Higher Order Methods and Functions
In previous sections we have seen the utility of passing functions to methods and returning functions from methods. In this section we'll see the usefulness of function composition.
Composition, in the mathematical rather than artistic sense, means creating something more complex by combining simpler parts. We could say we compose the numbers 1 and 1, using addition, to produce 2. Composing functions means creating a function that connects the output of one component function to the input of another component function. Written in terms of types, function composition joins functions of type A => B
and B => C
to produce a function type A => C
. In Scala we use the andThen
method to do this.
Here's an example. We start by defining two functions. The first adds a drop shadow to an Image
. The second places an Image
beside a copy of itself mirrored around the yaxis.
val dropShadow = (image: Image) =>
image.on(image.strokeColor(Color.black).fillColor(Color.black).at(5, 5))
val mirrored = (image: Image) =>
image.beside(image.transform(Transform.horizontalReflection))
Both functions have type Image => Image
, so we can compose them together. We do this using the andThen
method to create a function that connects the output of the first function to the input of the second function.
val composed = mirrored.andThen(dropShadow)
Below we see the image created by the program
val star = Image
.star(100, 50, 5)
.fillColor(Color.fireBrick)
.strokeColor(Color.dodgerBlue)
.strokeWidth(7.0)
dropShadow(star)
.beside(mirrored(star))
.beside(composed(star))
This shows how the composed function applies the output of the first function to the second function: we first mirror the star and then add a drop shadow.
Let's see how we can apply function composition to our examples of parametric curves. One limitation of the parametric cures we've created so far is that their size is fixed. For example when we defined parametricCircle
we fixed the radius at 200.
def parametricCircle(angle: Angle): Point =
Point.polar(200, angle)
What if we want to create circles of different radius? We could use a method that returns a function like so.
def parametricCircle(radius: Double): Angle => Point =
(angle: Angle) => Point.polar(radius, angle)
This would be a reasonable solution but we're going to explore a different approach using our new tool of function composition. Our approach will be this:

each parametric curve will be of some default size that we'll loosely define as "usually between 0 and 1"; and

we'll define a function
scale
that will change the size as appropriate.
A quick example will make this more concrete. Let's redefine parametricCircle
so the radius is 1.
val parametricCircle: Angle => Point =
(angle: Angle) => Point(1.0, angle)
Now we can define scale
.
def scale(factor: Double): Point => Point =
(point: Point) => Point(point.r * factor, point.angle)
We can use function composition to create circles of different sizes as follows.
val circle100 = parametricCircle.andThen(scale(100))
val circle200 = parametricCircle.andThen(scale(200))
val circle300 = parametricCircle.andThen(scale(300))
We can use the same approach for our spiral, adjusting the function slightly so the radius of the spiral varies from about 0.36 at 0 degrees to 1 at 360 degrees.
val parametricSpiral: Angle => Point =
(angle: Angle) => Point(Math.exp(angle.toTurns  1), angle)
Then we can compose with scale
to produce spirals of different size.
val spiral100 = parametricSpiral.andThen(scale(100))
val spiral200 = parametricSpiral.andThen(scale(200))
val spiral300 = parametricSpiral.andThen(scale(300))
What else can we do with function composition?
Our parametric functions have type Angle => Point
. We can compose these with functions of type Point => Image
and with this setup we can make the "dots" from which we build our images depend on the point.
Here's an example where the dots get bigger as the angle increases.
val growingDot: Point => Image =
(pt: Point) => Image.circle(pt.angle.toTurns * 20).at(pt)
val growingCircle = parametricCircle
.andThen(scale(100))
.andThen(growingDot)
Exercise: Drawing Curves
If we want to draw this function we'll need to change drawCurve
so the parameter has type Angle => Image
instead of Angle => Point
. In other words we want the following skeleton.
def drawCurve(points: Int, curve: Angle => Image): Image =
???
Implement drawCurve
.
The answer is a small modification of the original drawCurve
. We drop the marker
parameter and the type of the curve
parameter changes. The rest follows from this.
def drawCurve(points: Int, curve: Angle => Image): Image = {
val step = Angle.one / points
def loop(count: Int): Image = {
val angle = step * count
count match {
case 0 => Image.empty
case n =>
curve(angle).on(loop(n  1))
}
}
loop(points)
}
Having implemented drawCurve
we can start drawing pictures. For example, below we have the output of growingCircle
above.
More Uses of Composition
At this point we can do a lot. Let's see another example. Remember the concentric circles exercise we used as an example:
def concentricCircles(count: Int, size: Int): Image =
count match {
case 0 => Image.empty
case n => Image.circle(size).on(concentricCircles(n1, size + 5))
}
This pattern allows us to create many different images
by changing the use of Image.circle
to another shape.
However, each time we provide a new replacement for Image.circle
,
we also need a new definition of concentricCircles
to go with it.
We can make concentricCircles
completely general by supplying
the replacement for Image.circle
as a parameter.
Here we've renamed the method to concentricShapes
, as we're no longer restricted to drawing circles,
and made singleShape
responsible for drawing an appropriately sized shape.
def concentricShapes(count: Int, singleShape: Int => Image): Image =
count match {
case 0 => Image.empty
case n => singleShape(n).on(concentricShapes(n1, singleShape))
}
Now we can reuse the same definition of concentricShapes
to produce plain circles, squares of different hue,
circles with different opacity, and so on.
