Assignment 5: Producing Playful Polygons

Write a procedure, (turtle-polygon! turtle side-length sides), that uses a turtle to draw a regular polygon with the specified number of sides, with each side of the specified length.

Important: Your procedure must return the turtle to its original position and angle. Your procedure must not change the turtle's brush or color.

Hint: Use repeat. The turtle must turn a total of 360 degrees to return to its original angle.

For example,

(turtle-polygon! t 100 3)

(turtle-polygon! t 100 4)

(turtle-polygon! t 60 5)

(turtle-polygon! t 40 6)

Write a procedure, (turtle-spin-polygon! turtle side-length sides angle copies), that draws the given number of copies of the specified polygon by calling your turtle-polygon! procedure, with the turtle turned an angle of angle between polygons.

Important: As in problem one, your procedure must return the turtle to its original position and orientation.

For example,

(turtle-spin-polygon! t 50 4 15 10)

(turtle-spin-polygon! t 50 4 20 5)

(turtle-spin-polygon! t 50 4 5 20)

(turtle-spin-polygon! t 50 4 -30 5)

Write a procedure, (turtle-scale-polygon! turtle initial-side-length sides scale-factor copies), that draws the given number of copies of the specified polygon, with each copy drawn with a side length scale-factor times the the previous side length.

For example, if the initial side length is ten, and the scale factor is two, this procedure would draw polygons with side lengths 10, 20, 40, 80, 160, ....

Similarly, if the initial side length is 100, and the scale factor is 0.9, the procedure would draw polygons with side lengths 100, 90, 81, 72.9, ....

Important: As in the previous problems, your procedure must return the turtle to its original position and orientation.

Hint: The expt function will be useful for finding the ratio of each polygon's side length to the original side length. For example, for scale factor two, the ratios would be 1, 2, 4, 8, 16, 32, and so on.

Here are some visual issues.

(turtle-scale-polygon! t 1 5 2 8)

(turtle-scale-polygon! t 1 5 1.2 30)

(turtle-scale-polygon! t 100 5 0.9 20)

A potential deficiency of each of the prior two procedures is that multiple polygons are joined at a vertex. The problem is, of course, that our basic polygon procedure draws polygons starting at a particular point, rather than centered at a certain point.

Copy the following procedure to your definitions pane.

(define turtle-centered-polygon! 
  (lambda (turtle radius sides)
    (let ((interior-angle (/ (* 180 (- sides 2)) sides)))
      (turtle-up! turtle)
      (turtle-forward! turtle radius)
      (turtle-down! turtle)
      (turtle-turn! turtle (- 180 (/ interior-angle 2)))
      (turtle-polygon! turtle (* 2 radius (sin (/ pi sides))) sides)
      (turtle-turn! turtle (/ interior-angle 2))
      (turtle-up! turtle)
      (turtle-forward! turtle radius)
      (turtle-turn! turtle 180)
      (turtle-down! turtle))))    

Experiment with this procedure to learn what it does. (Hint: Note that this procedure assumes your turtle-polygon! draws the first side of the polygon before turning.)

a. Document this procedure using the 6Ps.

b. Explain why each expression in the procedure's body is necessary to achieve the result that you see. That is, explain each step of the algorithm in English.

Hint: You may find it useful to comment out each expression using a semicolon, one at a time, to see how the result of leaving out that expression differs from the desired result.)

Write the procedure (turtle-spin-centered-polygon! turtle radius sides angle copies).

This procedure should be very similar to turtle-spin-polygon! from problem 2, except that it will call the turtle-centered-polygon! procedure from Problem 4 instead of your turtle-polygon! procedure from Problem 1.

Your first polygon procedure draws the outline of a polygon. What if, instead of drawing a simple polygon, we wanted to position the turtle at each of the vertices of a polygon as preparation for some other action? If that other action is provided in the form of a procedure that takes a turtle as a parameter and leaves the turtle in the same position it started in, the task is similar to our original polygon creating procedure.

Write a procedure (turtle-action-polygon! turtle side-length sides action!), that applies the procedure action! to turtle at the vertex of a polygon with the specified number of sides, with each side of the specified length.

Important: As in the previous problems, your procedure must return the turtle to its original position and orientation.

(turtle-action-polygon! t 65 5 (lambda (turtle) (turtle-spin-polygon! turtle 20 4 15 5)))

(turtle-action-polygon! t 8 50 (lambda (turtle) (turtle-polygon! turtle 5 20 )))

Write a program that systematically generates an interesting image composed of a series of polygons. In addition to spinning and scaling polygons, you should incorporate at least one additional element, such as changing the polygons' positions, changing colors or brushes, or even changing the number of sides between different polygons.