The Hough transform is commonly used for detecting linear features within an image. A line is mapped to a peak within parameter space corresponding to the parameters of the line. By analysing the shape of the peak, or peak locus, within parameter space, it is possible to also use the line Hough transform to detect or analyse arbitrary (non-parametric) curves. It is shown that there is a one-to-one relationship between the curve in image space, and the peak locus in parameter space, enabling the complete curve to be reconstructed from its peak locus. In this paper, we determine the patterns of the peak locus for closed curves (including circles and ellipses), linear segments, inflection points, and corners. It is demonstrated that the curve shape can be simplified by ignoring parts of the peak locus. One such simplification is to derive the convex hull of shapes directly from the representation within the Hough transform. It is also demonstrated that the parameters of elliptical blobs can be measured directly from the Hough transform.