The development of generalisation (simplification) methods for the geometry of features in digital cartography in most cases involves the improvement of existing algorithms without their validation with respect to the similarity of feature geometry before and after the process. It also consists of the assessment of results from the algorithms, i.e., characteristics that are indispensable for automatic generalisation. The preparation of a fully automatic generalisation for spatial data requires certain standards, as well as unique and verifiable algorithms for particular groups of features. This enables cartographers to draw features from these databases to be used directly on the maps. As a result, collected data and their generalised unique counterparts at various scales should constitute standardised sets, as well as their updating procedures. This paper proposes a solution which consists in contractive self-mapping (contractor for scale s = 1) that fulfils the assumptions of the Banach fixed-point theorem. The method of generalisation of feature geometry that uses the contractive self-mapping approach is well justified due to the fact that a single update of source data can be applied to all scales simultaneously. Feature data at every scale s < 1 are generalised through contractive mapping, which leads to a unique solution. Further generalisation of the feature is carried out on larger scale spatial data (not necessarily source data), which reduces the time and cost of the new elaboration. The main part of this article is the theoretical presentation of objectifying the complex process of the generalisation of the geometry of a feature. The use of the inherent characteristics of metric spaces, narrowing mappings, Lipschitz and Cauchy conditions, Salishchev measures, and Banach theorems ensure the uniqueness of the generalisation process. Their application to generalisation makes this process objective, as it ensures that there is a single solution for portraying the generalised features at each scale. The present study is dedicated to researchers concerned with the theory of cartography.

中文翻译：

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特征几何通用化的统一方法

这 在大多数情况下，数字制图要素几何的泛化（简化）方法的开发涉及对现有算法的改进，而无需就处理前后的特征几何相似性进行验证。它还包括对算法结果的评估，即对于自动概括必不可少的特征。对于空间数据的全自动概括的准备需要某些标准，以及针对特定特征组的独特且可验证的算法。这使制图人员可以从这些数据库中绘制要素，以直接在地图上使用。因此，收集的数据及其在各种规模上的广义唯一对应物应构成标准化的集合及其更新程序。本文提出了一种解决方案，该解决方案包括可压缩自映射（标度s = 1的承包商），该解决方案满足Banach不动点定理的假设。由于可以同时将原始数据的单个更新同时应用于所有比例，因此使用收缩自映射方法的特征几何一般化方法是很合理的。s <1的每个尺度上的特征数据都通过压缩映射进行了概括，从而得出了独特的解决方案。在更大范围的空间数据（不一定是源数据）上进行特征的进一步概括，这减少了新工作的时间和成本。本文的主要部分是对特征几何图形概化的复杂过程进行客观化的理论介绍。使用度量空间的内在特征，缩小映射，Lipschitz和Cauchy条件，Salishchev度量和Banach定理可确保泛化过程的唯一性。它们在一般化中的应用使此过程成为目标，因为它确保有一个单一的解决方案来描绘每个尺度的一般化特征。本研究致力于与制图理论有关的研究人员。