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In this paper, the influence of the tangent-plane continuity of two surfaces along a common linkage curve on (1) the Dupin indicatrices of the surfaces in terms of squared Dupin indicatrices; and (2) the Gaussian and mean curvatures of the surfaces in terms of ratios of differences between Gaussian and mean curvatures is studied. Alternative sets of curvature continuity conditions in terms of intrinsic geometry of surfaces are developed. Relationships between the equalities of Gaussian and mean curvatures and curvature continuity of the surfaces along the linkage curve are studied. For two surfaces which are tangent-plane continuous along a linkage curve, the following two criteria individually guarantee their curvature continuity along the linkage curve: (1) The Gaussian Curvature Criterion: the Gaussian curvatures along the linkage curve are the same, if the linkage curve does not contain straight lines or binormal generators of the two surfaces; (2) The Mean Curvature Criterion: the mean curvatures are the same along the linkage curve. Also an alternative proof for the Linkage Curve Theorem is provided. These criteria can be used in shape interrogations, such as in visual detection of curvature discontinuity between two surfaces.