The Data Singular and the Data Isotropic Loci for Affine Cones

The generic number of critical points of the Euclidean distance function from a data point to a variety is called the Euclidean distance degree. The two special loci of the data points where the number of critical points is smaller then the ED degree are called the Euclidean Distance Data Singular Locus and the Euclidean Distance Data Isotropic Locus. In this article we present connections between these two special loci of an affine cone and its dual cone.