HasCl Laboratory Exercise

By working through the steps in this handout, you will develop a HasCl program to draw a picture made up of many overlapping squares of different sizes and colors.

The following data type, which is defined in the Graphics module in Funnie, represents a graphical object to be displayed on the screen:
```
data Graphic = Square(Num, Num, Num, Color) | Over(Graphic, Graphic) | Blank
```
This says that a Graphic value is either a single square or the combination of one Graphic over another, or nothing at all. A square needs four parameters--the value Square(x, y, w, c) describes a square whose lower-left corner is at the point (x, y), with width w and color c. The first three parameters, x, y, and w, are all numbers expressing positions or distances relative to the coordinate system of the graphics window: the lower-left corner is at (- 50, - 50), and the upper-right corner is at (50, 50), regardless of the actual size of the window. The color, c, is specified as a value of the following data type, which is also defined in the Graphics module:
```
data Color = RGB(Num, Num, Num)
```
The value RGB(r, g, b) specifies a color with red, green, and blue components given by the numbers r, g, and b, respectively, where each component ranges from 0 (completely absent) to 255 (full strength). The Graphics module also defines some useful constants of type Color; for example, red is the value RGB(255,0,0) (look in the module browser in Funnie to see what other colors are defined).
For this item, all we will do is display a single square. Following the description above, we can form a green square, 30 units on each side, with lower-left corner at (10, 20), by using the expression Square(10, 20, 30, RGB(0, 255, 0)). If you type this in, the system should pop-up a graphics window containing the square.
For testing, it will be convenient to define some sample squares. Enter each of the following in a separate definition window:
```
redGiant = Square(-20, -10, 50, RGB(255, 0, 0))
whiteDwarf = Square(10, 30, 20, RGB(255, 255, 255))
purpleMedium = Square(0, -20, 35, RGB(128, 0, 128))
```
Now you can easily draw a square by entering its name, e.g., redGiant, in a Funnie expression window. However, this only lets you draw one square at a time. To combine several squares in one picture, we need the Over constructor of the Graphic data type, as follows: Over(redGiant, whiteDwarf). Notice how the squares are combined, and see what happens if you reverse the order of the squares.
Now let's create a list of Squares. We will need a function which takes a list of squares and combines them all into a single Graphic. As usual when working with a list of things, we will define the function by recursion. Here is the base case, which uses the special value Blank to produce a blank picture:
```
showSquareList([ ]) = Blank
```
The recursive case will have the following form:
```
showSquareList([s : ss]) =
```
When this pattern matches, s will be the first square on the list, and ss will be a list of the remaining squares. Write an appropriate right-hand side for this case of the function. The square s is already a Graphic; you will need to use showSquareList to produce another Graphic from ss. These two Graphics will then need to be combined into one with Over. When you have defined the function, try it out with showSquareList([redGiant, whiteDwarf]).
We will also want a function to generate a list of squares, so that we don't have to type them all in by hand. Here is an example:
```
diagonalSquares(x, y, d, rgb, 0) = [ ]
diagonalSquares(x, y, d, rgb, n) = [Square(x, y, d, rgb) :
                                    diagonalSquares(x+d/2, y+d/2, d, rgb, n-1)]
```
After entering this function, evaluate showSquareList(diagonalSquares(-30, -30, 10, white, 9)). The cases for this function mean that diagonalSquares(x, y, d, rgb, n) will produce a list of n squares (because the list is empty when n is 0, and it gets one extra element for each recursive call as n counts down to 0). The first one will have its corner at (x,y), with width d and color rgb. Succeeding squares will be offset by adding d/2 to the x and y coordinates of the corner; the effect will be that each square will be centered on the upper-right corner of the previous one. You should get a picture that looks like a white staircase.
A simple modification to the previous code gives us a chain of squares of different sizes. Create a new function named vanishingSquares which is similar to diagonalSquares except replace the size argument d in the recursive call (to vanishingSquares, of course) with the expression d*3/4. To try this out, you will probably want to start with a larger initial square; something like this should work: showSquareList(vanishingSquares(-30, -30, 20, white, 7))
Now we can approach our original goal of creating an interesting picture by replacing the single recursive call in the previous item with four separate calls, one at each of the four corners of the central square. Each of these recursive calls will produce a list of squares, so we will need to append all of the lists together, using ++. Here is a skeleton of the code for you to fill in:
```
squareDesign(x, y, d, rgb, 0) = 
squareDesign(x, y, d, rgb, n) = [Square(x, y, d, rgb) :
                                 squareDesign(x-d/4, y-d/4, d/2, rgb, n-1) ++
                                 squareDesign(x-d/4, y+3*d/4, d/2, rgb, n-1) ++
                                 squareDesign(                             ) ++
                                 squareDesign(                               )]
```
When you have successfully compiled the finished function, try evaluating the following expression: showSquareList(squareDesign(-20, -20, 40, white, 4)). You can change the 4 to 5 to draw one more level, but it will take too long if you try to do 6 or more levels (each level has four times as many squares, so there are 1024 squares at level 6).
The last step will be to change the color as well as the size. Modify the recursive calls so that the upper-left and lower-right corners have only half the red component of the current square, while the upper-right and lower-left corners have only half the green component; leave the blue component untouched. The easiest way to do this is to write two auxilliary functions, lessRed and lessGreen, each of type (Color) -> Color. Here is one:
```
lessRed(RGB(r, g, b)) = RGB(r/2, g, b)
```
You will need to enter this and a similar definition for lessGreen, then write a new function similar to squareDesign that calls these functions in the appropriate places. Call this new function colorDesign, and test it.
As a final problem, how could you change the code to draw the larger squares on the bottom, instead of on top? Whichever function needs to be modified, name the new version by appending a 2; for example, if you modify colorDesign, then your new function should be named colorDesign2.