We investigate the classification of quasihomogeneous polynomials in two variables with real coefficients under semialgebraic bi-Lipschitz equivalence in a neighborhood of the origin in \(\mathbb R^2\) . Building on the work of Birbrair, Fernandes, and Panazzolo, our approach is based on reducing the problem to the Lipschitz classification of associated single variable polynomial functions, called height functions. We establish conditions under which semialgebraic bi-Lipschitz equivalence of quasihomogeneous polynomials corresponds to the Lipschitz equivalence of their height functions. To achieve this, we develop the theory of \(\beta \) -transforms and inverse \(\beta \) -transforms. As an application, we examine a family of quasihomogeneous polynomials previously used by Henry and Parusiński to show that the bi-Lipschitz equivalence of analytic function germs \((\mathbb R^2,0)\rightarrow (\mathbb R,0)\) admits continuous moduli. Our results show that semialgebraic bi-Lipschitz equivalence of real quasihomogeneous polynomials in two variables also admits continuous moduli, thereby contributing to a finer understanding of the complexity of Lipschitz classification problems in real algebraic geometry.
