This paper explores a novel mathematical approach to extract archaeological insights from ensembles of similar artifact shapes. We show that by considering all the shape information in a find collection, it is possible to identify shape patterns that would be difficult to discern by considering the artifacts individually or by classifying shapes into predefined archaeological types and analyzing the associated distinguishing characteristics. Recently, series of high-resolution digital representations of artifacts have become available, and we explore their potential on a set of 3D models of ancient Greek and Roman sundials, with the aim of providing alternatives to the traditional archaeological method of ``trend extraction by ordination'' (typology). In the proposed approach, each 3D shape is represented as a point in a shape space -- a high-dimensional, curved, non-Euclidean space. By performing regression in shape space, we find that for Roman sundials, the bend of the sundials' shadow-receiving surface changes with the location's latitude. This suggests that, apart from the inscribed hour lines, also a sundial's shape was adjusted to the place of installation. As an example of more advanced inference, we use the identified trend to infer the latitude at which a sundial, whose installation location is unknown, was placed. We also derive a novel method for differentiated morphological trend assertion, building upon and extending the theory of geometric statistics and shape analysis. Specifically, we present a regression-based method for statistical normalization of shapes that serves as a means of disentangling parameter-dependent effects (trends) and unexplained variability.

翻译：本文探索了一种新颖的数学方法，用于从相似文物形状的集合中提取考古学见解。我们证明，通过考虑考古发现集合中的所有形状信息，可以识别出那些难以通过单独观察文物、或通过将形状分类为预定义的考古类型并分析相关区分特征来辨别的形状模式。近年来，高分辨率文物数字表征序列的出现为研究提供了可能，我们以一组古希腊与古罗马日晷的三维模型为基础，探索其潜力，旨在为传统考古学中“通过排序提取趋势”（类型学）的方法提供替代方案。在所提出的方法中，每个三维形状被表示为形状空间中的一个点——该空间为高维、弯曲且非欧几里得空间。通过在形状空间中进行回归分析，我们发现：对于古罗马日晷而言，其受光面的弯曲程度随安装地点纬度的变化而变化。这表明，除了刻划的时线外，日晷的形状本身也根据安装地点进行了调整。作为更高级推理的示例，我们利用识别出的趋势推断了一个安装地点未知的日晷所放置的纬度。此外，我们还推导出一种用于差异化形态趋势断言的新方法，该方法在几何统计学与形状分析理论基础上进行了构建与拓展。具体而言，我们提出了一种基于回归的形状统计归一化方法，作为分离参数相关效应（趋势）与未解释变异的手段。