A 3D shape of an object is N-fold rotational-symmetric 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 paper, 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 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.

中文翻译：

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3D 场景中的旋转对称及其 2D 图像

如果对象的 3D 形状在绕轴 360/N 度旋转时保持不变，则该对象的 3D 形状是 N 重旋转对称的。人类观察者对视网膜图像的 2D 旋转对称性敏感，但他们对 2D 镜像对称性的敏感度要低，后者涉及跨轴反射的不变性。请注意，对 2D 图像和 3D 形状的镜像对称性的感知已经得到了很好的研究，其中已经表明观察者对 3D 形状的镜像对称性很敏感，并且 3D 镜像对称性起着至关重要的作用从其 2D 图像中真实感知 3D 形状。另一方面，旋转对称的感知，尤其是 3D 旋转对称，却很少受到研究。在本文中，我们推导出 2D 和 3D 旋转对称的几何特性，并将它们与镜像对称的几何特性进行比较。然后，我们基于这种比较讨论镜像对称和旋转对称之间的感知差异。我们发现旋转对称具有许多与镜像对称几何特性相似的几何特性，但请注意，3D 旋转对称形状的 2D 投影在计算上比 3D 镜像对称形状的 2D 投影更复杂. 这种计算难度可能使人类视觉系统对 3D 形状的旋转对称性比其镜像对称性更不敏感。我们发现旋转对称具有许多与镜像对称几何特性相似的几何特性，但请注意，3D 旋转对称形状的 2D 投影在计算上比 3D 镜像对称形状的 2D 投影更复杂. 这种计算难度可能使人类视觉系统对 3D 形状的旋转对称性比其镜像对称性更不敏感。我们发现旋转对称具有许多与镜像对称几何特性相似的几何特性，但请注意，3D 旋转对称形状的 2D 投影在计算上比 3D 镜像对称形状的 2D 投影更复杂. 这种计算难度可能使人类视觉系统对 3D 形状的旋转对称性比其镜像对称性更不敏感。