The axial curvature for corank 1 singular surfaces

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For singular corank 1 surfaces in $\mathbb{R^3}$, we introduce a distinguished normal vector called the axial vector. Using this vector and the curvature parabola, we define a new type of curvature called the axial curvature, which generalizes the singular curvature for frontal type singularities. We then study contact properties of the surface with respect to the plane orthogonal to the axial vector and show how they are related to the axial curvature. Finally, for certain fold type singularities, we relate the axial curvature with the Gaussian curvature of an appropriate blow up.