The restoration of paintings on elements in cultural heritage buildings (fundamentally, churches) involves two structural problems: capturing the geometry of the construction element and its development. In many cases, the geometries are regular (e.g., cylinders, spheres, elliptical domes). However, there are cases in which the elements cannot be adapted to any known geometry, much less one that can be mathematically developed. The development of surfaces becomes essential for the restoration of paintings over "flat elements" (over which work is performed on the ground) that are subsequently transferred to the real surface (ceilings). The mathematical transformations that allow regular geometries to be developed are widely known (cartographic projections). However, when the geometry is irregular, there is no development. This study presents a new methodology based on differential rectification and its application for the development of oculi in the Los Santos Juanes Church (Valencia), whose geometry is completely irregular both in shape and as a result of construction defects (and damage caused by fire). The present study focuses on the restoration of paintings damaged by fire.

The National Institute of Archaeology and History manages the documentation and preservation of the patrimony of Mexico, a country with an area of ??1.959.000 square kilometers, which is almost four times the size of Spain. To attend on all this patrimony, INAH has several departments. The National Coordination of Historical Monuments manages the monuments and architecture built from the colonial era. Inside the Coordination, the Technical Support Division manages the teams and the tools necessary for the survey. El Instituto Nacional de Arqueología e Historia gestiona la documentación y conservación del patrimonio de México un país con una extensión de 1.959000 kilómetros cuadrados, que casi cuadriplica la extensión de España. Para atender todo este Patrimonio el INAH cuenta con diversos departamentos. La Coordinación Nacional de Monumentos Históricos gestiona los monumentos y arquitecturas construidos a partir de la época virreinal. Dentro de la Coordinación, la Dirección de Apoyo Técnico gestiona los equipos e instrumentos necesarios para el levantamiento.