Generation of real algebraic loci by complex detours

We analyze the locus generation algorithm of the dynamic geometry software Cinderella, and examine how it uses complex detours to resolve singularities. We show that the algorithm terminates if its step width is small enough. In this case, the algorithm is independent of the orientation of its complex detours. We introduce a variation of the algorithm that generates entire real connected components of real algebraic loci. Several examples illustrate its use for organic generation of real algebraic loci. Another example shows how we can apply the algorithm to simulate mechanical linkages. Apparently, the use of complex detours produces physically reasonable motion of such linkages.