jupytersketcher/doc/pub/tutorial/html/_sources/main_sketcher.txt

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.. Document title:

Pysketcher: Create Principal Sketches of Physics Problems
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

:Authors: Hans Petter Langtangen
:Date: Jan 22, 2016


.. The below box could be typeset as .. admonition: Attention
   but we have decided not to do so since the admon needs a title
   (the box formatting is therefore just ignored)

This document is derived from Chapter 9 in the book
`A Primer on Scientific Programming with Python <http://www.amazon.com/Scientific-Programming-Computational-Science-Engineering/dp/3642549586/ref=sr_1_2?s=books&ie=UTF8&qid=1407225588&sr=1-2&keywords=langtangen>`__, by H. P. Langtangen,
4th edition, Springer, 2014.


*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...

.. !split

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 labels (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.
Programming as a superior alternative to interactive drawing is
the mantra of this section.

.. _sketcher:fig:inclinedplane:

.. figure:: wheel_on_inclined_plane.png
   :width: 400

   *Sketch of a physics problem*

Basic construction of sketches
------------------------------

Before attacking real-life sketches as in Figure :ref:`sketcher:fig:inclinedplane`
we focus on the significantly simpler drawing shown
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:: vehicle0_dim.png
   :width: 600

   *Sketch of a simple figure*

When the sketch is defined in terms of computer code, it is natural to
parameterize geometric features, such as the radius of the wheel (:math:`R`),
the center point of the left wheel (:math:`w_1`), as well as the height (:math:`H`) and
length (:math:`L`) of the main part. The simple vehicle in
Figure :ref:`sketcher:fig:vehicle0` is quickly drawn in almost any interactive
tool. However, if we want to change the radius of the wheels, you need a
sophisticated drawing tool to avoid redrawing the whole figure, while
in computer code this is a matter of changing the :math:`R` parameter and
rerunning the program.
For example, Figure :ref:`sketcher:fig:vehicle0b` shows
a variation of the drawing in
Figure :ref:`sketcher:fig:vehicle0` obtained by just setting
:math:`R=0.5`, :math:`L=5`, :math:`H=2`, and :math:`R=2`. Being able
to quickly change geometric sizes is key to many problem settings in
physics and engineering, but then a program must define the geometry.

.. _sketcher:fig:vehicle0b:

.. figure:: vehicle_v2.png
   :width: 500

   *Redrawing a figure with other geometric parameters*

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:

.. code-block:: python

        from pysketcher import *
        
        drawing_tool.set_coordinate_system(
            xmin=0, xmax=10, ymin=-1, ymax=8)

Instead of working with lengths expressed by specific numbers it is
highly recommended to use variables to parameterize lengths as
this makes it easier to change dimensions later.
Here we introduce some key lengths for the radius of the wheels,
distance between the wheels, etc.:

.. code-block:: python

        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)

With the drawing area in place we can make the first ``Circle`` object
in an intuitive fashion:

.. code-block:: python

        wheel1 = Circle(center=(w_1, R), radius=R)

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 to the right according to a
displacement vector :math:`(L,0)`:

.. code-block:: python

        wheel2 = wheel1.copy()
        wheel2.translate((L,0))

The two rectangles are also 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 code 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 forth in the hierarchical tree structure of
figure elements,
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

        wheels.set_linestyle('solid')
        wheels.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:: python

        # 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 making up the figure
is hierarchical, similar to a family, where
each object has a parent and a number of children. Do a
``print fig`` to display the relations:

.. code-block:: text

        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``

In this listing there are also objects not defined by the
programmer: ``rectangle`` and ``arc``. These are of type ``Curve``
and automatically generated by the classes ``Rectangle`` and ``Circle``.

More detailed information can be printed by

.. code-block:: python

        print fig.show_hierarchy('std')

yielding the output

.. code-block:: text

        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:: text

        Terminal> dot -Tpng -o fig.png fig.dot

.. _sketcher:fig:vehicle0:hier1:

.. figure:: 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:: 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 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:: 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 drawing 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.

A figure that moves in time is conveniently realized by the
function ``animate``:

.. code-block:: python

        animate(fig, tp, action)

Here, ``fig`` is the entire figure, ``tp`` is an array of
time points, and ``action`` is a user-specified function that changes
``fig`` at a specific time point. Typically, ``action`` will move
parts of ``fig``.

In the present case we can define the movement through a velocity
function ``v(t)`` and displace the figure ``v(t)*dt`` for small time
intervals ``dt``. A possible velocity function is

.. code-block:: python

        def v(t):
            return -8*R*t*(1 - t/(2*R))
        

Our action function for horizontal displacements ``v(t)*dt`` becomes

.. code-block:: python

        def move(t, fig):
            x_displacement = dt*v(t)
            fig['vehicle'].translate((x_displacement, 0))

Since our velocity is negative for :math:`t\in [0,2R]` the displacement is
to the left.

The ``animate`` function will for each time point ``t`` in ``tp`` erase
the drawing, call ``action(t, fig)``, and show the new figure by
``fig.draw()`` and ``drawing_tool.display()``.
Here we choose a resolution of the animation corresponding to
25 time points in the time interval :math:`[0,2R]`:

.. code-block:: python

        import numpy
        tp = numpy.linspace(0, 2*R, 25)
        dt = tp[1] - tp[0]  # time step
        
        animate(fig, tp, move, pause_per_frame=0.2)

The ``pause_per_frame`` adds a pause, here 0.2 seconds, between
each frame in the animation.

