Computer algebra and geometry – some interactions

Algebraic curves and surfaces have been studied intensively in algebraic geometry for centuries. Thus, there exists a huge amount of theoretical knowledge about these geometric objects. Recently, algebraic curves and surfaces play an important and ever increasing rôle in computer aided geometric design, computer vision, computer aided manufacturing, coding theory, and cryptography, just to name a few application areas. Consequently, theoretical results need to be adapted to practical needs. We need efficient algorithms for generating, representing, manipulating, analyzing, rendering algebraic curves and surfaces. Exact computer algebraic methods can be employed effectively for dealing with these geometric problems.