yielding the output .. code-block:: python ground (Wall): wall (Curve): 4 coords fillcolor='white' fillpattern='/' vehicle (Composition): body (Composition): over (Rectangle): rectangle (Curve): 5 coords under (Rectangle): rectangle (Curve): 5 coords wheels (Composition): wheel1 (Circle): arc (Curve): 181 coords wheel2 (Circle): arc (Curve): 181 coords Here we can see the class type for each figure object, how many coordinates that are involved in basic figures (``Curve`` objects), and special settings of the basic figure (fillcolor, line types, etc.). For example, ``wheel2`` is a ``Circle`` object consisting of an ``arc``, which is a ``Curve`` object consisting of 181 coordinates (the points needed to draw a smooth circle). The ``Curve`` objects are the only objects that really holds specific coordinates to be drawn. The other object types are just compositions used to group parts of the complete figure. One can also get a graphical overview of the hierarchy of figure objects that build up a particular figure ``fig``. Just call ``fig.graphviz_dot('fig')`` to produce a file ``fig.dot`` in the *dot format*. This file contains relations between parent and child objects in the figure and can be turned into an image, as in Figure :ref:`sketcher:fig:vehicle0:hier1`, by running the ``dot`` program: .. code-block:: console Terminal> dot -Tpng -o fig.png fig.dot .. _sketcher:fig:vehicle0:hier1: .. figure:: figs-sketcher/vehicle0_hier1.png :width: 500 *Hierarchical relation between figure objects* The call ``fig.graphviz_dot('fig', classname=True)`` makes a ``fig.dot`` file where the class type of each object is also visible, see Figure :ref:`sketcher:fig:vehicle0:hier2`. The ability to write out the object hierarchy or view it graphically can be of great help when working with complex figures that involve layers of subfigures. .. _sketcher:fig:vehicle0:hier2: .. figure:: figs-sketcher/vehicle0_hier2.png :width: 500 *Hierarchical relation between figure objects, including their class names* Any of the objects can in the program be reached through their names, e.g., .. code-block:: python fig['vehicle'] fig['vehicle']['wheels'] fig['vehicle']['wheels']['wheel2'] fig['vehicle']['wheels']['wheel2']['arc'] fig['vehicle']['wheels']['wheel2']['arc'].x # x coords fig['vehicle']['wheels']['wheel2']['arc'].y # y coords fig['vehicle']['wheels']['wheel2']['arc'].linestyle fig['vehicle']['wheels']['wheel2']['arc'].linetype Grabbing a part of the figure this way is very handy for changing properties of that part, for example, colors, line styles (see Figure :ref:`sketcher:fig:vehicle0:v2`): .. code-block:: python fig['vehicle']['wheels'].set_filled_curves('blue') fig['vehicle']['wheels'].set_linewidth(6) fig['vehicle']['wheels'].set_linecolor('black') fig['vehicle']['body']['under'].set_filled_curves('red') fig['vehicle']['body']['over'].set_filled_curves(pattern='/') fig['vehicle']['body']['over'].set_linewidth(14) fig['vehicle']['body']['over']['rectangle'].linewidth = 4 The last line accesses the ``Curve`` object directly, while the line above, accesses the ``Rectangle`` object which will then set the linewidth of its ``Curve`` object, and other objects if it had any. The result of the actions above is shown in Figure :ref:`sketcher:fig:vehicle0:v2`. .. _sketcher:fig:vehicle0:v2: .. figure:: figs-sketcher/vehicle0.png :width: 700 *Left: Basic line-based drawing. Right: Thicker lines and filled parts* We can also change position of parts of the figure and thereby make animations, as shown next. Animation: Translating the Vehicle ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Can we make our little vehicle roll? A first attempt will be to fake rolling by just displacing all parts of the vehicle. The relevant parts constitute the ``fig['vehicle']`` object. This part of the figure can be translated, rotated, and scaled. A translation along the ground means a translation in :math:`x` direction, say a length :math:`L` to the right: .. code-block:: python fig['vehicle'].translate((L,0)) You need to erase, draw, and display to see the movement: .. code-block:: python drawing_tool.erase() fig.draw() drawing_tool.display() Without erasing the old position of the vehicle will remain in the figure so you get two vehicles. Without ``fig.draw()`` the new coordinates of the vehicle will not be communicated to the drawing tool, and without calling display the updated drawing will not be visible. Let us make a velocity function and move the object according to that velocity in small steps of time: .. code-block:: python def v(t): return -8*R*t*(1 - t/(2*R)) animate(fig, tp, action) For small time steps ``dt`` the corresponding displacement is well approximated by ``dt*v(t)`` (we could integrate the velocity to obtain the exact position, but we would anyway need to calculate the displacement from time step to time step). The ``animate`` function takes as arguments some figure ``fig``, a set of time points ``tp``, and a user function ``action``, and then a new figure is drawn for each time point and the user can through the provided ``action`` function modify desired parts of the figure. Here the ``action`` function will move the vehicle: .. code-block:: python def move_vehicle(t, fig): x_displacement = dt*v(t) fig['vehicle'].translate((x_displacement, 0)) Defining a set of time points for the frames in the animation and performing the animation is done by .. code-block:: python import numpy tp = numpy.linspace(0, 2*R, 25) dt = tp[1] - tp[0] # time step animate(fig, tp, move_vehicle, pause_per_frame=0.2) The ``pause_per_frame`` adds a pause, here 0.2 seconds, between each frame. We can also make a movie file of the animation: .. code-block:: python files = animate(fig, tp, move_vehicle, moviefiles=True, pause_per_frame=0.2) The ``files`` variable holds a string with the family of files constituting the frames in the movie, here ``'tmp_frame*.png'``. Making a movie out of the individual frames can be done in many ways. A simple approach is to make an animated GIF file with help of ``convert``, a program in the ImageMagick software suite: .. code-block:: console Terminal> convert -delay 20 tmp_frame*.png anim.gif Terminal> animate anim.gif # play movie The delay between frames governs the speed of the movie. The ``anim.gif`` file can be embedded in a web page and shown as a movie the page is loaded into a web browser (just insert ```` in the HTML code to play the GIF animation). The tool ``ffmpeg`` can alternatively be used, e.g., .. code-block:: console Terminal> ffmpeg -i "tmp_frame_%04d.png" -b 800k -r 25 \ -vcodec mpeg4 -y -qmin 2 -qmax 31 anim.mpeg An easy-to-use interface to movie-making tools is provided by the SciTools package: .. code-block:: python from scitools.std import movie # HTML page showing individual frames movie(files, encoder='html', output_file='anim.html') # Standard GIF file movie(files, encoder='convert', output_file='anim.gif') # AVI format movie('tmp_*.png', encoder='ffmpeg', fps=4, output_file='anim.avi') # requires ffmpeg package # MPEG format movie('tmp_*.png', encoder='ffmpeg', fps=4, output_file='anim2.mpeg', vodec='mpeg2video') # or movie(files, encoder='ppmtompeg', fps=24, output_file='anim.mpeg') # requires the netpbm package When difficulties with encoders and players arise, the simple web page for showing a movie, here ``anim.html`` (generated by the first ``movie`` command above), is a safe method that you always can rely on. You can try loading ``anim.html`` into a web browser, after first having run the present example in the file ``vehicle0.py` `_. Alternatively, you can view a ready-made `movie `_. .. _sketcher:vehicle1:anim: Animation: Rolling the Wheels ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ It is time to show rolling wheels. To this end, we make somewhat more complicated wheels with spokes as on a bicyle, formed by two crossing lines, see Figure :ref:`sketcher:fig:vehicle1`. The construction of the wheels will now involve a circle and two lines: .. code-block:: python wheel1 = Composition({ 'wheel': Circle(center=(w_1, R), radius=R), 'cross': Composition({'cross1': Line((w_1,0), (w_1,2*R)), 'cross2': Line((w_1-R,R), (w_1+R,R))})}) wheel2 = wheel1.copy() wheel2.translate((L,0)) Observe that ``wheel1.copy()`` copies all the objects that make up the first wheel, and ``wheel2.translate`` translates all the copied objects. .. _sketcher:fig:vehicle1: .. figure:: figs-sketcher/vehicle1.png :width: 400 *Wheels with spokes to show rotation* The ``move_vehicle`` function now needs to displace all the objects in the entire vehicle and also rotate the ``cross1`` and ``cross2`` objects in both wheels. The rotation angle follows from the fact that the arc length of a rolling wheel equals the displacement of the center of the wheel, leading to a rotation angle .. code-block:: python angle = - x_displacement/R With ``w_1`` tracking the :math:`x` coordinate of the center of the front wheel, we can rotate that wheel by .. code-block:: python w1 = fig['vehicle']['wheels']['wheel1'] from math import degrees w1.rotate(degrees(angle), center=(w_1, R)) The ``rotate`` function takes two parameters: the rotation angle (in degrees) and the center point of the rotation, which is the center of the wheel in this case. The other wheel is rotated by .. code-block:: python w2 = fig['vehicle']['wheels']['wheel2'] w2.rotate(degrees(angle), center=(w_1 + L, R)) That is, the angle is the same, but the rotation point is different. The update of the center point is done by ``w_1 += displacement[0]``. The complete ``move_vehicle`` function then becomes .. code-block:: python w_1 = w_1 + L # start position def move_vehicle(t, fig): x_displacement = dt*v(t) fig['vehicle'].translate((x_displacement, 0)) # Rotate wheels global w_1 w_1 += x_displacement # R*angle = -x_displacement angle = - x_displacement/R w1 = fig['vehicle']['wheels']['wheel1'] w1.rotate(degrees(angle), center=(w_1, R)) w2 = fig['vehicle']['wheels']['wheel2'] w2.rotate(degrees(angle), center=(w_1 + L, R)) The complete example is found in the file ``vehicle1.py` `_. You may run this file or watch a `ready-made movie `_. The advantages with making figures this way through programming, rather than using interactive drawing programs, are numerous. For example, the objects are parameterized by variables so that various dimensions can easily be changed. Subparts of the figure, possible involving a lot of figure objects, can change color, linetype, filling or other properties through a *single* function call. Subparts of the figure can be rotated, translated, or scaled. Subparts of the figure can also be copied and moved to other parts of the drawing area. However, the single most important feature is probably the ability to make animations governed by mathematical formulas or data coming from physics simulations of the problem sketched in the drawing, as very simplistically shown in the example above. Inner Workings of the Pysketcher Tool ===================================== We shall now explain how we can, quite easily, realize software with the capabilities demonstrated in the previous examples. Each object in the figure is represented as a class in a class hierarchy. Using inheritance, classes can inherit properties from parent classes and add new geometric features. Class programming is a key technology for realizing Pysketcher. As soon as some classes are established, more are easily added. Enhanced functionality for all the classes is also easy to implement in common, generic code that can immediately be shared by all present and future classes. The fundamental data structure involved in the ``pysketcher`` package is a hierarchical tree, and much of the material on implementation issues targets how to traverse tree structures with recursive function calls in object hierarchies. This topic is of key relevance in a wide range of other applications as well. In total, the inner workings of Pysketcher constitute an excellent example on the power of class programming. Example of Classes for Geometric Objects ---------------------------------------- We introduce class ``Shape`` as superclass for all specialized objects in a figure. This class does not store any data, but provides a series of functions that add functionality to all the subclasses. This will be shown later. Simple Geometric Objects ~~~~~~~~~~~~~~~~~~~~~~~~ One simple subclass is ``Rectangle``: .. code-block:: python class Rectangle(Shape): def __init__(self, lower_left_corner, width, height): p = lower_left_corner # short form x = [p[0], p[0] + width, p[0] + width, p[0], p[0]] y = [p[1], p[1], p[1] + height, p[1] + height, p[1]] self.shapes = {'rectangle': Curve(x,y)} Any subclass of ``Shape`` will have a constructor which takes geometric information about the shape of the object and creates a dictionary ``self.shapes`` with the shape built of simpler shapes. The most fundamental shape is ``Curve``, which is just a collection of :math:`(x,y)` coordinates in two arrays ``x`` and ``y``. Drawing the ``Curve`` object is a matter of plotting ``y`` versus ``x``. The ``Rectangle`` class illustrates how the constructor takes information about the lower left corner, the width and the height, and creates coordinate arrays ``x`` and ``y`` consisting of the four corners, plus the first one repeated such that plotting ``x`` and ``y`` will form a closed four-sided rectangle. This construction procedure demands that the rectangle will always be aligned with the :math:`x` and :math:`y` axis. However, we may easily rotate the rectangle about any point once the object is constructed. Class ``Line`` constitutes a similar example: .. code-block:: python class Line(Shape): def __init__(self, start, end): x = [start[0], end[0]] y = [start[1], end[1]] self.shapes = {'line': Curve(x, y)} Here we only need two points, the start and end point on the line. However, we may add some useful functionality, e.g., the ability to give an :math:`x` coordinate and have the class calculate the corresponding :math:`y` coordinate: .. code-block:: python def __call__(self, x): """Given x, return y on the line.""" x, y = self.shapes['line'].x, self.shapes['line'].y self.a = (y[1] - y[0])/(x[1] - x[0]) self.b = y[0] - self.a*x[0] return self.a*x + self.b Unfortunately, this is too simplistic because vertical lines cannot be handled (infinite ``self.a``). The source code of ``Line`` therefore provides a more general solution at the cost of significantly longer code with more tests. A circle gives us somewhat increased complexity. Again we represent the geometric object by a ``Curve`` object, but this time the ``Curve`` object needs to store a large number of points on the curve such that a plotting program produces a visually smooth curve. The points on the circle must be calculated manually in the constructor of class ``Circle``. The formulas for points :math:`(x,y)` on a curve with radius :math:`R` and center at :math:`(x_0, y_0)` are given by .. math:: x &= x_0 + R\cos (t),\\ y &= y_0 + R\sin (t), where :math:`t\in [0, 2\pi]`. A discrete set of :math:`t` values in this interval gives the corresponding set of :math:`(x,y)` coordinates on the circle. The user must specify the resolution, i.e., the number of :math:`t` values, or equivalently, points on the circle. The circle's radius and center must of course also be specified. We can write the ``Circle`` class as .. code-block:: python class Circle(Shape): def __init__(self, center, radius, resolution=180): self.center, self.radius = center, radius self.resolution = resolution t = linspace(0, 2*pi, resolution+1) x0 = center[0]; y0 = center[1] R = radius x = x0 + R*cos(t) y = y0 + R*sin(t) self.shapes = {'circle': Curve(x, y)} As in class ``Line`` we can offer the possibility to give an angle :math:`\theta` (equivalent to :math:`t` in the formulas above) and then get the corresponding :math:`x` and :math:`y` coordinates: .. code-block:: python def __call__(self, theta): """Return (x, y) point corresponding to angle theta.""" return self.center[0] + self.radius*cos(theta), \ self.center[1] + self.radius*sin(theta) There is one flaw with this method: it yields illegal values after a translation, scaling, or rotation of the circle. A part of a circle, an arc, is a frequent geometric object when drawing mechanical systems. The arc is constructed much like a circle, but :math:`t` runs in :math:`[\theta_0, \theta_1]`. Giving :math:`\theta_1` and :math:`\theta_2` the slightly more descriptive names ``start_angle`` and ``arc_angle``, the code looks like this: .. code-block:: python class Arc(Shape): def __init__(self, center, radius, start_angle, arc_angle, resolution=180): self.center = center self.radius = radius self.start_angle = start_angle*pi/180 # radians self.arc_angle = arc_angle*pi/180 self.resolution = resolution t = linspace(self.start_angle, self.start_angle + self.arc_angle, resolution+1) x0 = center[0]; y0 = center[1] R = radius x = x0 + R*cos(t) y = y0 + R*sin(t) self.shapes = {'arc': Curve(x, y)} Having the ``Arc`` class, a ``Circle`` can alternatively befined as a subclass specializing the arc to a circle: .. code-block:: python class Circle(Arc): def __init__(self, center, radius, resolution=180): Arc.__init__(self, center, radius, 0, 360, resolution) A wall is about drawing a curve, displacing the curve vertically by some thickness, and then filling the space between the curves by some pattern. The input is the ``x`` and ``y`` coordinate arrays of the curve and a thickness parameter. The computed coordinates will be a polygon: going along the originally curve and then back again along the vertically displaced curve. The relevant code becomes .. code-block:: python class CurveWall(Shape): def __init__(self, x, y, thickness): # User's curve x1 = asarray(x, float) y1 = asarray(y, float) # Displaced curve (according to thickness) x2 = x1 y2 = y1 + thickness # Combine x1,y1 with x2,y2 reversed from numpy import concatenate x = concatenate((x1, x2[-1::-1])) y = concatenate((y1, y2[-1::-1])) wall = Curve(x, y) wall.set_filled_curves(color='white', pattern='/') self.shapes = {'wall': wall} Class Curve ~~~~~~~~~~~ Class ``Curve`` sits on the coordinates to be drawn, but how is that done? The constructor just stores the coordinates, while a method ``draw`` sends the coordinates to the plotting program to make a