Okay, now perhaps I should have thought about this before giving the
“Circle
Drawing” quiz, but… how do you summarize circle drawing? “Nice job,
it’s a
circle!” Or: “Oooh, sorry… but you drew a square. Better luck next
time.”
More seriously, there are many serious things that can be said about
drawing
in the digital realm. The problems associated with trying to draw a
simple
shape in ASCII do not disappear when you graduate to higher-density
pixels.
And the correct answer very much depends on the specification. Take, for
example, this look at font rendering. Which answer is correct
depends
very much on your specifications and goals. For fonts, there may be many
subjective criteria; for circles, there should be far less.
But, for this quiz, not zero. I didn’t fully specify exactly how I
wanted “a
circle of radius 7” drawn. Seeing how the mathematical radius of such
circle
would be 14, some might want their circles drawing within a 14x14 area.
However, others (including my own example in the original description)
pick
the circle center in the middle of an ASCII character center, then
measure
out 7 units in each direction, which fills a 15x15 area.
Which is correct? Depends on what you want… If you want something
close to
the mathematical ideal, you want the latter. However, if you’re
attempting
to integrate circles into a larger system of shapes, it may be that the
former fits your purposes better. In any case, as I (intentionally)
didn’t
specify in the original presentation, no one loses any points here.
Given that, let’s take a look at the solution from Jon G.. His
solution doesn’t include the aspect ratio correction, but that gives us
a
good look at the core algorithm. Here it is, holding back on the helper
methods for the moment:
class Circle
def initialize(radius)
@radius = radius.to_i
end
def draw
(0..@radius*2).each do |x|
(0..@radius*2).each do |y|
print distance_from_center(x,y).round == @radius ? '#' : '.'
end
puts
end
end
end
Circle.new(ARGV.shift).draw
A nice little Circle class encapsulates the code, storing only the
radius
during initialization. Some solutions, like Jon’s, didn’t keep a canvas
internally, while other solutions did. At this degree of simplicity,
keeping
a canvas or not is of little concern. In a larger application, speed and
memory concerns would be an important factor for keeping a canvas or
recalculating each draw.
To draw, two loops are used, nested, to iterate over a 2D grid. At each
cell, the cell’s distance from the center is computed and compared to
the
radius. When equal (i.e. on the circle), our hash symbol is output; when
off
the circle, a period (to represent empty space). Simple and quite
effective.
Now let’s look at distance_from_center:
def distance_from_center(x,y)
a = calc_side(x)
b = calc_side(y)
return Math.sqrt(a**2 + b**2)
end
def calc_side(z)
z < @radius ? (@radius - z) : (z - @radius)
end
Given coordinates (x, y) within the circumscribed square, those
coordinates
are adjusted relative to the center of the circle via calc_side. The
adjusted coordinates are the legs of a right triangle, with the
hypotenuse
calculated via the square-root of the sum of the squares of the legs.
Standard basic geometry.
I might make a couple minor changes, though, to Jon’s methods here, just
to
make things even simpler.
def draw
(-@radius..@radius).each do |x|
(-@radius..@radius).each do |y|
print distance_from_center(x,y).round == @radius ? '#' : '.'
end
puts
end
end
def distance_from_center(x,y)
return Math.sqrt(x**2 + y**2)
end
In draw, instead of looping from zero to the radius, loop from
negative
radius to positive radius. You cover the same range, and x and y are
now
exactly what a and b would have been as calculated by calc_side,
which
can now be removed.
It was good to see most folks supporting the aspect ratio, which
essentially
involved two parts. First, making sure that the canvas (or iterated
area)
was adjusted (in one dimension or the other; either choice was okay
without
a better specification). Second, when examining coordinates as the
canvas
was filled, the coordinates had to be also adjusted.
Finally, kudos to Andrea F. for bringing Bresenham into the mix.
Bresenham’s line algorithm is a well known algorithm in the
computer
graphics field. Not the first line drawer nor the last, it did the job
quite
well and was quite fast, using only integer numbers and operations – no
floating point. The technique is adaptable to more than just lines, as
Andrea’s solution shows.