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Given a parametric polynomial curve $\gamma:[a,b]\rightarrow \mathbb{R}^n$,how can we sample a random point $\mathfrak{x}\in \mathrm{im}(\gamma)$ in sucha way that it is distributed uniformly with respect to the arc-length?Unfortunately, we cannot sample exactly such a point-even assuming we canperform exact arithmetic operations. So we end up with the following question:how does the method we choose affect the quality of the approximate sample weobtain? In practice, there are many answers. However, in theory, there arestill gaps in our understanding. In this paper, we address this question fromthe point of view of complexity theory, providing bounds in terms of the sizeof the desired error.

On the Error of Random Sampling: Uniformly Distributed Random Points on Parametric Curves