There is currently a gap in theory for point patterns that lie on the surface of objects, with researchers focusing on patterns that lie in a Euclidean space, typically planar and spatial data. Methodology for planar and spatial data thus relies on Euclidean geometry and is therefore inappropriate for analysis of point patterns observed in non-Euclidean spaces. Recently, there has been extensions to the analysis of point patterns on a sphere, however, many other shapes are left unexplored. This is in part due to the challenge of defining the notion of stationarity for a point process existing on such a space due to the lack of rotational and translational isometries. Here, we construct functional summary statistics for Poisson processes defined on convex shapes in three dimensions. Using the Mapping Theorem, a Poisson process can be transformed from any convex shape to a Poisson process on the unit sphere which has rotational symmetries that allow for functional summary statistics to be constructed. We present the first and second order properties of such summary statistics and demonstrate how they can be used to construct a test statistics to determine whether an observed pattern exhibits complete spatial randomness or spatial preference on the original convex space. We compare this test statistic with one constructed from an analogue $L$-function for inhomogeneous point processes on the sphere. A study of the Type I and II errors of our test statistics are explored through simulations on ellipsoids of varying dimensions.

当前，在对象表面上的点模式在理论上存在空白，研究人员将注意力集中在位于欧几里得空间中的模式，通常是平面和空间数据。因此，平面和空间数据的方法论依赖于欧几里得几何，因此不适用于分析在非欧几里得空间中观察到的点模式。近来，对球体上的点图案的分析有了扩展，但是还有许多其他形状尚未探索。这部分是由于定义平稳性的挑战由于缺乏旋转和平移的等距性，存在于这种空间上的点处理。在这里，我们为在三个维度上凸形上定义的泊松过程构造函数汇总统计量。使用映射定理，泊松过程可以从单位球面上的任何凸形转换为泊松过程，该泊松过程具有旋转对称性，可以构造功能汇总统计量。我们介绍了此类摘要统计信息的一阶和二阶属性，并演示了如何使用它们来构建测试统计信息，以确定观察到的模式在原始凸空间上是否展现出完整的空间随机性或空间偏好。我们将该测试统计量与通过类似物构建的统计量进行比较$\mathrm{大号}$函数用于球上的非均匀点过程。通过对不同尺寸的椭球进行模拟，探索了对我们的测试统计数据的I型和II型误差的研究。