A Univariate Resultant-Based Algorithm For Computing Offsets To Algebraic Curves

An offset to an algebraic curve is the point locus that is driven out normally from the original curve in equal distance and direction.. Offsets are widely used in geometric modeling, engineering and NC machining etc., so the problems about offset curves become one of researching hotspot in Computer Aided Geometric Design. Offsets to rational curves is not always rational since the square root involved in the expression of the unit normal vector. In order to construct rational offsets, people proposed the concept of PH curves. Offsets to PH curves is rational by using the properties of PH curves. On the other hand, algebraic curves as a basic element of the geometric modeling, we need to discuss the algorithm for computing offsets to algebraic curves and their properties. At present, the approach for computing the offset based on Groebner basis or multivariate resultant. Although the method based on Groebner basis to give exact representations for offset curves and surfaces, it is time consuming and runs efficiently low. At the same time, it is hard to analyze the properties of offsets while we use Groebner basis method. The method based on multivariate resultant runs efficiently. However, it always suffers the extraneous factors.In this paper we present an efficient univariate resultant-based method for computing offsets to plane algebraic curves. Moreover, based on the analysis of some special points (singular points and points at infinity) and the properties of the univariate resultant, the extraneous factors can also be removed. Thus, the expression of offsets to algebraic curves is obtained.