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 six-P style of documentation.

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 two procedures, (turtle-spin-centered-polygon! turtle radius sides angle copies) and (turtle-scale-centered-polygon! turtle initial-radius sides scale-factor copies).

These procedures should be very similar to turtle-spin-polygon! and turtle-scale-polygon! from problems 2 and 3, except that they will call the turtle-centered-polygon! procedure from Problem 4 instead of your turtle-polygon! procedure from Problem 1.

Write a procedure, (turtle-spin-and-scale-centered-polygon! turtle initial-radius sides angle scale-factor copies), that makes the given number of copies of the polygon, scaling and spinning each subsequent polygon. Again, call the turtle-centered-polygon! procedure from Problem 4.

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.