The finite element method has for some time been a popular choice for approximating PDEs over a geometry domain. We usually use a FEM mesh that matches the geometry description. By FEM mesh we mean the mesh where we find the solution to the PDE. In order to use finite element method where the geometry description does not match the FEM mesh, we need a technique to modify the FEM around the geometry description. An essential part of such techniques is to find the intersection points between the geometry description and the FEM mesh. We say that FEM mesh is cut when the geometry description does not match the FEM mesh, this is also referred to as CutFEM. In this thesis, we have contributed to a suitable algorithm for finding intersection points between the geometry description and the FEM mesh. We base our algorithm on Newton's method to detect these intersection points. The results show that the intersection points are very accurate for regular triangles. The intersection points are found directly on the limit surface of Loop's subdivision surfaces.