The present thesis is the outcome of studies conducted by the author from November, 2008 to September, 2011, under the supervision of Dr. Bernard Mourrain. Our motivation has been to combine modern algebraic tools with emerging techniques in computer aided geometric design in order, on one hand, to bring closer the fields of algebraic geometry and applied or algorithmic geometry and, on the other hand, to take a step towards advancing today's CAD/CAM industry. This exciting idea evolved in the previous European research projects GAIA and GAIA II (2000-2005) and continued in 2008 with the launch of the SAGA project. The text is developed in a way that chapters can be read individually, since each one contains extended preliminary details. However, the reader is invited to follow the natural flow of the text, in order to experience the bottom-up approach, that we also adopted in implementing these techniques.