  In 1986 William P. Thurston introduced the celebrated (asymmetric) Lipschitz
distance on the Teichmueller space of a (closed or punctured) surface. In this
paper we extend his work to the Teichmueller space of a surface with boundary
endowed the arc distance. In this new setting we construct a large family of
geodesics, which generalize Thurston's stretch lines. We prove that the
Teichmueller space of a surface with boundary, endowed with the arc distance,
is a geodesic metric space. Furthermore, the arc distance is induced by a
Finsler metric. As a corollary, we describe a new class of geodesics in the
Teichmueller space of a closed/punctured surface that are not stretch lines in
the sense of Thurston.