We can also ask ``animate`` to store each frame in a file:

.. code-block:: python

        files = animate(fig, tp, move_vehicle, moviefiles=True,
                        pause_per_frame=0.2)

The ``files`` variable, here ``'tmp_frame_%04d.png'``,
is the printf-specification used to generate the individual
plot files. We can use this specification to make a video
file via ``ffmpeg`` (or ``avconv`` on Debian-based Linux systems such
as Ubuntu). Videos in the Flash and WebM formats can be created
by

.. code-block:: text

        Terminal> ffmpeg -r 12 -i tmp_frame_%04d.png -vcodec flv mov.flv
        Terminal> ffmpeg -r 12 -i tmp_frame_%04d.png -vcodec libvpx mov.webm

An animated GIF movie can also be made using the ``convert`` program
from the ImageMagick software suite:

.. code-block:: text

        Terminal> convert -delay 20 tmp_frame*.png mov.gif
        Terminal> animate mov.gif  # play movie

The delay between frames, in units of 1/100 s,
governs the speed of the movie.
To play the animated GIF file in a web page, simply insert
``<img src="mov.gif">`` in the HTML code.

The individual PNG frames can be directly played in a web
browser by running

.. code-block:: text

        Terminal> scitools movie output_file=mov.html fps=5 tmp_frame*

or calling

.. code-block:: python

        from scitools.std import movie
        movie(files, encoder='html', output_file='mov.html')

in Python. Load the resulting file ``mov.html`` into a web browser
to play the movie.

Try to run `vehicle0.py <http://tinyurl.com/ot733jn/vehicle0.py>`__ and
then load ``mov.html`` into a browser, or play one of the ``mov.*``
video files.  Alternatively, you can view a ready-made `movie <http://tinyurl.com/oou9lp7/mov-tut/vehicle0.html>`__.

.. _sketcher:vehicle1:anim:

Animation: rolling the wheels
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

It is time to show rolling wheels. To this end, we add spokes to the
wheels, 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:: vehicle1.png
   :width: 400

   *Wheels with spokes to illustrate rolling*

The ``move`` 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 += x_displacement``.
The complete ``move`` function with translation of the entire
vehicle and rotation of the wheels then becomes

.. code-block:: python

        w_1 = w_1 + L   # start position
        
        def move(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 <http://tinyurl.com/ot733jn/vehicle1.py>`__. You may run this file or watch a `ready-made movie <http://tinyurl.com/oou9lp7/mov-tut/vehicle1.html>`__.

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, as shown in the example above.

.. !split

.. _sketcher:ex:pendulum:

A simple pendulum
=================

.. _sketcher:ex:pendulum:basic:

The basic physics sketch
------------------------

We now want to make a sketch of simple pendulum from physics, as shown
in Figure :ref:`sketcher:ex:pendulum:fig1`. A body with mass :math:`m` is attached
to a massless, stiff rod, which can rotate about a point, causing the
pendulum to oscillate.

A suggested work flow is to
first sketch the figure on a piece of paper and introduce a coordinate
system. A simple coordinate system is indicated in Figure
:ref:`sketcher:ex:pendulum:fig1wgrid`. In a code we introduce variables
``W`` and ``H`` for the width and height of the figure (i.e., extent of
the coordinate system) and open the program like this:

.. code-block:: python

        from pysketcher import *
        
        H = 7.
        W = 6.
        
        drawing_tool.set_coordinate_system(xmin=0, xmax=W,
                                           ymin=0, ymax=H,
                                           axis=True)
        drawing_tool.set_grid(True)
        drawing_tool.set_linecolor('blue')

Note that when the sketch is ready for "production", we will (normally)
set ``axis=False`` to remove the coordinate system and also remove the
grid, i.e., delete or
comment out the line ``drawing_tool.set_grid(True)``.
Also note that we in this example let all lines be blue by default.

.. _sketcher:ex:pendulum:fig1:

.. figure:: pendulum1.png
   :width: 300

   *Sketch of a simple pendulum*

.. _sketcher:ex:pendulum:fig1wgrid:

.. figure:: pendulum1_wgrid.png
   :width: 400

   *Sketch with assisting coordinate system*

The next step is to introduce variables for key quantities in the sketch.
Let ``L`` be the length of the pendulum, ``P`` the rotation point, and let
``a`` be the angle the pendulum makes with the vertical (measured in degrees).
We may set

.. code-block:: python

        L = 5*H/7          # length
        P = (W/6, 0.85*H)  # rotation point
        a = 40             # angle

Be careful with integer division if you use Python 2! Fortunately, we
started out with ``float`` objects for ``W`` and ``H`` so the expressions above
are safe.

What kind of objects do we need in this sketch? Looking at
Figure :ref:`sketcher:ex:pendulum:fig1` we see that we need

1. a vertical, dashed line

2. an arc with no text but dashed line to indicate the *path* of the
   mass

3. an arc with name :math:`\theta` to indicate the *angle*

4. a line, here called *rod*, from the rotation point to the mass

5. a blue, filled circle representing the *mass*

6. a text :math:`m` associated with the mass

7. an indicator of the pendulum's *length* :math:`L`, visualized as
   a line with two arrows tips and the text :math:`L`

8. a gravity vector with the text :math:`g`

Pysketcher has objects for each of these elements in our sketch.
We start with the simplest element: the vertical line, going from
``P`` to ``P`` minus the length :math:`L` in :math:`y` direction:

.. code-block:: python

        vertical = Line(P, P-point(0,L))

The class ``point`` is very convenient: it turns its two coordinates into
a vector with which we can compute, and is therefore one of the most
widely used Pysketcher objects.

The path of the mass is an arc that can be made by
Pysketcher's ``Arc`` object:

.. code-block:: python

        path = Arc(P, L, -90, a)

The first argument ``P`` is the center point, the second is the
radius (``L`` here), the next argument is the start angle, here
it starts at -90 degrees, while the next argument is the angle of
the arc, here ``a``.
For the path of the mass, we also need an arc object, but this
time with an associated text. Pysketcher has a specialized object
for this purpose, ``Arc_wText``, since placing the text manually can
be somewhat cumbersome.

.. code-block:: python

        angle = Arc_wText(r'$\theta$', P, L/4, -90, a, text_spacing=1/30.)

