  The ``analyst's traveling salesman theorem'' of geometric measure theory
characterizes those subsets of Euclidean space that are contained in curves of
finite length. This result, proven for the plane by Jones (1990) and extended
to higher-dimensional Euclidean spaces by Okikiolu (1991), says that a bounded
set $K$ is contained in some curve of finite length if and only if a certain
``square beta sum'', involving the ``width of $K$'' in each element of an
infinite system of overlapping ``tiles'' of descending size, is finite.
  In this paper we characterize those {\it points} of Euclidean space that lie
on {\it computable} curves of finite length by formulating and proving a
computable extension of the analyst's traveling salesman theorem. Our extension
says that a point in Euclidean space lies on some computable curve of finite
length if and only if it is ``permitted'' by some computable ``Jones
constriction''. A Jones constriction here is an explicit assignment of a
rational cylinder to each of the above-mentioned tiles in such a way that, when
the radius of the cylinder corresponding to a tile is used in place of the
``width of $K$'' in each tile, the square beta sum is finite. A point is
permitted by a Jones constriction if it is contained in the cylinder assigned
to each tile containing the point. The main part of our proof is the
construction of a computable curve of finite length traversing all the points
permitted by a given Jones constriction. Our construction uses the main ideas
of Jones's ``farthest insertion'' construction, but our algorithm for computing
the curve must work exclusively with the Jones constriction itself, because it
has no direct access to the (typically uncomputable) points permitted by the
Jones constriction.