graph. Or more precisely, to avoid a lot of (e.g.) Matplotlib-specific plotting commands we have created a small layer with a simple programming interface to plotting programs. This makes it straightforward to change from Matplotlib to another plotting program. The programming interface is represented by the ``drawing_tool`` object and has a few functions: * ``plot_curve`` for sending a curve in terms of :math:`x` and :math:`y` coordinates to the plotting program, * ``set_coordinate_system`` for specifying the graphics area, * ``erase`` for deleting all elements of the graph, * ``set_grid`` for turning on a grid (convenient while constructing the plot), * ``set_instruction_file`` for creating a separate file with all plotting commands (Matplotlib commands in our case), * a series of ``set_X`` functions where ``X`` is some property like ``linecolor``, ``linestyle``, ``linewidth``, ``filled_curves``. This is basically all we need to communicate to a plotting program. Any class in the ``Shape`` hierarchy inherits ``set_X`` functions for setting properties of curves. This information is propagated to all other shape objects that make up the figure. Class ``Curve`` stores the line properties together with the coordinates of its curve and propagates this information to the plotting program. When saying ``vehicle.set_linewidth(10)``, all objects that make up the ``vehicle`` object will get a ``set_linewidth(10)`` call, but only the ``Curve`` object at the end of the chain will actually store the information and send it to the plotting program. A rough sketch of class ``Curve`` reads .. code-block:: python class Curve(Shape): """General curve as a sequence of (x,y) coordintes.""" def __init__(self, x, y): self.x = asarray(x, dtype=float) self.y = asarray(y, dtype=float) self.linestyle = None self.linewidth = None self.linecolor = None self.fillcolor = None self.fillpattern = None self.arrow = None def draw(self): drawing_tool.plot_curve( self.x, self.y, self.linestyle, self.linewidth, self.linecolor, self.arrow, self.fillcolor, self.fillpattern) def set_linewidth(self, width): self.linewidth = width det set_linestyle(self, style): self.linestyle = style ... Compound Geometric Objects ~~~~~~~~~~~~~~~~~~~~~~~~~~ The simple classes ``Line``, ``Arc``, and ``Circle`` could define the geometric shape through just one ``Curve`` object. More complicated figure elements are built from instances of various subclasses of ``Shape``. Classes used for professional drawings soon get quite complex in composition and have a lot of geometric details, so here we prefer to make a very simple composition: the already drawn vehicle from Figure ref:ref:`sketcher:fig:vehicle0`. That is, instead of composing the drawing in a Python code we make a class ``Vehicle0`` for doing the same thing, and derive it from ``Shape``. The ``Shape`` hierarchy is found in the ``pysketcher`` package, so to use these classes or derive a new one, we need to import ``pysketcher``. The constructor of class ``Vehicle0`` performs approximately the same statements as in the example program we developed for making the drawing in Figure ref:ref:`sketcher:fig:vehicle0`. .. code-block:: python class Vehicle0(Shape): def __init__(self, w_1, R, L, H): wheel1 = Circle(center=(w_1, R), radius=R) wheel2 = wheel1.copy() wheel2.translate((L,0)) under = Rectangle(lower_left_corner=(w_1-2*R, 2*R), width=2*R + L + 2*R, height=H) over = Rectangle(lower_left_corner=(w_1, 2*R + H), width=2.5*R, height=1.25*H) wheels = Composition( {'wheel1': wheel1, 'wheel2': wheel2}) body = Composition( {'under': under, 'over': over}) vehicle = Composition({'wheels': wheels, 'body': body}) xmax = w_1 + 2*L + 3*R ground = Wall(x=[R, xmax], y=[0, 0], thickness=-0.3*R) self.shapes = {'vehicle': vehicle, 'ground': ground} Any subclass of ``Shape`` *must* define the ``shapes`` attribute, otherwise the inherited ``draw`` method (and a lot of other methods too) will not work. The painting of the vehicle could be offered by a method: .. code-block:: python def colorful(self): wheels = self.shapes['vehicle']['wheels'] wheels.set_filled_curves('blue') wheels.set_linewidth(6) wheels.set_linecolor('black') under = self.shapes['vehicle']['body']['under'] under.set_filled_curves('red') over = self.shapes['vehicle']['body']['over'] over.set_filled_curves(pattern='/') over.set_linewidth(14) The usage of the class is simple: after having set up an appropriate coordinate system a s previously shown, we can do .. code-block:: python vehicle = Vehicle0(w_1, R, L, H) vehicle.draw() drawing_tool.display() The color from