Abstract: The procedure of converting the parametric equation of a curve or a surface into the implicit equation is called implicitization. Implicitization is a classic elimination problem in algebraic geometry and has found important applications in computer aided geometric design — an inter-discipline in computer science and scientific computing. The classic methods for implicitization were reviewed first, and then the state of the art techniques in curve and surface implicitization were surveryed, especially the recent advances of the moving curves/surfaces method and μ basis method developed in the last decade.

Key words: parametric curve, parametric surface, implicitization, resultant, Groebner basis, Wus method, moving curves/surfaces, μ-basis