The arguments are as for ``Arc`` above, but the first one is the desired
text. Remember to use a raw string since we want a LaTeX greek letter
that contains a backslash.
The ``text_spacing`` argument must often be tweaked. It is recommended
to create only a few objects before rendering the sketch and then
adjust spacings as one goes along.

The rod is simply a line from ``P`` to the mass. We can easily
compute the position of the mass from basic geometry considerations,
but it is easier and safer to look up this point in other objects
if it is already computed. In the present case,
the ``path`` object stored its start and
end points, so ``path.geometric_features()['end']`` is the end point
of the path, which is the position of the mass. We can therefore
create the rod simply as a line from ``P`` to this already computed end point:

.. code-block:: python

        mass_pt = path.geometric_features()['end']
        rod = Line(P, mass_pt)

The mass is a circle filled with color:

.. code-block:: python

        mass = Circle(center=mass_pt, radius=L/20.)
        mass.set_filled_curves(color='blue')

To place the :math:`m` correctly, we go a small distance in the direction of
the rod, from the center of the circle. To this end, we need to
compute the direction. This is easiest done by computing a vector
from ``P`` to the center of the circle and calling ``unit_vec`` to make
a unit vector in this direction:

.. code-block:: python

        rod_vec = rod.geometric_features()['end'] - \ 
                  rod.geometric_features()['start']
        unit_rod_vec = unit_vec(rod_vec)
        mass_symbol = Text('$m$', mass_pt + L/10*unit_rod_vec)

Again, the distance ``L/10`` is something one has to experiment with.

The next object is the length measure with the text :math:`L`. Such length
measures are represented by Pysketcher's ``Distance_wText`` object.
An easy construction is to first place this length measure along the
rod and then translate it a little distance (``L/15``) in the
normal direction of the rod:

.. code-block:: python

        length = Distance_wText(P, mass_pt, '$L$')
        length.translate(L/15*point(cos(radians(a)), sin(radians(a))))

For this translation we need a unit vector in the normal direction
of the rod, which is from geometric considerations given by
:math:`(\cos a, \sin a)`, when :math:`a` is the angle of the pendulum.
Alternatively, we could have found the normal vector as a vector that
is normal to ``unit_rod_vec``: ``point(-unit_rod_vec[1],unit_rod_vec[0])``.

The final object is the gravity force vector, which is so common
in physics sketches that Pysketcher has a ready-made object: ``Gravity``,

.. code-block:: python

        gravity = Gravity(start=P+point(0.8*L,0), length=L/3)

Since blue is the default color for
lines, we want the dashed lines (for ``vertical`` and ``path``) to be black
and with linewidth 1. These properties can be set one by one for each
object, but we can also make a little helper function:

.. code-block:: python

        def set_dashed_thin_blackline(*objects):
            """Set linestyle of objects to dashed, black, width=1."""
            for obj in objects:
                obj.set_linestyle('dashed')
                obj.set_linecolor('black')
                obj.set_linewidth(1)
        
        set_dashed_thin_blackline(vertical, path)

Now, all objects are in place, so it remains to compose the final
figure and draw the composition:

.. code-block:: python

        fig = Composition(
            {'body': mass, 'rod': rod,
             'vertical': vertical, 'theta': angle, 'path': path,
             'g': gravity, 'L': length, 'm': mass_symbol})
        
        fig.draw()
        drawing_tool.display()
        drawing_tool.savefig('pendulum1')

The body diagram
----------------

Now we want to isolate the mass and draw all the forces that act on it.
Figure :ref:`sketcher:ex:pendulum:fig2wgrid` shows the desired result, but
embedded in the coordinate system.
We consider three types of forces: the gravity force, the force from the
rod, and air resistance. The body diagram is key for deriving the
equation of motion, so it is illustrative to add useful mathematical
quantities needed in the derivation, such as the unit vectors in polar
coordinates.

.. _sketcher:ex:pendulum:fig2wgrid:

.. figure:: pendulum5_wgrid.png
   :width: 300

   *Body diagram of a simple pendulum*

We start by listing the objects in the sketch:

1. a text :math:`(x_0,y_0)` representing the rotation  point ``P``

2. unit vector :math:`\boldsymbol{i}_r` with text

3. unit vector :math:`\boldsymbol{i}_\theta` with text

4. a dashed vertical line

5. a dashed line along the rod

6. an arc with text :math:`\theta`

7. the gravity force with text :math:`mg`

8. the force in the rod with text :math:`S`

9. the air resistance force with text :math:`\sim |v|v`

The first object, :math:`(x_0,y_0)`, is simply a plain text where we have
to experiment with its position. The unit vectors in polar coordinates
may be drawn using the Pysketcher's ``Force`` object since it has an
arrow with a text. The first three objects can then be made as follows:

.. code-block:: python

        x0y0 = Text('$(x_0,y_0)$', P + point(-0.4,-0.1))
        ir = Force(P, P + L/10*unit_vec(rod_vec),
                   r'$\boldsymbol{i}_r$', text_pos='end',
                   text_spacing=(0.015,0))
        ith = Force(P, P + L/10*unit_vec((-rod_vec[1], rod_vec[0])),
                   r'$\boldsymbol{i}_{\theta}$', text_pos='end',
                    text_spacing=(0.02,0.005))

Note that tweaking of the position of ``x0y0`` use absolute coordinates, so
if ``W`` or ``H`` is changed in the beginning of the figure, the tweaked position
will most likely not look good. A better solution would be to express
the tweaked displacement ``point(-0.4,-0.1)`` in terms of ``W`` and ``H``.
The ``text_spacing`` values in the ``Force`` objects also use absolute
coordinates. Very often, this is much more convenient when adjusting
the objects, and global size parameters like ``W`` and ``H`` are in practice
seldom changed, so the solution above is quite typical.

The vertical, dashed line, the dashed rod, and the arc for :math:`\theta`
are made by

.. code-block:: python

        rod_start = rod.geometric_features()['start']  # Point P
        vertical2 = Line(rod_start, rod_start + point(0,-L/3))
        set_dashed_thin_blackline(vertical2)
        set_dashed_thin_blackline(rod)
        angle2 = Arc_wText(r'$\theta$', rod_start, L/6, -90, a,
                           text_spacing=1/30.)

