Improved Arithmetic of Complex Fans

Complex fans are sets of complex numbers whose magnitudes and angles range in closed intervals. The fact that the sum of two fans is a disordered shape gives rise to the need for computational methods to find the minimal enclosing fan. Cases where the sum of two fans contains the origin of the complex plane as a boundary point are of special interest. The result of the addition is then enclosed by circles in current methods, but under certain circumstances this turns out to be an overestimate. The focus of this article is the diagnosis and treatment of such cases.