Functions for Structures

Chapter: Functions for Structures

In the last chapter when we computed the magnitude of a we squared each element of the vector before adding. Now consider an operation called normalization. Normalizing a vector produces a new vector which points in the same direction, but with a magnitude (length) of 1. Mathematicians refer to vectors of length one as unit vectors. Normalizing is an operation often used in writing low-level computer graphics programs. To normalize a vector we simply divide each element of the vector by the magnitude.

Scheme has a handy function called map which takes a function and a list, invokes the function on each element of the list, and returns the list of results. Some examples:

> (map car '((a b) (c d) (e f)))
(a c e)
> (map ((curry +) 2) '(1 5 2 4 3))
(3 7 4 6 5)
> (define vec '(10 11 4 8))
> (map add1 vec)
(11 12 5 9)
> (map (lambda (x) (* x 2)) vec)
(20 22 8 16)

Q. 8: What is the result of (map (lambda (k) k) '(9 10 11 12))

Q. 9

Here's a tricky one. What is the result of

(map (lambda (f) ((compose ((curry *) 2) f) 3))
     (cons add1 (cons sub1 (cons (lambda (z) (* z 3)) ()))))

Now, we can use map to normalize the vector (10.5 10.1 3.5 2.0).

> (define vec '(10.5 10.1 3.5 2.0))
> (define the-mag (apply mag vec))
> (map (lambda (item) (/ item the-mag)) vec)
(0.6946030627129489
   0.6681419936572175
   0.23153435423764962
   0.13230534527865692)
>

To accomplish this without using map, we would use the normal flat-recursion technique:

(define the-mag ...)

(define normalize-v
  (lambda (v)
    (if (null? v)
	'()
	(cons (/ (car v) the-mag) (normalize (cdr v))))))

Q. 10: Although map is provided as a Scheme primitive, it is easy to write. Can you write it?

Exercise 2

Define a procedure norm which takes a list of reals as its argument and returns a normalized list. Recall normalizing is done by dividing each element of the list by the magnitude. Use mag from Exercise 2 to compute the magnitude and map to perform the normalizing.

Exercise 3

Define a procedure and-map which, like map takes a procedure and a list. The procedure is applied to each element in turn and returns either #t or #f depending on the element of the list it is applied to. and-map returns #t if all results returned true and returns #f on the first occurrence of a false result.

> (and-map atom? '(a b c () g))
#t
> (and-map pair? '((a b) 2 (g h) 4 5))
#f

rms@cs.oberlin.edu