The distributions of toroidal data, often viewed as an extension of circular distributions, do not consider the intrinsic geometry of a curved torus. For the first time, Diaconis et al. (2013)[Diaconis, P., Holmes, S., & Shahshahani, M. (2013). Sampling from a manifold. Advances in modern statistical theory and applications: a Festschrift in honor of Morris L. Eaton, 10, 102-125.] introduce uniform distribution on the surface of a curved torus with respect to its surface area. But the suggested acceptance-rejection method of sampling from it rejects approximately half of the data. We propose a probabilistic transformation for sampling from the same distribution without losing data. In addition, we introduce a new genesis of random samples from some popular circular distributions using histogram-based acceptance-rejection sampling that uses a very thin envelope. The idea leads to generalizing for sampling from distributions on the surface of a curved torus with a high acceptance rate.Apart from reducing computational cost in the inferential study of different toroidal distributions, uniform sampling from the surface of a curve torus will be helpful to understand any unknown distribution on it.

翻译：环形数据的分布通常被视为圆分布的一种扩展，但未考虑弯曲环面的内在几何结构。Diaconis等人（2013）[Diaconis, P., Holmes, S., & Shahshahani, M. (2013). Sampling from a manifold. Advances in modern statistical theory and applications: a Festschrift in honor of Morris L. Eaton, 10, 102-125.] 首次引入了相对于弯曲环面表面积的均匀分布。然而，他们所提出的接受-拒绝采样方法会拒绝约一半的数据。我们提出了一种概率变换方法，可在不损失数据的情况下从同一分布中采样。此外，我们引入了一种基于直方图的接受-拒绝采样方法，利用极薄包络线从一些常见的圆分布中生成随机样本的新方法。这一思路可推广到从弯曲环面表面分布中采样，且具有高接受率。除了降低不同环形分布推断研究的计算成本外，从弯曲环面表面均匀采样还有助于理解该表面上的任何未知分布。