A Comprehensive Derivation of the Conditions of Conformality, Equivalence, and Equidistance of Normal Cylindrical Projections

Isaac Ramos; Leonard Silveira; Vinícius Martins; Robert Silva; Andréa Seixas; Sílvio Garnés; Lucas Calado; Paulo Airoldi

doi:10.1590/SciELOPreprints.16769

##article.authors##

Isaac Ramos Federal University of Pampa https://orcid.org/0009-0009-9657-4060
- Conceptualization
- Data Curation
- Formal Analysis
- Investigation
- Methodology
- Project Administration
- Supervision
- Validation
- Visualization
- Writing – Original Draft Preparation
- Writing – Review & Editing
Leonard Silveira Federal University of Pampa https://orcid.org/0000-0003-1780-135X
- Conceptualization
- Formal Analysis
- Methodology
Vinícius Martins Federal University of Pampa https://orcid.org/0000-0001-8532-4247
- Formal Analysis
- Investigation
- Methodology
- Software
- Validation
- Visualization
Robert Silva Federal University of Pampa https://orcid.org/0000-0002-0277-5980
- Writing – Original Draft Preparation
- Writing – Review & Editing
- Supervision
- Investigation
Andréa Seixas Federal University of Pernambuco https://orcid.org/0000-0002-5879-4902
- Conceptualization
- Formal Analysis
- Investigation
- Methodology
- Writing – Original Draft Preparation
- Writing – Review & Editing
Sílvio Garnés Federal University of Pernambuco https://orcid.org/0000-0002-0098-6645
- Formal Analysis
- Methodology
- Writing – Review & Editing
Lucas Calado Federal University of Pernambuco https://orcid.org/0000-0002-2456-8288
- Conceptualization
- Investigation
- Writing – Original Draft Preparation
- Writing – Review & Editing
- Formal Analysis
- Methodology
Paulo Airoldi Federal University of Pampa https://orcid.org/0009-0009-9648-9915
- Methodology
- Writing – Review & Editing

DOI:

https://doi.org/10.1590/SciELOPreprints.16769

Keywords:

Mathematical Cartography, Mapping, Distortion

Abstract

Map projections play a significant role in representing the Earth’s surface by converting coordinates defined on a curved reference surface into coordinates on a plane. Since no single map projection can preserve all geometric attributes at the same time, distortions related to distance, shape, and area inevitably arise, making the analysis of map projection properties essential in mathematical cartography. This study develops a detailed and rigorous derivation of the conditions for conformality, equivalence, and equidistance based on the first Gaussian fundamental quantities. As an illustrative case, the normal cylindrical projection tangent to the equator is considered, due to its historical relevance and its connection with well-known projections such as Mercator, Lambert cylindrical equal-area, and Plate Carrée. The derivations are formulated through a differential-geometric approach, linking surface metrics directly to cartographic distortions. The obtained results show that the classical properties of map projections can be systematically described using the first Gaussian fundamental quantities, thereby offering an integrated interpretation of their metric behavior. In this way, the paper emphasizes the importance of differential geometry for cartography and provides a broad mathematical and pedagogical discussion of the geometric principles that support map projections.

Article in Portuguese: https://doi.org/10.1590/SciELOPreprints.14560