Figure :ref:`sketcher:fig:vehicle0:v2` is realized by .. code-block:: python drawing_tool.erase() vehicle.colorful() vehicle.draw() drawing_tool.display() A complete code defining and using class ``Vehicle0`` is found in the file ``vehicle2.py` `_. The ``pysketcher`` package contains a wide range of classes for various geometrical objects, particularly those that are frequently used in drawings of mechanical systems. Adding Functionality via Recursion ================================== The really powerful feature of our class hierarchy is that we can add much functionality to the superclass ``Shape`` and to the "bottom" classes ``Curve``, and all other classes for all types of geometrical shapes immediately get the new functionality. To explain the idea we first have to look at the ``draw`` method, which all classes in the ``Shape`` hierarchy must have. The inner workings of the ``draw`` method explain the secrets of how a series of other useful operations on figures can be implemented. Basic Principles of Recursion ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ We work with two types of class hierarchies: one Python class hierarchy, with ``Shape`` as superclass, and one *object hierarchy* of figure elements in a specific figure. A subclass of ``Shape`` stores its figure in the ``self.shapes`` dictionary. This dictionary represents the object hierarchy of figure elements for that class. We want to make one ``draw`` call for an instance, say our class ``Vehicle0``, and then we want this call to be propagated to *all* objects that are contained in ``self.shapes`` and all is nested subdictionaries. How is this done? The natural starting point is to call ``draw`` for each ``Shape`` object in the ``self.shapes`` dictionary: .. code-block:: python def draw(self): for shape in self.shapes: self.shapes[shape].draw() This general method can be provided by class ``Shape`` and inherited in subclasses like ``Vehicle0``. Let ``v`` be a ``Vehicle0`` instance. Seemingly, a call ``v.draw()`` just calls .. code-block:: python v.shapes['vehicle'].draw() v.shapes['ground'].draw() However, in the former call we call the ``draw`` method of a ``Composition`` object whose ``self.shapes`` attributed has two elements: ``wheels`` and ``body``. Since class ``Composition`` inherits the same ``draw`` method, this method will run through ``self.shapes`` and call ``wheels.draw()`` and ``body.draw()``. Now, the ``wheels`` object is also a ``Composition`` with the same ``draw`` method, which will run through the ``shapes`` dictionary, now containing the ``wheel1`` and and ``wheel2`` objects. The ``wheel1`` object is a ``Circle``, so calling ``wheel1.draw()`` calls the ``draw`` method in class ``Circle``, but this is the same ``draw`` method as shown above. This method will therefore traverse the circle's ``shapes`` dictionary, which we have seen consists of one ``Curve`` element. The ``Curve`` object holds the coordinates to be plotted so here ``draw`` really needs to do something "physical", namely send the coordinates to the plotting program. The ``draw`` method is outlined in the short listing of class ``Curve`` shown previously. We can go to any of the other shape objects that appear in the figure hierarchy and follow their ``draw`` calls in the similar way. Every time, a ``draw`` call will invoke a new ``draw`` call, until we eventually hit a ``Curve`` object in the "botton" of the figure hierarchy, and then that part of the figure is really plotted (or more precisely, the coordinates are sent to a plotting program). When a method calls itself, such as ``draw`` does, the calls are known as *recursive* and the programming principle is referred to as *recursion*. This technique is very often used to traverse hierarchical structures like the figure structures we work with here. Even though the hierarchy of objects building up a figure are of different types, they all inherit the same ``draw`` method and therefore exhibit the same behavior with respect to drawing. Only the ``Curve`` object has a different ``draw`` method, which does not lead to more recursion. Without this different ``draw`` method in class ``Curve``, the repeated ``draw`` calls would go on forever. Explaining Recursion ~~~~~~~~~~~~~~~~~~~~ Understanding recursion is usually a challenge. To get a better idea of how recursion works, we have equipped class ``Shape`` with a method ``recurse`` which just visits all the objects in the ``shapes`` dictionary and prints out a message for each object. This feature allows us to trace the execution and see exactly where we are in