A two-dimensional shape is generally represented with line drawings or object contours in a digital image. Shapes can be divided into two types, namely ordered and unordered shapes. An ordered shape is an ordered set of points, while an unordered shape is an unordered set. As a result, each type typically uses different attributes to define the local descriptors involved in representing the local distributions of points sampled from the shape. Throughout this paper, we focus on unordered shapes. Since most local descriptors of unordered shapes are not scale-invariant, we usually make the shapes in an image data set the same size through scale normalization, before applying shape matching procedures. Shapes obtained through scale normalization are suitable for such descriptors if the original whole shapes are similar. However, they are not suitable if parts of each original shape are drawn using different scales. Thus, in this paper, we present a scale-invariant descriptor constructed by von Mises distributions to deal with such shapes. Since this descriptor has the merits of being both scale-invariant and a probability distribution, it does not require scale normalization and can employ an arbitrary measure of probability distributions in matching shape points. In experiments on shape matching and retrieval, we show the effectiveness of our descriptor, compared to several conventional descriptors.