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It is essential for robotic tactile sensors to provide capability of shape discrimination from static touch. For this purpose, the principal curvatures and corresponding principal directions play important roles which define the local shape of the object. This paper proposes a novel acoustic tactile sensing system which identifies a set of principal curvatures and directions on the object surface by utilizing acoustical waves. The major difficulty, however, existing in estimating these parameters is that the wavefront reflected by the paraboloidal surface cannot be described in the linear combination of the plane-waves such that the non-linearity exists between time-of-flight (TOF) and the parameters defining the surface. Avoiding the difficulties, the proposed sensing system utilizing the differences of TOF between primary and secondary reflections.