Note how we reuse the earlier defined object ``rod``.

The forces are constructed as shown below.

.. code-block:: python

        mg_force  = Force(mass_pt, mass_pt + L/5*point(0,-1),
                          '$mg$', text_pos='end')
        rod_force = Force(mass_pt, mass_pt - L/3*unit_vec(rod_vec),
                          '$S$', text_pos='end',
                          text_spacing=(0.03, 0.01))
        air_force = Force(mass_pt, mass_pt -
                          L/6*unit_vec((rod_vec[1], -rod_vec[0])),
                          '$\sim|v|v$', text_pos='end',
                          text_spacing=(0.04,0.005))

Note that the drag force from the air is directed perpendicular to
the rod, so we construct a unit vector in this direction directly from
the ``rod_vec`` vector.

All objects are in place, and we can compose a figure to be drawn:

.. code-block:: python

        body_diagram = Composition(
            {'mg': mg_force, 'S': rod_force, 'rod': rod,
             'vertical': vertical2, 'theta': angle2,
             'body': mass, 'm': mass_symbol})
        
        body_diagram['air'] = air_force
        body_diagram['ir'] = ir
        body_diagram['ith'] = ith
        body_diagram['origin'] = x0y0

Here, we exemplify that we can start out with a composition as a
dictionary, but (as in ordinary Python dictionaries) add new
elements later when desired.

.. FIGURE: [fig-tut/pendulum1.png, width=300 frac=0.5] Sketch of a simple pendulum.

.. _sketcher:ex:pendulum:anim:

Animated body diagram
---------------------

We want to make an animated body diagram so that we can see how forces
develop in time according to the motion. This means that we must
couple the sketch at each time level to a numerical solution for
the motion of the pendulum.

Function for drawing the body diagram
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

The previous flat program for making sketches of the pendulum is not
suitable when we want to make a sketch at a lot of different points
in time, i.e., for a lot of different angles that the pendulum makes
with the vertical. We therefore need to draw the body diagram in
a function where the angle is a parameter. We also supply arrays
containing the (numerically computed) values of the angle :math:`\theta` and
the forces at various time levels, plus the desired time point and level
for this particular sketch:

.. code-block:: python

        from pysketcher import *
        
        H = 15.
        W = 17.
        
        drawing_tool.set_coordinate_system(xmin=0, xmax=W,
                                           ymin=0, ymax=H,
                                           axis=False)
        
        def pendulum(theta, S, mg, drag, t, time_level):
        
            drawing_tool.set_linecolor('blue')
            import math
            a = math.degrees(theta[time_level])
            L = 0.4*H         # length
            P = (W/2, 0.8*H)  # rotation point
        
            vertical = Line(P, P-point(0,L))
            path = Arc(P, L, -90, a)
            angle = Arc_wText(r'$\theta$', P, L/4, -90, a, text_spacing=1/30.)
        
            mass_pt = path.geometric_features()['end']
            rod = Line(P, mass_pt)
        
            mass = Circle(center=mass_pt, radius=L/20.)
            mass.set_filled_curves(color='blue')
            rod_vec = rod.geometric_features()['end'] - \ 
                      rod.geometric_features()['start']
            unit_rod_vec = unit_vec(rod_vec)
            mass_symbol = Text('$m$', mass_pt + L/10*unit_rod_vec)
        
            length = Distance_wText(P, mass_pt, '$L$')
            # Displace length indication
            length.translate(L/15*point(cos(radians(a)), sin(radians(a))))
            gravity = Gravity(start=P+point(0.8*L,0), length=L/3)
        
            def set_dashed_thin_blackline(*objects):
                """Set linestyle of objects to dashed, black, width=1."""
                for obj in objects:
                    obj.set_linestyle('dashed')
                    obj.set_linecolor('black')
                    obj.set_linewidth(1)
        
            set_dashed_thin_blackline(vertical, path)
        
            fig = Composition(
                {'body': mass, 'rod': rod,
                 'vertical': vertical, 'theta': angle, 'path': path,
                 'g': gravity, 'L': length})
        
            #fig.draw()
            #drawing_tool.display()
            #drawing_tool.savefig('tmp_pendulum1')
        
            drawing_tool.set_linecolor('black')
        
            rod_start = rod.geometric_features()['start']  # Point P
            vertical2 = Line(rod_start, rod_start + point(0,-L/3))
            set_dashed_thin_blackline(vertical2)
            set_dashed_thin_blackline(rod)
            angle2 = Arc_wText(r'$\theta$', rod_start, L/6, -90, a,
                               text_spacing=1/30.)
        
            magnitude = 1.2*L/2   # length of a unit force in figure
            force = mg[time_level]  # constant (scaled eq: about 1)
            force *= magnitude
            mg_force  = Force(mass_pt, mass_pt + force*point(0,-1),
                              '', text_pos='end')
            force = S[time_level]
            force *= magnitude
            rod_force = Force(mass_pt, mass_pt - force*unit_vec(rod_vec),
                              '', text_pos='end',
                              text_spacing=(0.03, 0.01))
            force = drag[time_level]
            force *= magnitude
            #print('drag(%g)=%g' % (t, drag[time_level]))
            air_force = Force(mass_pt, mass_pt -
                              force*unit_vec((rod_vec[1], -rod_vec[0])),
                              '', text_pos='end',
                              text_spacing=(0.04,0.005))
        
            body_diagram = Composition(
                {'mg': mg_force, 'S': rod_force, 'air': air_force,
                 'rod': rod,
                 'vertical': vertical2, 'theta': angle2,
                 'body': mass})
        
            x0y0 = Text('$(x_0,y_0)$', P + point(-0.4,-0.1))
            ir = Force(P, P + L/10*unit_vec(rod_vec),
                       r'$\boldsymbol{i}_r$', text_pos='end',
                       text_spacing=(0.015,0))
            ith = Force(P, P + L/10*unit_vec((-rod_vec[1], rod_vec[0])),
                       r'$\boldsymbol{i}_{\theta}$', text_pos='end',
                        text_spacing=(0.02,0.005))
        
