Criteria for sharksfin and deltoid singularities from the plane into the plane and their applications

We give criteria for sharksfin and deltoid singularities from the plane into the plane. We also give geometric meanings for the conditions in the criterion of a sharksfin. As applications, we investigate such singularities appearing on an orthogonal projection of a Whitney umbrella, and a sharksfin appearing on planar motions with 2-degrees of freedom.