# 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. By composing functions we mean to create a function that connects the output of one component function to the input of another component function.

Here's an example. We use the `andThen` method to create a function that connects the output of the first function to the input of the second function.

``````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))

In Figure hof:composed we see the output of the program

``````val star = Image
.star(100, 50, 5)
.fillColor(Color.fireBrick)
.strokeColor(Color.dodgerBlue)
.strokeWidth(7.0)
.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 function 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 =

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 when 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: Sample {-}

If we want to draw this function we'll need to change `sample` so the parametric has type `Angle => Image` instead of `Angle => Point`. In other words we want the following skeleton.

``````def sample(samples: Int, curve: Angle => Image): Image =
???``````

Implement `sample`.

<div class="solution">

The answer is a small modification of the original `sample`. We drop the `dot` parameter and the type of the `curve` parameter changes. The rest follows from this.

``````def sample(samples: Int, curve: Angle => Image): Image = {
val step = Angle.one / samples
def loop(count: Int): Image = {
val angle = step * count
count match {
case 0 => Image.empty
case n =>
curve(angle).on(loop(n - 1))
}
}

loop(samples)
}``````

</div>

Once we've implemented `sample` we can start drawing pictures. For example, in Figure hof:growing-circle 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(n-1, 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(n-1, singleShape))
}``````

Now we can re-use 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 _)``````

### Exercises {-}

#### The Colour and the Shape {-}

Starting with the code below we are going to write color and shape functions to produce the image shown in Figure hof:colors-and-shapes. ``````def concentricShapes(count: Int, singleShape: Int => Image): Image =
count match {
case 0 => Image.empty
case n => singleShape(n).on(concentricShapes(n-1, 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:

1. write an appropriate definition of `singleShape` for each of the three shapes in the target image; and
1. call `concentricShapes` three times, passing in the appropriate definition of `singleShape` each time and putting the results `beside` 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 Figure hof:colors-and-shapes-step1. 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 Figure hof:colors-and-shapes-step2 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) => ???``````

<div class="solution"> 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(n-1, 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)
}

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)
}

concentricShapes(10, rainbowCircle)
.beside(
.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(n-1, singleShape)
}

def colored(shape: Int => Image, color: Int => Color): Int => Image =
(n: Int) =>
shape(n).strokeWidth(10).strokeColor(color(n))

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))

concentricShapes(10, colored(circle, spinning))
.beside(
.beside(concentricShapes(10, colored(square, spinning)))
)
//   l = On(
//     t = StrokeColor(
//       image = StrokeWidth(image = Circle(d = 340.0), width = 10.0),
//       color = HSLA(
//         h = Angle(9.42477796076938),
//         s = Normalized(get = 0.5),
//         l = Normalized(get = 0.5),
//         a = Normalized(get = 1.0)
//       )
//     ),
//     b = On(
//       t = StrokeColor(
//         image = StrokeWidth(image = Circle(d = 316.0), width = 10.0),
//         color = HSLA(
//           h = Angle(8.901179185171081),
//           s = Normalized(get = 0.5),
//           l = Normalized(get = 0.5),
//           a = Normalized(get = 1.0)
//         )
//       ),
//       b = On(
//         t = StrokeColor(
//           image = StrokeWidth(image = Circle(d = 292.0), width = 10.0),
//           color = HSLA(
//             h = Angle(8.377580409572781),
//             s = Normalized(get = 0.5),
//             l = Normalized(get = 0.5),
//             a = Normalized(get = 1.0)
//           )
//         ),
//         b = On(
//           t = StrokeColor(
//             image = StrokeWidth(image = Circle(d = 268.0), width = 10.0),
//             color = HSLA(
//               h = Angle(7.853981633974483),
//               s = Normalized(get = 0.5),
//               l = Normalized(get = 0.5),
//               a = Normalized(get = 1.0)
//             )
//           ),
//           b = On(
//             t = StrokeColor(
//               image = StrokeWidth(image = Circle(d = 244.0), width = 10.0),
//               color = HSLA(
//                 h = Angle(7.330382858376184),
//                 s = Normalized(get = 0.5),
//                 l = Normalized(get = 0.5),
//                 a = Normalized(get = 1.0)
// ...``````

</div>

#### More Shapes {-}

The `concentricShapes` methods takes an `Int => Image` function, and we can construct such as function using `sample`, the parametric curves we created earlier, and the various utilities we have created along the way. There is an example is Figure hof:concentric-dotty-circle. The code to create this is below.

``````def dottyCircle(n: Int): Image =
sample(
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.

Exercises→