            #body_diagram['ir'] = ir
            #body_diagram['ith'] = ith
            #body_diagram['origin'] = x0y0
        
            drawing_tool.erase()
            body_diagram.draw(verbose=0)
            #drawing_tool.display('Body diagram')
            drawing_tool.savefig('tmp_%04d.png' % time_level, crop=False)
            # No cropping: otherwise movies will be very strange

Equations for the motion and forces
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

The modeling of the motion of a pendulum is most conveniently done in
polar coordinates since then the unknown force in the rod is separated
from the equation determining the motion :math:`\theta(t)`.
The position vector for the mass is

.. math::
         \boldsymbol{r} = x_0\boldsymbol{i} + y_0\boldsymbol{j} + L{\boldsymbol{i}_r}{\thinspace .}

The corresponding acceleration becomes

.. math::
         \ddot{\boldsymbol{r}} = L\ddot{\theta}{\boldsymbol{i}_{\theta}} - L\dot{\theta^2}{{\boldsymbol{i}_r}}{\thinspace .}

.. Note: the extra braces help to render the equation correctly in sphinx!

There are three forces on the mass: the gravity force
:math:`mg\boldsymbol{j} = mg(-\cos\theta\,{\boldsymbol{i}_r} + \sin\theta\,\boldsymbol{i}_{\theta})`, the force in the rod
:math:`-S{\boldsymbol{i}_r}`, and the drag force because of air resistance:

.. math::
         -\frac{1}{2} C_D \varrho \pi R^2 |v|v\,\boldsymbol{i}_{\theta},

where :math:`C_D\approx 0.4` is the drag coefficient for a sphere, :math:`\varrho`
is the density of air, :math:`R` is the radius of the mass, and :math:`v` is the
velocity (:math:`v=L\dot\theta`). The drag force acts in :math:`-\boldsymbol{i}_{\theta}` direction
when :math:`v>0`.

Newton's second law of motion for the pendulum now becomes

.. math::
         mL\ddot\theta\boldsymbol{i}_{\theta} - mL\dot\theta^2{\boldsymbol{i}_r} = -mg(-\cos\theta\,{\boldsymbol{i}_r} +
        \sin\theta\,\boldsymbol{i}_{\theta})
        -S{\boldsymbol{i}_r} - \frac{1}{2} C_D \varrho \pi R^2 L^2|\dot\theta|\dot\theta\boldsymbol{i}_{\theta},

which gives two component equations

.. math::
   :label: sketcher:ex:pendulum:anim:eq:ith
        
        mL\ddot\theta + \frac{1}{2} C_D \varrho \pi R^2 L^2|\dot\theta|\dot\theta +
        mg\sin\theta = 0,
        
        

.. math::
   :label: sketcher:ex:pendulum:anim:eq:ir
          
        S = mL\dot\theta^2 + mg\cos\theta
        {\thinspace .}
        

It is almost always convenient to scale such equations. Introducing
the dimensionless time

.. math::
         \bar t = \frac{t}{t_c},\quad t_c = \sqrt{\frac{L}{g}},

leads to

.. math::
   :label: sketcher:ex:pendulum:anim:eq:ith:s
        
        \frac{d^2\theta}{d\bar t^2} +
        \alpha\left\vert\frac{d\theta}{d\bar t}\right\vert\frac{d\theta}{d\bar t} +
        \sin\theta = 0,
        
        

.. math::
   :label: sketcher:ex:pendulum:anim:eq:ir:s
          
        \bar S = \left(\frac{d\theta}{d\bar t}\right)^2
        + \cos\theta,
        
        

where :math:`\alpha` is a dimensionless drag coefficient

.. math::
         \alpha = \frac{C_D\varrho\pi R^2L}{2m},

and :math:`\bar S` is the scaled force

.. math::
         \bar S = \frac{S}{mg}{\thinspace .}

We see that :math:`\bar S = 1` for the equilibrium position :math:`\theta=0`, so this
scaling of :math:`S` seems appropriate.

The parameter :math:`\alpha` is about
the ratio of the drag force and the gravity force:

.. math::
         \frac{|\frac{1}{2} C_D\varrho \pi R^2 |v|v|}{|mg|}\sim
        \frac{C_D\varrho \pi R^2 L^2 t_c^{-2}}{mg}
        \left|\frac{d\bar\theta}{d\bar t}\right|\frac{d\bar\theta}{d\bar t}
        \sim \frac{C_D\varrho \pi R^2 L}{2m}\theta_0^2 = \alpha \theta_0^2{\thinspace .}

(We have that :math:`\theta(t)/d\theta_0` is in :math:`[-1,1]`, so we expect
since :math:`\theta_0^{-1}d\bar\theta/d\bar t` to be around unity.)

The next step is to write a numerical solver for
:eq:`sketcher:ex:pendulum:anim:eq:ith:s`-:eq:`sketcher:ex:pendulum:anim:eq:ir:s`. To
this end, we use the `Odespy <https://github.com/hplgit/odespy>`__
package. The system of second-order ODEs must be expressed as a system
of first-order ODEs. We realize that the unknown :math:`\bar S` is decoupled
from :math:`\theta` in the sense that we can first use
:eq:`sketcher:ex:pendulum:anim:eq:ith:s` to solve for :math:`\theta` and
then compute :math:`\bar S` from :eq:`sketcher:ex:pendulum:anim:eq:ir:s`.
The first-order ODEs become

.. math::
   :label: _auto1
        
        \frac{d\omega}{d\bar t} = -\alpha\left\vert\omega\right\vert\omega
        - \sin\theta,
        
        

.. math::
   :label: _auto2
          
        \frac{d\theta}{d\bar t} = \omega{\thinspace .}
        
        

Then we compute

.. math::
   :label: _auto3
        
        \bar S = \omega^2 + \cos\theta{\thinspace .}
        
        

The dimensionless air resistance force can also be computed:

.. math::
   :label: _auto4
        
        -\alpha|\omega|\omega{\thinspace .}
        
        

Since we scaled the force :math:`S` by :math:`mg`, :math:`mg` is the natural force scale,
and the :math:`mg` force itself is then unity.