the hierarchy and which objects that are visited. The ``recurse`` method is very similar to ``draw``: .. code-block:: python def recurse(self, name, indent=0): # print message where we are (name is where we come from) for shape in self.shapes: # print message about which object to visit self.shapes[shape].recurse(indent+2, shape) The ``indent`` parameter governs how much the message from this ``recurse`` method is intended. We increase ``indent`` by 2 for every level in the hierarchy. This makes it easy to see on the printout how far down in the hierarchy we are. A typical message written by ``recurse`` when ``name`` is ``body`` and the ``shapes`` dictionary contains two entries, ``over`` and ``under``, will be .. code-block:: python Composition: body.shapes has entries 'over', 'under' call body.shapes["over"].recurse("over", 6) The number of leading blanks on each line corresponds to the value of ``indent``. The code printing out such messages looks like .. code-block:: python def recurse(self, name, indent=0): space = ' '*indent print space, '%s: %s.shapes has entries' % \ (self.__class__.__name__, name), \ str(list(self.shapes.keys()))[1:-1] for shape in self.shapes: print space, print 'call %s.shapes["%s"].recurse("%s", %d)' % \ (name, shape, shape, indent+2) self.shapes[shape].recurse(shape, indent+2) Let us follow a ``v.recurse('vehicle')`` call in detail, ``v`` being a ``Vehicle0`` instance. Before looking into the output from ``recurse``, let us get an overview of the figure hierarchy in the ``v`` object (as produced by ``print v``) .. code-block:: python ground wall vehicle body over rectangle under rectangle wheels wheel1 arc wheel2 arc The ``recurse`` method performs the same kind of traversal of the hierarchy, but writes out and explains a lot more. The data structure represented by ``v.shapes`` is known as a *tree*. As in physical trees, there is a *root*, here the ``v.shapes`` dictionary. A graphical illustration of the tree (upside down) is shown in Figure :ref:`sketcher:fig:Vehicle0:hier2`. From the root there are one or more branches, here two: ``ground`` and ``vehicle``. Following the ``vehicle`` branch, it has two new branches, ``body`` and ``wheels``. Relationships as in family trees are often used to describe the relations in object trees too: we say that ``vehicle`` is the parent of ``body`` and that ``body`` is a child of ``vehicle``. The term *node* is also often used to describe an element in a tree. A node may have several other nodes as *descendants*. .. _sketcher:fig:Vehicle0:hier2: .. figure:: figs-sketcher/Vehicle0_hier2.png :width: 600 *Hierarchy of figure elements in an instance of class `Vehicle0`* Recursion is the principal programming technique to traverse tree structures. Any object in the tree can be viewed as a root of a subtree. For example, ``wheels`` is the root of a subtree that branches into ``wheel1`` and ``wheel2``. So when processing an object in the tree, we imagine we process the root and then recurse into a subtree, but the first object we recurse into can be viewed as the root of the subtree, so the processing procedure of the parent object can be repeated. A recommended next step is to simulate the ``recurse`` method by hand and carefully check that what happens in the visits to ``recurse`` is consistent with the output listed below. Although tedious, this is a major exercise that guaranteed will help to demystify recursion. Also remember that it requires some efforts to understand recursion. A part of the printout of ``v.recurse('vehicle')`` looks like .. code-block:: python Vehicle0: vehicle.shapes has entries 'ground', 'vehicle' call vehicle.shapes["ground"].recurse("ground", 2) Wall: ground.shapes has entries 'wall' call ground.shapes["wall"].recurse("wall", 4) reached "bottom" object Curve call vehicle.shapes["vehicle"].recurse("vehicle", 2) Composition: vehicle.shapes has entries 'body', 'wheels' call vehicle.shapes["body"].recurse("body", 4) Composition: body.shapes has entries 'over', 'under' call body.shapes["over"].recurse("over", 6) Rectangle: over.shapes has entries 'rectangle' call over.shapes["rectangle"].recurse("rectangle", 8) reached "bottom" object Curve call body.shapes["under"].recurse("under", 6) Rectangle: under.shapes has entries 'rectangle' call under.shapes["rectangle"].recurse("rectangle", 8) reached "bottom" object Curve ... This example should clearly demonstrate the principle that we can start at any object in the tree and do a recursive set of calls with that object as root. .. _sketcher:scaling: Scaling, Translating, and Rotating a