
# #ifdef PRIMER_BOOK
===== Example of Classes for Geometric Objects =====
# #else
======= Inner Workings of the Pysketcher Tool =======
# #endif

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.

# #ifndef PRIMER_BOOK

idx{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 =====
# #endif

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:
!bc pycod
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)}
!ec

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 $(x,y)$
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:
!bc pycod
class Line(Shape):
    def __init__(self, start, end):
        x = [start[0], end[0]]
        y = [start[1], end[1]]
        self.shapes = {'line': Curve(x, y)}
!ec
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 $x$ coordinate and have the class calculate the
corresponding $y$ coordinate:

!bc pycod
    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
!ec
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 $(x,y)$ on a curve with radius $R$
and center at $(x_0, y_0)$ are given by

!bt
\begin{align*}
x &= x_0 + R\cos (t),\\
y &= y_0 + R\sin (t),
\end{align*}
!et
where $t\in [0, 2\pi]$. A discrete set of $t$ values in this
interval gives the corresponding set of $(x,y)$ coordinates on
the circle. The user must specify the resolution as the number
of $t$ values. The circle's radius and center must of course
also be specified.

We can write the `Circle` class as

!bc pycod
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)}
!ec
As in class `Line` we can offer the possibility to give an angle
$\theta$ (equivalent to $t$ in the formulas above)
and then get the corresponding $x$ and $y$ coordinates:

!bc pycod
    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)
!ec
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 $t$ runs in $[\theta_s, \theta_s + \theta_a]$. Giving
$\theta_s$ and $\theta_a$ the slightly more descriptive names
`start_angle` and `arc_angle`, the code looks like this:

!bc pycod
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)}
!ec

Having the `Arc` class, a `Circle` can alternatively be defined as
a subclass specializing the arc to a circle:
!bc pycod
class Circle(Arc):
    def __init__(self, center, radius, resolution=180):
        Arc.__init__(self, center, radius, 0, 360, resolution)
!ec


=== 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 $x$ and $y$ 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

!bc pycod
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
    ...
!ec

=== 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}.

!bc pycod
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}
!ec

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:

!bc pycod
    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)
!ec

The usage of the class is simple: after having set up an appropriate
coordinate system as previously shown, we can do

!bc pycod
vehicle = Vehicle0(w_1, R, L, H)
vehicle.draw()
drawing_tool.display()
!ec
and go on the make a painted version by

!bc pycod
drawing_tool.erase()
vehicle.colorful()
vehicle.draw()
drawing_tool.display()
!ec
A complete code defining and using class `Vehicle0` is found in the file
"`vehicle2.py`": "${src_path_pysketcher}/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 =====

idx{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:
!bc pycod
def draw(self):
    for shape in self.shapes:
        self.shapes[shape].draw()
!ec
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
!bc pycod
v.shapes['vehicle'].draw()
v.shapes['ground'].draw()
!ec
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`:

!bc pycod
    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)
!ec
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

!bc dat
     Composition: body.shapes has entries 'over', 'under'
     call body.shapes["over"].recurse("over", 6)
!ec
The number of leading blanks on each line corresponds to the value of
`indent`. The code printing out such messages looks like

!bc pycod
    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)
!ec

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`)

!bc dat
ground
    wall
vehicle
    body
        over
            rectangle
        under
            rectangle
    wheels
        wheel1
            arc
        wheel2
            arc
!ec
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*.

FIGURE: [fig-tut/Vehicle0_hier2, width=600 frac=0.8] Hierarchy of figure elements in an instance of class `Vehicle0`. label{sketcher:fig:Vehicle0:hier2}

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

!bc dat
 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
...
!ec
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.


===== Scaling, Translating, and Rotating a Figure =====
label{sketcher:scaling}

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 $n$
coordinates $(x_i,y_i)$ that make up a curve, scaling by a factor $a$
means that we multiply all the $x$ and $y$ coordinates by $a$:

!bt
\[
x_i \leftarrow ax_i,\quad y_i\leftarrow ay_i,
\quad i=0,\ldots,n-1\thinspace .
\]
!et
Here we apply the arrow as an assignment operator.
The corresponding Python implementation in
class `Curve` reads

!bc pycod
class Curve:
    ...
    def scale(self, factor):
        self.x = factor*self.x
        self.y = factor*self.y
!ec
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,

!bc pycod
class Curve:
    ...
    def scale(self, factor):
        self.x *= factor
        self.y *= factor
!ec
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:

!bc pycod
class Shape:
    ...
    def scale(self, factor):
        for shape in self.shapes:
            self.shapes[shape].scale(factor)
!ec
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 $(x_i, y_i)$ can be translated $v_0$ units in
the $x$ direction and $v_1$ units in the $y$ direction using the formulas

!bt
\begin{equation*}
x_i\leftarrow x_i+v_0,\quad y_i\leftarrow y_i+v_1,
\quad i=0,\ldots,n-1\thinspace .
\end{equation*}
!et
The natural specification of the translation is in terms of the
vector $v=(v_0,v_1)$.
The corresponding Python implementation in class `Curve` becomes

!bc pycod
class Curve:
    ...
    def translate(self, v):
        self.x += v[0]
        self.y += v[1]
!ec
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:

!bc pycod
class Shape:
    ....
    def translate(self, v):
        for shape in self.shapes:
            self.shapes[shape].translate(v)
!ec

=== Rotation ===

Rotating a figure is more complicated than scaling and translating.
A counter clockwise rotation of $\theta$ degrees for a set of
coordinates $(x_i,y_i)$ is given by

!bt
\begin{align*}
 \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 .
\end{align*}
!et
This rotation is performed around the origin. If we want the figure
to be rotated with respect to a general point $(x,y)$, we need to
extend the formulas above:

!bt
\begin{align*}
 \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 .
\end{align*}
!et
The Python implementation in class `Curve`, assuming that $\theta$
is given in degrees and not in radians, becomes

!bc pycod
    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
!ec
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 Section ref{sketcher:vehicle1:anim}.
