The number of directions determined by less than $q$ points

In this article we prove a theorem about the number of directions determined by less then $q$ affine points, similar to the result of Blokhuis et al. (in J. Comb. Theory Ser. A 86(1), 187-196, 1999) on the number of directions determined by $q$ affine points.
Comment: 12 pages. NOTICE: this is the authors' version of a work that was submitted for publication in J. Algebr. Combin. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms are not reflected in this document. The final publication is available at Springer via doi:10.1007/s10801-012-0357-1