Figure ------------------------------------------- With recursion, as explained in the previous section, we can within minutes equip *all* classes in the ``Shape`` hierarchy, both present and future ones, with the ability to scale the figure, translate it, or rotate it. This added functionality requires only a few lines of code. Scaling ~~~~~~~ We start with the simplest of the three geometric transformations, namely scaling. For a ``Curve`` instance containing a set of :math:`n` coordinates :math:`(x_i,y_i)` that make up a curve, scaling by a factor :math:`a` means that we multiply all the :math:`x` and :math:`y` coordinates by :math:`a`: .. math:: x_i \leftarrow ax_i,\quad y_i\leftarrow ay_i, \quad i=0,\ldots,n-1\thinspace . Here we apply the arrow as an assignment operator. The corresponding Python implementation in class ``Curve`` reads .. code-block:: python class Curve: ... def scale(self, factor): self.x = factor*self.x self.y = factor*self.y Note here that ``self.x`` and ``self.y`` are Numerical Python arrays, so that multiplication by a scalar number ``factor`` is a vectorized operation. An even more efficient implementation is to make use of in-place multiplication in the arrays, as this saves the creation of temporary arrays like ``factor*self.x``, which is then assigned to ``self.x``: .. code-block:: python class Curve: ... def scale(self, factor): self.x *= factor self.y *= factor In an instance of a subclass of ``Shape``, the meaning of a method ``scale`` is to run through all objects in the dictionary ``shapes`` and ask each object to scale itself. This is the same delegation of actions to subclass instances as we do in the ``draw`` (or ``recurse``) method. All all objects, except ``Curve`` instances, can share the same implementation of the ``scale`` method. Therefore, we place the ``scale`` method in the superclass ``Shape`` such that all subclasses inherit the method. Since ``scale`` and ``draw`` are so similar, we can easily implement the ``scale`` method in class ``Shape`` by copying and editing the ``draw`` method: .. code-block:: python class Shape: ... def scale(self, factor): for shape in self.shapes: self.shapes[shape].scale(factor) This is all we have to do in order to equip all subclasses of ``Shape`` with scaling functionality! Any piece of the figure will scale itself, in the same manner as it can draw itself. Translation ~~~~~~~~~~~ A set of coordinates :math:`(x_i, y_i)` can be translated :math:`v_0` units in the :math:`x` direction and :math:`v_1` units in the :math:`y` direction using the formulas .. math:: x_i\leftarrow x_i+v_0,\quad y_i\leftarrow y_i+v_1,\quad i=0,\ldots,n-1\thinspace . The natural specification of the translation is in terms of a vector :math:`v=(v_0,v_1)`. The corresponding Python implementation in class ``Curve`` becomes .. code-block:: python class Curve: ... def translate(self, v): self.x += v[0] self.y += v[1] The translation operation for a shape object is very similar to the scaling and drawing operations. This means that we can implement a common method ``translate`` in the superclass ``Shape``. The code is parallel to the ``scale`` method: .. code-block:: python class Shape: .... def translate(self, v): for shape in self.shapes: self.shapes[shape].translate(v) Rotation ~~~~~~~~ Rotating a figure is more complicated than scaling and translating. A counter clockwise rotation of :math:`\theta` degrees for a set of coordinates :math:`(x_i,y_i)` is given by .. math:: \bar x_i &\leftarrow x_i\cos\theta - y_i\sin\theta,\\ \bar y_i &\leftarrow x_i\sin\theta + y_i\cos\theta\thinspace . This rotation is performed around the origin. If we want the figure to be rotated with respect to a general point :math:`(x,y)`, we need to extend the formulas above: .. math:: \bar x_i &\leftarrow x + (x_i -x)\cos\theta - (y_i -y)\sin\theta,\\ \bar y_i &\leftarrow y + (x_i -x)\sin\theta + (y_i -y)\cos\theta\thinspace . The Python implementation in class ``Curve``, assuming that :math:`\theta` is given in degrees and not in radians, becomes .. code-block:: python def rotate(self, angle, center): angle = radians(angle) x, y = center c = cos(angle); s = sin(angle) xnew = x + (self.x - x)*c - (self.y - y)*s ynew = y + (self.x - x)*s + (self.y - y)*c self.x = xnew self.y = ynew The ``rotate`` method in class ``Shape`` is identical to the ``draw``, ``scale``, and ``translate`` methods except that we recurse into ``self.rotate(angle, center)``. We have already seen the ``rotate`` method in action when animating the rolling wheel at the end of the section :ref:`sketcher:vehicle1:anim`.