By updating :math:`\omega` in the first equation, we can use an Euler-Cromer
scheme on Odespy (all other schemes are independent of whether the
:math:`\theta` or :math:`\omega` equation comes first).

Numerical solution
~~~~~~~~~~~~~~~~~~

An appropriate solver is

.. code-block:: python

        def simulate_pendulum(alpha, theta0, dt, T):
            import odespy
        
            def f(u, t, alpha):
                omega, theta = u
                return [-alpha*omega*abs(omega) - sin(theta),
                        omega]
        
            import numpy as np
            Nt = int(round(T/float(dt)))
            t = np.linspace(0, Nt*dt, Nt+1)
            solver = odespy.RK4(f, f_args=[alpha])
            solver.set_initial_condition([0, theta0])
            u, t = solver.solve(t,
                                terminate=lambda u, t, n: abs(u[n,1]) < 1E-3)
            omega = u[:,0]
            theta = u[:,1]
            S = omega**2 + np.cos(theta)
            drag = -alpha*np.abs(omega)*omega
            return t, theta, omega, S, drag

Animation
~~~~~~~~~

We can finally traverse the time array and draw a body diagram
at each time level. The resulting sketches are saved to files
``tmp_%04d.png``, and these files can be combined to videos:

.. code-block:: python

        def animate():
            # Clean up old plot files
            import os, glob
            for filename in glob.glob('tmp_*.png') + glob.glob('movie.*'):
                os.remove(filename)
            # Solve problem
            from math import pi, radians, degrees
            import numpy as np
            alpha = 0.4
            period = 2*pi
            T = 12*period
            dt = period/40
            a = 70
            theta0 = radians(a)
            t, theta, omega, S, drag = simulate_pendulum(alpha, theta0, dt, T)
            mg = np.ones(S.size)
            # Visualize drag force 5 times as large
            drag *= 5
            print('NOTE: drag force magnified 5 times!!')
        
            # Draw animation
            import time
            for time_level, t_ in enumerate(t):
                pendulum(theta, S, mg, drag, t_, time_level)
                time.sleep(0.2)
        
            # Make videos
            prog = 'ffmpeg'
            filename = 'tmp_%04d.png'
            fps = 6
            codecs = {'flv': 'flv', 'mp4': 'libx264',
                      'webm': 'libvpx', 'ogg': 'libtheora'}
            for ext in codecs:
                lib = codecs[ext]
                cmd = '%(prog)s -i %(filename)s -r %(fps)s ' % vars()
                cmd += '-vcodec %(lib)s movie.%(ext)s' % vars()
                print(cmd)
                os.system(cmd)

.. raw:: html
        
        <div>
        <video  loop controls width='640' height='365' preload='none'>
            <source src='https://github.com/hplgit/pysketcher/raw/master/doc/pub/tutorial/mov-tut/pendulum/movie.mp4'  type='video/mp4;  codecs="avc1.42E01E, mp4a.40.2"'>
            <source src='https://github.com/hplgit/pysketcher/raw/master/doc/pub/tutorial/mov-tut/pendulum/movie.webm' type='video/webm; codecs="vp8, vorbis"'>
            <source src='https://github.com/hplgit/pysketcher/raw/master/doc/pub/tutorial/mov-tut/pendulum/movie.ogg'  type='video/ogg;  codecs="theora, vorbis"'>
        </video>
        </div>
        <p><em>The drag force is magnified 5 times (compared to :math:`mg` and :math:`S`!</em></p>
        
        <!-- Issue warning if in a Safari browser -->
        <script language="javascript">
        if (!!(window.safari)) {
          document.write("<div style=\"width: 95%%; padding: 10px; border: 1px solid #100; border-radius: 4px;\"><p><font color=\"red\">The above movie will not play in Safari - use Chrome, Firefox, or Opera.</font></p></div>")}
        </script>
        

.. !split

Basic shapes
============

This section presents many of the basic shapes in Pysketcher:
``Axis``, ``Distance_wText``, ``Rectangle``, 	``Triangle``, ``Arc``,
``Spring``, ``Dashpot``, and ``Wavy``.
Each shape is demonstrated with a figure and a
unit test that shows how the figure is constructed in Python code.
These demos rely heavily on the method ``draw_dimensions`` in
the shape classes, which annotates the basic drawing of the shape
with the various geometric parameters that govern the shape.

Axis
----

The ``Axis`` object gives the possibility draw a single axis to
notify a coordinate system. Here is an example where we
draw :math:`x` and :math:`y` axis of three coordinate systems of different
rotation:

| 
| 

.. figure:: Axis.png
   :width: 500

| 
| 

The corresponding code looks like this:

.. code-block:: python

        def test_Axis():
            drawing_tool.set_coordinate_system(
                xmin=0, xmax=15, ymin=-7, ymax=8, axis=True,
                instruction_file='tmp_Axis.py')
            # Draw normal x and y axis with origin at (7.5, 2)
            # in the coordinate system of the sketch: [0,15]x[-7,8]
            x_axis = Axis((7.5,2), 5, 'x', rotation_angle=0)
            y_axis = Axis((7.5,2), 5, 'y', rotation_angle=90)
            system = Composition({'x axis': x_axis, 'y axis': y_axis})
            system.draw()
            drawing_tool.display()
        
            # Rotate this system 40 degrees counter clockwise
            # and draw it with dashed lines
            system.set_linestyle('dashed')
            system.rotate(40, (7.5,2))
            system.draw()
            drawing_tool.display()
        
            # Rotate this system another 220 degrees and show
            # with dotted lines
            system.set_linestyle('dotted')
            system.rotate(220, (7.5,2))
            system.draw()
            drawing_tool.display()
        
            drawing_tool.display('Axis')

Distance with text
------------------

The object ``Distance_wText`` is used to display an arrow, to indicate
a distance in a sketch, with an additional text in the middle of the arrow.

