Problem 1 (The Ridiculous)
Write a Scheme program which creates a graphics object which is a reasonable likeness of Cartman, Stan, Kyle, or Kenny. It does not have to be perfect, just recognizable. I got these depictions off the web. You may be able to find better ones. If you are a person with good taste and find Southpark offensive, then you are welcome to draw Snoopy, Charlie Brown, Linus, or Lucy instead. Hint: Figuring out how to draw an ellipse (or a portion thereof) at an arbitrary position and orientation will get you 90% of the way there. The following code will return an ellipse with width-to-height ratio, a/b:
(define pi (acos -1))
(define rad->deg
(lambda (rad)
(* rad (/ 360 (* 2 pi)))))
(define deg->rad
(lambda (deg)
(* deg (/ (* 2 pi) 360))))
(define sind
(lambda (x)
(sin (deg->rad x))))
(define cosd
(lambda (x)
(cos (deg->rad x))))
(define atand
(lambda (dy dx)
(rad->deg (atan dy dx))))
(define ellipse
(lambda (a b)
(ellipse-helper a b 360)))
(define ellipse-helper
(lambda (a b t)
(if (= t 0)
(adjoin)
(adjoin
(straight 1)
(bend (- (atand (* b (cosd t))
(* (- a) (sind t)))
(atand (* b (cosd (sub1 t)))
(* (- a) (sind (sub1 t))))))
(ellipse-helper a b (sub1 t))))))
Problem 2
The following program draws a spiral:
(define spiral
(lambda (angle len n)
(if (= n 0)
(adjoin)
(adjoin (bend angle)
(straight len)
(spiral angle (add1 len) (sub1 n))))))
Rewrite spiral so that it is tail-recursive. You are free to
use any helper functions you may require.
Problem 3
Write a Scheme program which, given an s-expression, draws the corresponding pair-tree. Hint: This is a typical tree-recursion on pairs, i.e., the same recursion on car and cdr. You might find a function similiar to the following helpful for assembling the tree:
(define tree-step
(lambda (trunk left-branch right-branch)
(adorn trunk
(adjoin (bend -20) left-branch)
(adjoin (bend 20) right-branch))))
To get full credit for Problem 3 (i.e., to get an A), you will want to
make your trees as nice looking as possible. One way to do this is to
decrease the trunk length and branch angle as the recursion depth
increases. Use the plumbing graphics function, text, to label
the leaves of the tree.
Problem 4 (The Sublime)
Write a Scheme program which draws either the Gosper-Peano Curve (more challenging) or the Sierpinski Arrowhead (less challenging). To get full credit for Problem 4 (i.e., to get an A) you should do the Gosper-Peano Curve. The generators for these curves can be found on p. 70 and p. 142 (respectively) of Benoit Mandelbrot's The Fractal Geometry of Nature. Hint: You may use my code for the Monkey Tree as a model for the Peano-Gosper Curve and Barak Pearlmutter's code for the Koch Snowflake as a model for the Sierpinski Arrowhead.The Monkey Tree (The Fractal Geometry of Nature, p. 31)
(define d (/ (sqrt 3) 3))
(define monkey-step
(lambda (side rside)
(adjoin
(bend -60)
(mirror side)
(mirror rside)
(bend 60)
side
(bend 60)
(mirror rside)
(bend 150)
(scale-size d (mirror rside))
(scale-size d (mirror side))
(bend -60)
(scale-size d rside)
(bend -60)
(scale-size d rside)
(scale-size d side)
(bend -90)
rside
side)))
(define rmonkey-step
(lambda (side rside)
(adjoin
rside
side
(bend 90)
(scale-size d rside)
(scale-size d side)
(bend 60)
(scale-size d side)
(bend 60)
(scale-size d (mirror rside))
(scale-size d (mirror side))
(bend -150)
(mirror side)
(bend -60)
rside
(bend -60)
(mirror side)
(mirror rside)
(bend 60))))
(define monkey-tree
(lambda (n len)
(if (= n 0)
(straight len)
(monkey-step (monkey-tree (sub1 n) len)
(rmonkey-tree (sub1 n) len)))))
(define rmonkey-tree
(lambda (n len)
(if (= n 0)
(straight len)
(rmonkey-step (monkey-tree (sub1 n) len)
(rmonkey-tree (sub1 n) len)))))
The Koch Snowflake (The Fractal Geometry of Nature, p. 43)
(define koch-step
(lambda (side angle)
(adjoin side
(bend (- angle))
side
(bend (* 2 angle))
side
(bend (- angle))
side)))
(define koch
(lambda (n len angle)
(if (= n 0)
(straight len)
(koch-step (koch (- n 1) len angle)
angle))))