Revision of 2.2.4 Example: A Picture Language from 17 July 2009 - 11:25pm

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This section presents a simple language for drawing pictures that illustrates the power of data abstraction and closure, and also exploits higher-order procedures in an essential way. The language is designed to make it easy to experiment with patterns such as the ones in figure 2.9, which are composed of repeated elements that are shifted and scaled.[22] In this language, the data objects being combined are represented as procedures rather than as list structure. Just as cons, which satisfies the closure property, allowed us to easily build arbitrarily complicated list structure, the operations in this language, which also satisfy the closure property, allow us to easily build arbitrarily complicated patterns.

The picture language

When we began our study of programming in section 1.1, we emphasized the importance of describing a language by focusing on the language’s primitives, its means of combination, and its means of abstraction. We’ll follow that framework here.

To combine images, we use various operations that construct new painters from given painters. For example, the beside operation takes two painters and produces a new, compound painter that draws the first painter’s image in the left half of the frame and the second painter’s image in the right half of the frame. Similarly, below takes two painters and produces a compound painter that draws the first painter’s image below the second painter’s image. Some operations transform a single painter to produce a new painter. For example, flip-vert takes a painter and produces a painter that draws its image upside-down, and flip-horiz produces a painter that draws the original painter’s image left-to-right reversed.

Figure 2.12 shows the drawing of a painter called wave4 that is built up in two stages starting from wave:

(define wave2 (beside wave (flip-vert wave)))
(define wave4 (below wave2 wave2))

Figure 2.12:  Creating a complex figure, starting from the wave painter of figure 2.10.

(define wave2                         (define wave4   (beside wave (flip-vert wave)))       (below wave2 wave2))

In building up a complex image in this manner we are exploiting the fact that painters are closed under the language’s means of combination. The beside or below of two painters is itself a painter; therefore, we can use it as an element in making more complex painters. As with building up list structure using cons, the closure of our data under the means of combination is crucial to the ability to create complex structures while using only a few operations.

Once we can combine painters, we would like to be able to abstract typical patterns of combining painters. We will implement the painter operations as Scheme procedures. This means that we don’t need a special abstraction mechanism in the picture language: Since the means of combination are ordinary Scheme procedures, we automatically have the capability to do anything with painter operations that we can do with procedures. For example, we can abstract the pattern in wave4 as

(define (flipped-pairs painter)
  (let ((painter2 (beside painter (flip-vert painter))))
    (below painter2 painter2)))

and define wave4 as an instance of this pattern:

(define wave4 (flipped-pairs wave))

We can also define recursive operations. Here’s one that makes painters split and branch towards the right as shown in figures 2.13 and  2.14:

(define (right-split painter n)
  (if (= n 0)
      painter
      (let ((smaller (right-split painter (- n 1))))
        (beside painter (below smaller smaller)))))

Figure 2.13:  Recursive plans for right-split and corner-split.

right-split n                   corner-split n

We can produce balanced patterns by branching upwards as well as towards the right (see exercise 2.44 and figures 2.13 and  2.14):

(define (corner-split painter n)
  (if (= n 0)
      painter
      (let ((up (up-split painter (- n 1)))
            (right (right-split painter (- n 1))))
        (let ((top-left (beside up up))
              (bottom-right (below right right))
              (corner (corner-split painter (- n 1))))
          (beside (below painter top-left)
                  (below bottom-right corner))))))

Figure 2.14:  The recursive operations right-split and corner-split applied to the painters wave and rogers. Combining four corner-split figures produces symmetric square-limit designs as shown in figure 2.9.

(right-split wave 4)         (right-split rogers 4)                  (corner-split wave 4)         (corner-split rogers 4)

By placing four copies of a corner-split appropriately, we obtain a pattern called square-limit, whose application to wave and rogers is shown in figure 2.9:

(define (square-limit painter n)
  (let ((quarter (corner-split painter n)))
    (let ((half (beside (flip-horiz quarter) quarter)))
      (below (flip-vert half) half))))

(define right-split (split beside below))
(define up-split (split below beside))

(define (make-frame origin edge1 edge2)
  (list origin edge1 edge2))

(define (make-frame origin edge1 edge2)
  (cons origin (cons edge1 edge2)))