The maximum number and its distribution of singular points for parametric piecewise algebraic curves

The piecewise algebraic curve, as the zero set of a bivariate spline function, is a generalization of the classical algebraic curve. Based on the previous method presented by Lai et al. (2009), we show that computing singular points of parametric piecewise algebraic curves amounts to solving parametric piecewise polynomial systems. In this article, we give a method to compute the maximum number and its distribution of singular points for a given parametric piecewise algebraic curve. This method also produces necessary and sufficient conditions of its parameters must be satisfied. An illustrated example shows that our method is flexible.