Abstract
The lambda-point map between two Legendre plane curves, which is a map from the plane into the plane, is introduced. The singularity of this map is studied through this paper and many known plane map singularities are realized as special cases of this construction. Precisely, the corank one and corank two singularities of the lambda-point map between two Legendre plane curves are investigated and the geometric conditions for this map to have corank one singularities, such as fold, cusp, swallowtail, lips, and beaks are obtained. Additionally, the geometric conditions for the lambda-point map to have a sharksfin singularity, which is a corank two singularity, are obtained.