Quiz 104 -- Solution

====================

Here is what turtle.rb looked like before I messed with it to produce

Quiz 104.

<code>

# An implementation of Turtle Procedure Notation (TPN) as described in

# H. Abelson & A. diSessa, "Turtle Geometry", MIT Press, 1981.

#

# Turtles navigate by traditional geographic coordinates: X-axis

pointing

# east, Y-axis pointing north, and angles measured clockwise from the

# Y-axis (north) in degrees.

class Turtle

include Math

DEG = Math:

I / 180.0

ORIGIN = [0.0, 0.0]

alias run instance_eval

attr_accessor :track

attr_reader

y, :heading

def degree

DEG

end

###

# Turtle primitives

###

</code>

I explicitly define a writer for @xy to get the Logo-like argument

checking that I wanted. Also, I decided to maintain @xy as an array

of floats to minimize the accumulation of position errors in long

tracks.

<code>

# Place the turtle at [x, y]. The turtle does not draw when it

changes

# position.

def xy=(coords)

if coords.size != 2

raise(ArgumentError, "turtle needs two coordinates")

end

x, y = coords

must_be_number(x, 'x-coordinate')

must_be_number(y, 'y-coordinate')

@xy = x.to_f, y.to_f

end

</code>

Similarly, I explicitly define a writer for @heading. But it's not

just for argument checking: I also use it to constrain @heading to

the interval [0.0, 360.0).

<code>

# Set the turtle's heading to <degrees>.

def heading=(degrees)

must_be_number(degrees, 'heading')

@heading = degrees.to_f

case

when @heading >= 360.0

@heading -= 360.0 while @heading >= 360.0

when @heading < 0.0

@heading += 360.0 while @heading < 0.0

end

@heading

end

# Raise the turtle's pen. If the pen is up, the turtle will not

draw;

# i.e., it will cease to lay a track until a pen_down command is

given.

def pen_up

@pen = :up

end

</code>

When the pen goes down, a new track segment must be added. Initially,

the segment contains only a single point. If the pen goes up before

another point is added to the segment, the segment ends up with just

one point. Such singleton segments are skipped when the track is

processed by in the view.

<code>

# Lower the turtle's pen. If the pen is down, the turtle will draw;

# i.e., it will lay a track until a pen_up command is given.

def pen_down

@pen = :down

@track << [@xy]

end

# Is the pen up?

def pen_up?

@pen == :up

end

# Is the pen down?

def pen_down?

@pen == :down

end

###

# Turtle commands

###

# Place the turtle at the origin, facing north, with its pen up.

# The turtle does not draw when it goes home.

def home

pen_up

self.xy = ORIGIN

self.heading = 0.0

end

# Home the turtle and empty out it's track.

def clear

home

self.track = []

end

alias initialize clear

# Turn right through the angle <degrees>.

def right(degrees)

must_be_number(degrees, 'turn')

self.heading = heading + degrees.to_f

end

# Turn left through the angle <degrees>.

def left(degrees)

right(-degrees)

end

</code>

This is one of two places in the code where it actually has to do

some trigonometry -- Turtle#toward below is the other.

<code>

# Move forward by <steps> turtle steps.

def forward(steps)

must_be_number(steps, 'distance')

angle = heading * DEG

x, y = xy

self.xy = [x + steps * sin(angle), y + steps * cos(angle)]

track.last << xy if pen_down?

end

# Move backward by <steps> turtle steps.

def back(steps)

forward(-steps)

end

# Move to the given point.

def go(pt)

self.xy = pt

track.last << xy if pen_down?

end

</code>

In Turtle#toward, the expression atan2(y2 - y1, x2 - x1) computes the

slope angle of the line between pt and xy. Math#atan2 is better here

than Math#atan because atan2 handles the four quadrant cases

automatically. Once the slope angle is known, it is easily converted

into a heading.

<code>

# Turn to face the given point.

def toward(pt)

x2, y2 = pt

must_be_number(x2, 'pt.x')

must_be_number(y2, 'pt.y')

x1, y1 = xy

set_h(90.0 - atan2(y2 - y1, x2 - x1) / DEG)

end

</code>

Turtle#distance is easy to implement providing one remembers the

existence of Math#hypot.

<code>

# Return the distance between the turtle and the given point.

def distance(pt)

x2, y2 = pt

must_be_number(x2, 'pt.x')

must_be_number(y2, 'pt.y')

x1, y1 = xy

hypot(x2 - x1, y2 - y1)

end

# Traditional abbreviations for turtle commands.

alias fd forward

alias bk back

alias rt right

alias lt left

alias pu pen_up

alias pd pen_down

alias pu? pen_up?

alias pd? pen_down?

alias set_h heading=

alias set_xy xy=

alias face toward

alias dist distance

private

# Raise an exception if <val> is not a number.

def must_be_number(val, name)

if !val.respond_to?(:to_f)

raise(ArgumentError, "#{name} must be a number")

end

end

end

</code>

Now that you've seen the code, let me discuss some of the

implementation decisions I made.

The first issue I had to deal with was how to reconcile the way

turtles measure angles with the way Ruby/Math measures angles.

Turtles, you recall, (following the conventions of geography/

navigation) measure angles clockwise from north in degrees, while the

Math module (following mathematical conventions) measures angles

counterclockwise from east in radians. Since the Turtle class

includes Math, there are advantages to following mathematical

conventions when maintaining the turtle's orientation internal to the

class, However, influenced by Logo, I chose to use the navigator's

notion of angle and to reconcile turtle angles to Math angles each

time I actually did some trig.

I also considered overriding the trig functions with methods that

would accept angles in degrees as their arguments. In the end, I

decided not to, but I still find myself thinking, from time to time,

that I should go back to the code and do it.

The next issue I settled was: what, if any, argument checking should

I do? I settled on accepting any argument that responds to to_f,

raising ArgumentError for those that don't, and providing Logo-like

error messages. The private method Turtle#must_be_number takes care

of this.

The last major issue was: how should I maintain the turtle's state?

That is, what instance variables should the class have? My choices were:

@xy turtle location

@heading turtle orientation

@pen pen state (up or down)

@track array needed to interface with Ruby/Tk

One last remark. Over the years I have built up a good-sized

collection of Logo turtle graphics programs. One of reasons I wanted

a Ruby turtle graphics capability was to convert this collection to

Ruby. I had the feeling that Ruby would prove to be a better Logo

than Logo. Well, I've performed the conversion and I'm convinced I

was right: the Ruby versions of the Logo programs are simpler, easier

to understand, and often considerably shorter than their Logo

counterparts.

Regards, Morton