Dynamic stability of cutting processes such as milling and turning is mainly restricted by the phenomenon of the regenerative effect, causing self-excited vibration, which is well known as machine-tool chatter. With the semidiscretization method for periodic delay-differential equations, there exists an appropriate method for determining the stability boundary curve in the domain of technological parameters. The stability boundary is implicitly defined as a level set of a function on the parameter domain, which makes the evaluation computationally expensive when using complete enumeration. In order to reduce computational cost, we first investigate two types of curve tracking algorithms finding them not appropriate for computing stability charts as they may get stuck at cusp points or near-branch zones. We then present a new curve tracking method, which overcomes these difficulties and makes it possible to compute stability boundary curves very efficiently.