Rotational-symmetry in a 3D scene and its 2D image

A 3D shape of an object is N-fold rotational-symmetrical if the shape is invariant for 360/N degree rotations about an axis. Human observers are sensitive to the 2D rotational-symmetry of a retinal image, but they are less sensitive than they are to 2D mirror-symmetry, which involves invariance to reflection across an axis. Note that perception of the mirror-symmetry of a 2D image and a 3D shape has been well studied, where it has been shown that observers are sensitive to the mirror-symmetry of a 3D shape, and that 3D mirror-symmetry plays a critical role in the veridical perception of a 3D shape from its 2D image. On the other hand, the perception of rotational-symmetry, especially 3D rotational-symmetry, has received very little study. In this study, we derive the geometrical properties of 2D and 3D rotational-symmetry and compare them to the geometrical properties of mirror-symmetry. Then, we discuss perceptual differences between mirror- and rotational- symmetry based on this comparison. We found that rotational-symmetry has many geometrical properties that are similar to the geometrical properties of mirror-symmetry, but note that the 2D projection of a 3D rotational-symmetrical shape is much more complex computationally than the 2D projection of a 3D mirror-symmetrical shape. This computational difficulty could make the human visual system less sensitive to the rotational-symmetry of a 3D shape than its mirror-symmetry.



Demo 1. A rotational-symmetrical pair of 3D curves produced from the pair of 2D curves in Figure 11A.

Demo 2. A rotational-symmetrical pair of 3D curves produced from the pair of 2D curves in Figure 12A.

Demo 3. A rotational-symmetrical pair of 3D curves produced from the pair of 2D curves in Figure 14A.

Demo 4. Orthographic views of the rotational-symmetrical pair of 3D curves in Demo 3.

Demo 5. A rotational-symmetrical pair of 3D curves produced from the pair of 2D curves in Figure 15.