5.8.1.5 Computing starting points for all branches

Starting points for tracing algebraic curves could be border points, turning points and singular points. Border points involve solution of a univariate polynomial equation, e.g. for border along , using (5.81)

Turning and singular point computation involve the first order partial derivatives:

Consequently, computation of turning points ( and ) is equivalent to solving a system of two nonlinear polynomial equations in two variables, and computation of singularities is equivalent to solving an overconstrained system of three nonlinear polynomial equations in two variables. Robust and efficient solution of these systems of nonlinear polynomial equations is addressed in Chap. 4.