Computing the intersection between a rational parametric curve and a rational parametric surface

In this paper, we present an algorithm to compute the intersection between a rational curve and a rational surface. Evaluating the parametric curve into the matrix representation of the parametric surface for implicitization, we get a matrix with one variable. We find the intersection from the matrix with the theory of real root isolation of univariate functions without computing its determinant as we have done in Jia et al. (2022). We compare our method with the state-of-the-art methods in Gershon (2022); Luu Ba (2014). The given examples show that our algorithms are efficient.