Flowers and Other Curves

In the previous section we saw that is was useful for methods to accept functions as parameters. In this section we'll see that it is useful for methods to return functions.

We've seen all the basic steps we need to make our flowers. Now we just need to know the curve that makes the flower shape! The shape I used is known as the rose curve. One example is shown in Figure hof:rose7.

An example of the rose curve.

The code for the parametric curve that gives this shape is below.

// Parametric equation for rose with k = 7
val rose7 = (angle: Angle) =>
  Point((angle * 7).cos * 200, angle)

You may wonder why I called this function rose7. It's because we can vary the shape by changing the value 7 to something else. We could make a method or function to which we pass the value of this parameter and this function would return a particular rose curve. Here's that idea in code.

def rose(k: Int): Angle => Point =
  (angle: Angle) => Point((angle * k).cos * 200, angle)

The rose method describes a family of curves. They all look similar, and we create individuals by choosing a particular value for the parameter k. In Figure hof:rose we show more rose curves, this time with k as 5, 8, and 9 respectively.

Examples of rose curves, with the parameter k chosen as 5, 8, or 9.

Let's look at some other interesting curves. In Figure hof:lissajous we show examples of a family of curves called Lissajous curves.

Examples of Lissajous curves, with the parameters a and b chosen as 1, 2, or 3.

The code for this is

def lissajous(a: Int, b: Int, offset: Angle): Angle => Point =
  (angle: Angle) =>
    Point(100 * ((angle * a) + offset).sin, 100 * (angle * b).sin)

The examples in Figure hof:lissajous use values of a and b of 1, 2, or 3, and the offset set to 90 degrees.

There are an unlimited number of functions we could use to create interesting curves. Let's see one more example, this time of what known as an epicycloid. An epicycloid is produced when we trace a point on a circle rotating around another circle. We can stack circles on top of circles, and change the speed at which they rotate, to produce many different curves. For our example we are going to fix the number and radius of the circles and allow their speed of rotation to vary. Here is the code:

def epicycloid(a: Int, b: Int, c: Int): Angle => Point =
  (angle: Angle) =>
    (Point(75, angle * a).toVec + Point(32, angle * b).toVec + Point(15, angle * c).toVec).toPoint

You might notice this code converts points to vectors and back again. This is a little technical detail (we cannot add points but we can add vectors) that isn't important if you aren't familiar with vectors.

In Figure hof:epicycloid we see three examples created by choosing the parameters a, b, c, as (1, 6, 14), (7, 13, 25), and (1, 7, -21) respectively.

Examples of epicycloid curves, with the parameters chosen as (1, 6, 14), (7, 13, 25), and (1, 7 -21).

Exercise {-}

We now have a lot of tools to play with. Your task here is simply to use some of the ideas we've just seen to make an image that you like. For inspiration you could use the image that we started the chapter with, but don't let it constrain you if you think of something else to explore.

Higher Order Methods and Functions→