The figure

| 
| 

.. figure:: Distance_wText.png
   :width: 500

| 
| 

was produced by this code:

.. code-block:: python

        def test_Distance_wText():
            drawing_tool.set_coordinate_system(
                xmin=0, xmax=10, ymin=0, ymax=6,
                axis=True, instruction_file='tmp_Distance_wText.py')
        
            fontsize=14
            t = r'$ 2\pi R^2 $'  # sample text
            examples = Composition({
                'a0': Distance_wText((4,5), (8, 5), t, fontsize),
                'a6': Distance_wText((4,5), (4, 4), t, fontsize),
                'a1': Distance_wText((0,2), (2, 4.5), t, fontsize),
                'a2': Distance_wText((0,2), (2, 0), t, fontsize),
                'a3': Distance_wText((2,4.5), (0, 5.5), t, fontsize),
                'a4': Distance_wText((8,4), (10, 3), t, fontsize,
                                     text_spacing=-1./60),
                'a5': Distance_wText((8,2), (10, 1), t, fontsize,
                                     text_spacing=-1./40, alignment='right'),
                'c1': Text_wArrow('text_spacing=-1./60',
                                  (4, 3.5), (9, 3.2),
                                  fontsize=10, alignment='left'),
                'c2': Text_wArrow('text_spacing=-1./40, alignment="right"',
                                  (4, 0.5), (9, 1.2),
                                  fontsize=10, alignment='left'),
                })
            examples.draw()
            drawing_tool.display('Distance_wText and text positioning')

Note the use of ``Text_wArrow`` to write an explaining text with an
associated arrow, here used to explain how
the ``text_spacing`` and ``alignment`` arguments can be used to adjust
the appearance of the text that goes with the distance arrow.

Rectangle
---------

.. figure:: Rectangle.png
   :width: 500

| 
| 

The above figure can be produced by the following code.

.. code-block:: python

        def test_Rectangle():
            L = 3.0
            W = 4.0
        
            drawing_tool.set_coordinate_system(
                xmin=0, xmax=2*W, ymin=-L/2, ymax=2*L,
                axis=True, instruction_file='tmp_Rectangle.py')
            drawing_tool.set_linecolor('blue')
            drawing_tool.set_grid(True)
        
            xpos = W/2
            r = Rectangle(lower_left_corner=(xpos,0), width=W, height=L)
            r.draw()
            r.draw_dimensions()
            drawing_tool.display('Rectangle')

Note that the ``draw_dimension`` method adds explanation of dimensions and various
important argument in the construction of a shape. It adapts the annotations
to the geometry of the current shape.

Triangle
--------

.. figure:: Triangle.png
   :width: 500

| 
| 

The code below produces the figure.

.. code-block:: python

        def test_Triangle():
            L = 3.0
            W = 4.0
        
            drawing_tool.set_coordinate_system(
                xmin=0, xmax=2*W, ymin=-L/2, ymax=1.2*L,
                axis=True, instruction_file='tmp_Triangle.py')
            drawing_tool.set_linecolor('blue')
            drawing_tool.set_grid(True)
        
            xpos = 1
            t = Triangle(p1=(W/2,0), p2=(3*W/2,W/2), p3=(4*W/5.,L))
            t.draw()
            t.draw_dimensions()
            drawing_tool.display('Triangle')

Here, the ``draw_dimension`` method writes the name of the corners at the
position of the corners, which does not always look nice (the present figure
is an example). For a high-quality sketch one would add some spacing
to the location of the p1, p2, and even p3 texts.

Arc
---

.. figure:: Arc.png
   :width: 400

| 
| 

An arc like the one above is produced by

.. code-block:: python

        def test_Arc():
            L = 4.0
            W = 4.0
        
            drawing_tool.set_coordinate_system(
                xmin=-W/2, xmax=W, ymin=-L/2, ymax=1.5*L,
                axis=True, instruction_file='tmp_Arc.py')
            drawing_tool.set_linecolor('blue')
            drawing_tool.set_grid(True)
        
            center = point(0,0)
            radius = L/2
            start_angle = 60
            arc_angle = 45
            a = Arc(center, radius, start_angle, arc_angle)
            a.draw()
        
            R1 = 1.25*radius
            R2 = 1.5*radius
            R = 2*radius
            a.dimensions = {
                'start_angle':
                Arc_wText(
                    'start_angle', center, R1, start_angle=0,
                    arc_angle=start_angle, text_spacing=1/10.),
                'arc_angle':
                Arc_wText(
                    'arc_angle', center, R2, start_angle=start_angle,
                    arc_angle=arc_angle, text_spacing=1/20.),
                'r=0':
                Line(center, center +
                     point(R*cos(radians(start_angle)),
                           R*sin(radians(start_angle)))),
                'r=start_angle':
                Line(center, center +
                     point(R*cos(radians(start_angle+arc_angle)),
                           R*sin(radians(start_angle+arc_angle)))),
                'r=start+arc_angle':
                Line(center, center +
                     point(R, 0)).set_linestyle('dashed'),
                'radius': Distance_wText(center, a(0), 'radius', text_spacing=1/40.),
                'center': Text('center', center-point(radius/10., radius/10.)),
                }
            for dimension in a.dimensions:
                if dimension.startswith('r='):
                    dim = a.dimensions[dimension]
                    dim.set_linestyle('dashed')
                    dim.set_linewidth(1)
                    dim.set_linecolor('black')
        
            a.draw_dimensions()
            drawing_tool.display('Arc')

