  We introduce a new geometric approach to Sturmian words by means of a mapping
that associates certain lines in the n x n -grid and sets of finite Sturmian
words of length n. Using this mapping, we give new proofs of the formulas
enumerating the finite Sturmian words and the palindromic finite Sturmian words
of a given length. We also give a new proof for the well-known result that a
factor of a Sturmian word has precisely two return words.
