Scale-invariant features are widely used for image retrieval and shape classification. The curvature of a planar curve is a fundamental feature and it is geometrically invariant with respect it the coordinate system. The curvature-based feature varies in position when multi-scale analysis is performed. Therefore, it is important to recognize the scale in order to detect the feature point. Numerous shape descriptors based on contour shapes have been developed in the field of pattern recognition and computer vision. A curvature scale-space (CSS) representation cannot be applied to a contour fragment and requires the tracking of feature points. In a gradient-based curvature computation, although the gradient computation considers the scale, the curvature is normalized with respect to not the scale but the contour length. The scale-invariant feature transform algorithm that detects feature points from an image solves similar problems by using the difference of Gaussian (DoG). It is difficult to apply the SIFT algorithm to a planar curve for feature extraction. In this paper, an automatic scale detection method for a contour fragment is proposed. The proposed method detects the appropriate scales and their positions on the basis of the difference of curvature (DoC) without the tracking of feature points. To calculate the differences, scale-normalized curvature is introduced. An advantage of the DoC algorithm is that the appropriate scale can be obtained from a contour fragment as a local feature. It then extends the application area. The validity of the proposed method is confirmed by experiments. The proposed method provides the most stable and robust scales of feature points among conventional methods such as curvature scale-space and gradient-based curvature.