All we have to do is pass in a suitable definition of singleShape
:
// Passing a function literal directly:
val blackCircles: Image =
concentricShapes(10, (n: Int) => Image.circle(50 + 5*n))
// Converting a method to a function:
def redCircle(n: Int): Image =
Image.circle(50 + 5*n).strokeColor(Color.red)
val redCircles: Image =
concentricShapes(10, redCircle _)
Exercise: The Colour and the Shape
Starting with the code below we are going to write color and shape functions to produce the image shown below.
def concentricShapes(count: Int, singleShape: Int => Image): Image =
count match {
case 0 => Image.empty
case n => singleShape(n).on(concentricShapes(n1, singleShape))
}
The concentricShapes
method is equivalent to the
concentricCircles
method from previous exercises.
The main difference is that we pass in
the definition of singleShape
as a parameter.
Let's think about the problem a little. We need to do two things:
 write an appropriate definition of
singleShape
for each of the three shapes in the target image; and
 call
concentricShapes
three times, passing in the appropriate definition ofsingleShape
each time and putting the resultsbeside
one another.
Let's look at the definition of the singleShape
parameter in more detail.
The type of the parameter is Int => Image
,
which means a function that accepts an Int
parameter and returns an Image
.
We can declare a method of this type as follows:
def outlinedCircle(n: Int): Image =
Image.circle(n * 10)
We can convert this method to a function, and pass it to concentricShapes
to create
an image of concentric black outlined circles:
concentricShapes(10, outlinedCircle _)
This produces the output shown in below.
The rest of the exercise is just a matter of copying, renaming, and customising this function to produce the desired combinations of colours and shapes:
def circleOrSquare(n: Int) =
if(n % 2 == 0) Image.rectangle(n*20, n*20) else Image.circle(n*10)
concentricShapes(10, outlinedCircle).beside(concentricShapes(10, circleOrSquare))
See below for the output.
For extra credit, when you've written your code to
create the sample shapes above, refactor it so you have two sets
of base functions: one to produce colours and one to produce shapes.
Combine these functions using a combinator as follows,
and use the result of the combinator as an argument to concentricShapes
def colored(shape: Int => Image, color: Int => Color): Int => Image =
(n: Int) => ???
The simplest solution is to define three singleShapes
as follows:
def concentricShapes(count: Int, singleShape: Int => Image): Image =
count match {
case 0 => Image.empty
case n => singleShape(n).on(concentricShapes(n1, singleShape))
}
def rainbowCircle(n: Int) = {
val color = Color.blue.desaturate(0.5.normalized).spin((n * 30).degrees)
val shape = Image.circle(50 + n*12)
shape.strokeWidth(10).strokeColor(color)
}
def fadingTriangle(n: Int) = {
val color = Color.blue.fadeOut((1  n / 20.0).normalized)
val shape = Image.triangle(100 + n*24, 100 + n*24)
shape.strokeWidth(10).strokeColor(color)
}
def rainbowSquare(n: Int) = {
val color = Color.blue.desaturate(0.5.normalized).spin((n * 30).degrees)
val shape = Image.rectangle(100 + n*24, 100 + n*24)
shape.strokeWidth(10).strokeColor(color)
}
val answer =
concentricShapes(10, rainbowCircle)
.beside(
concentricShapes(10, fadingTriangle)
.beside(concentricShapes(10, rainbowSquare))
)
However, there is some redundancy here:
rainbowCircle
and rainbowTriangle
, in particular,
use the same definition of color
.
There are also repeated calls to strokeWidth(10)
and
strokeColor(color)
that can be eliminated.
The extra credit solution factors these out into their own functions
and combines them with the colored
combinator:
def concentricShapes(count: Int, singleShape: Int => Image): Image =
count match {
case 0 => Image.empty
case n => singleShape(n) on concentricShapes(n1, singleShape)
}
def colored(shape: Int => Image, color: Int => Color): Int => Image =
(n: Int) =>
shape(n).strokeWidth(10).strokeColor(color(n))
def fading(n: Int): Color =
Color.blue.fadeOut((1  n / 20.0).normalized)
def spinning(n: Int): Color =
Color.blue.desaturate(0.5.normalized).spin((n * 30).degrees)
def size(n: Int): Double =
100 + 24 * n
def circle(n: Int): Image =
Image.circle(size(n))
def square(n: Int): Image =
Image.square(size(n))
def triangle(n: Int): Image =
Image.triangle(size(n), size(n))
val answer =
concentricShapes(10, colored(circle, spinning))
.beside(
concentricShapes(10, colored(triangle, fading))
.beside(concentricShapes(10, colored(square, spinning)))
)
Exercise: More Shapes
The concentricShapes
methods takes an Int => Image
function, and we can construct such as function using drawCurve
, the parametric curves we created earlier, and the various utilities we have created along the way. There is an example below.
The code to create this is below.
def dottyCircle(n: Int): Image =
drawCurve(
72,
parametricCircle.andThen(scale(100 + n * 24)).andThen(growingDot)
)
concentricShapes(10, colored(dottyCircle, spinning))
Use the techniques we've seen so far to create a picture of your choosing (perhaps similar to the flower with which we started the chapter). No solution here; there is no right or wrong answer.