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Adversary and invisibility are two fundamental but conflict characters of adversarial perturbations. Previous adversarial attacks on 3D point cloud recognition have often been criticized for their noticeable point outliers, since they just involve an "implicit constrain" like global distance loss in the time-consuming optimization to limit the generated noise. While point cloud is a highly structured data format, it is hard to constrain its perturbation with a simple loss or metric properly. In this paper, we propose a novel Point-Cloud Sensitivity Map to boost both the efficiency and imperceptibility of point perturbations. This map reveals the vulnerability of point cloud recognition models when encountering shape-invariant adversarial noises. These noises are designed along the shape surface with an "explicit constrain" instead of extra distance loss. Specifically, we first apply a reversible coordinate transformation on each point of the point cloud input, to reduce one degree of point freedom and limit its movement on the tangent plane. Then we calculate the best attacking direction with the gradients of the transformed point cloud obtained on the white-box model. Finally we assign each point with a non-negative score to construct the sensitivity map, which benefits both white-box adversarial invisibility and black-box query-efficiency extended in our work. Extensive evaluations prove that our method can achieve the superior performance on various point cloud recognition models, with its satisfying adversarial imperceptibility and strong resistance to different point cloud defense settings. Our code is available at: https://github.com/shikiw/SI-Adv.