Objects in our environment are subject to manifold transformations, either of the physical objects themselves or of the object images on the retina. Despite drastic effects on the objects' physical appearances, we are often able to identify stable objects across transformations and have strong subjective impressions of the transformations themselves. This suggests the brain is equipped with sophisticated mechanisms for inferring both object constancy, and objects' causal history. We employed a dot-matching task to study in geometrical detail the effects of rigid transformations on representations of shape and space. We presented an untransformed ‘base shape’ on the left side of the screen and its transformed counterpart on the right (rotated, scaled, or both). On each trial, a dot was superimposed at a given location on the contour (Experiment 1) or within and around the shape (Experiment 2). The participant's task was to place a dot at the corresponding location on the right side of the screen. By analyzing correspondence between responses and physical transformations, we tested for object constancy, causal history, and transformation of space. We find that shape representations are remarkably robust against rotation and scaling. Performance is modulated by the type and amount of transformation, as well as by contour saliency. We also find that the representation of space within and around a shape is transformed in line with the shape transformation, as if shape features establish an object-centered reference frame. These findings suggest robust mechanisms for the inference of shape, space and correspondence across transformations.