set-6

1. Implement this algorithm as a procedure reduce-terms that takes two term lists n and d as arguments and returns a list nn, dd, which are n and d reduced to lowest terms via the algorithm given above. Also write a procedure reduce-poly, analogous to add-poly, that checks to see if the two polys have the same variable. If so, reduce-poly strips off the variable and passes the problem to reduce-terms, then reattaches the variable to the two term lists supplied by reduce-terms.

2. Define a procedure analogous to reduce-terms that does what the original make-rat did for integers:

   (define (reduce-integers n d)
     (let ((g (gcd n d)))
       (list (/ n g) (/ d g))))

   and define reduce as a generic operation that calls apply-generic to dispatch to either reduce-poly (for polynomial arguments) or reduce-integers (for scheme-number arguments). You can now easily make the rational-arithmetic package reduce fractions to lowest terms by having make-rat call reduce before combining the given numerator and denominator to form a rational number. The system now handles rational expressions in either integers or polynomials. To test your program, try the example at the beginning of this extended exercise:

   (define p1 (make-polynomial 'x '((1 1)(0 1))))
   (define p2 (make-polynomial 'x '((3 1)(0 -1))))
   (define p3 (make-polynomial 'x '((1 1))))
   (define p4 (make-polynomial 'x '((2 1)(0 -1))))
   
   (define rf1 (make-rational p1 p2))
   (define rf2 (make-rational p3 p4))
   
   (add rf1 rf2)

   See if you get the correct answer, correctly reduced to lowest terms.

Make changes to the square limit of wave shown in figure 2.9 by working at each of the levels described above. In particular:

1. Add some segments to the primitive wave painter of exercise 2.49 (to add a smile, for example).
2. Change the pattern constructed by corner-split (for example, by using only one copy of the up-split and right-split images instead of two).
3. Modify the version of square-limit that uses square-of-four so as to assemble the corners in a different pattern. (For example, you might make the big Mr. Rogers look outward from each corner of the square.)