The Poisson kernel-based (PKB) distribution is a flexible and numerically convenient family on the unit sphere used for model-based clustering, uniformity testing, and directional data analysis. Available random variate generators rely on rejection from a von Mises-Fisher, angular central Gaussian, or projected Saw envelope and therefore require a rejection loop, tuning or root-finding, or a univariate setup phase. We show that none is needed. A uniformly random line through the interior point $ρμ$ meets the sphere in two points; selecting one with probability proportional to the opposite segment length returns exactly the target density. The sampler is exact, rejection- and tuning-free, uses one uniform direction and one Bernoulli draw, and costs $O(d)$ operations. We place the distribution in the power-Saw family of densities proportional to $\|x-ρμ\|^{-ξ}$ and show that, for $d\geq3$, the same construction is exact for one further exponent. This introduces the Newton kernel-based (NKB) distribution, whose normalizing constant is identically one. Across a wide grid of dimensions, concentrations, and sample sizes, the direct sampler is faster than the better of two previously recommended methods in every case, with the largest gains at moderate concentration.
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