Spring
------

.. figure:: Spring.png
   :width: 800

| 
| 

The code for making these two springs goes like this:

.. code-block:: python

        def test_Spring():
            L = 5.0
            W = 2.0
        
            drawing_tool.set_coordinate_system(
                xmin=0, xmax=7*W, ymin=-L/2, ymax=1.5*L,
                axis=True, instruction_file='tmp_Spring.py')
            drawing_tool.set_linecolor('blue')
            drawing_tool.set_grid(True)
        
            xpos = W
            s1 = Spring((W,0), L, teeth=True)
            s1_title = Text('Default Spring',
                            s1.geometric_features()['end'] + point(0,L/10))
            s1.draw()
            s1_title.draw()
            #s1.draw_dimensions()
            xpos += 3*W
            s2 = Spring(start=(xpos,0), length=L, width=W/2.,
                        bar_length=L/6., teeth=False)
            s2.draw()
            s2.draw_dimensions()
            drawing_tool.display('Spring')

Dashpot
-------

.. figure:: Dashpot.png
   :width: 600

| 
| 

This dashpot is produced by

.. code-block:: python

        def test_Dashpot():
            L = 5.0
            W = 2.0
            xpos = 0
        
            drawing_tool.set_coordinate_system(
                xmin=xpos, xmax=xpos+5.5*W, ymin=-L/2, ymax=1.5*L,
                axis=True, instruction_file='tmp_Dashpot.py')
            drawing_tool.set_linecolor('blue')
            drawing_tool.set_grid(True)
        
            # Default (simple) dashpot
            xpos = 1.5
            d1 = Dashpot(start=(xpos,0), total_length=L)
            d1_title = Text('Dashpot (default)',
                            d1.geometric_features()['end'] + point(0,L/10))
            d1.draw()
            d1_title.draw()
        
            # Dashpot for animation with fixed bar_length, dashpot_length and
            # prescribed piston_pos
            xpos += 2.5*W
            d2 = Dashpot(start=(xpos,0), total_length=1.2*L, width=W/2,
                         bar_length=W, dashpot_length=L/2, piston_pos=2*W)
            d2.draw()
            d2.draw_dimensions()
        
            drawing_tool.display('Dashpot')

Wavy
----

Looks strange. Fix x axis.

Stochastic curves
-----------------

The ``StochasticWavyCurve`` object offers three precomputed
graphics that have a random variation:

| 
| 

.. figure:: StochasticWavyCurve.png
   :width: 600

| 
| 

The usage is simple. The construction

.. code-block:: python

        curve = StochasticWavyCurve(curve_no=1, percentage=40)

picks the second curve (the three are numbered 0, 1, and 2),
and the first 40% of that curve. In case one desires another extent
of the axis, one can just scale the coordinates directly as these
are stored in the arrays ``curve.x[curve_no]`` and
``curve.y[curve_no]``.

.. !split

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.

.. index:: tree data structure

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``, specified by the coordinates of
the lower left corner and its width and height:

.. 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 that 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``.  For class ``Rectangle`` the ``x``
and ``y`` arrays contain the corner points of the rectangle in
counterclockwise direction, starting and ending with in the lower left
corner.

Class ``Line`` is also a simple class:

.. 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 want to 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 true source code of ``Line`` therefore
provides a more general solution at the cost of significantly longer
code with more tests.

A circle implies a 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 as the number
of :math:`t` values. 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_s, \theta_s + \theta_a]`. Giving
:math:`\theta_s` and :math:`\theta_a` 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.start_angle = radians(start_angle)
                self.arc_angle = radians(arc_angle)
        
                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 be defined 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)

Class curve
~~~~~~~~~~~

Class ``Curve`` sits on the coordinates to be drawn, but how is that
done? The constructor of class ``Curve`` 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 in class ``Curve`` 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 figure),

  * ``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 in the ``self.shapes`` dictionary. 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)
        
            def draw(self):
                drawing_tool.plot_curve(
                    self.x, self.y,
                    self.linestyle, self.linewidth, self.linecolor, ...)
        
            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 can the geometric
shape through just one ``Curve`` object. More complicated shapes 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:`sketcher:fig:vehicle0`.  That is, instead of composing the drawing
in a Python program as shown above, we make a subclass ``Vehicle0`` in
the ``Shape`` hierarchy for doing the same thing.

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:`sketcher:fig:vehicle0`.

.. code-block:: python

        from pysketcher import *
        
        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, as shown in the right part of
Figure :ref:`sketcher:fig:vehicle0:v2`, could in class ``Vehicle0``
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 as previously shown, we can do

.. code-block:: python

        vehicle = Vehicle0(w_1, R, L, H)
        vehicle.draw()
        drawing_tool.display()

and go on the make a painted version 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 <http://tinyurl.com/ot733jn/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
----------------------------------

.. index:: recursive function calls

The really powerful feature of our class hierarchy is that we can add
much functionality to the superclass ``Shape`` and to the "bottom" class
``Curve``, and then all other classes for various types of geometrical shapes
immediately get the new functionality. To explain the idea we may
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
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Note that we work with two types of hierarchies in the
present documentation: 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 ``self.shapes``, now containing
the ``wheel1`` 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 at the "bottom" 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.

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``
that 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, i.e., every row of objects in Figure
:ref:`sketcher:fig:Vehicle0:hier2`. This indentation 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 has the keys ``'over'`` and ``'under'``,
will be

.. code-block:: text

             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:: text

        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:: 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.

A part of the printout of ``v.recurse('vehicle')`` looks like

.. code-block:: text

         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,

.. code-block:: python

        class Curve:
            ...
            def scale(self, factor):
                self.x *= factor
                self.y *= factor

as this saves the creation of temporary arrays like ``factor*self.x``.

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 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 the
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`` follows the principle of the
``draw``, ``scale``, and ``translate`` methods.

We have already seen the ``rotate`` method in action when animating the
rolling wheel at the end of the section :ref:`sketcher:vehicle1:anim`.