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.

<div data-canvas-width="82.4750337835114"><p>The squinches isa very versatile construction type in the field of stonework. It is defined in its most elementary case as the covering of a triangular area with a conical surface, but, nevertheless, there are an infinite number of variants increasing the complexity of the layout and its execution.The studied case in this paper is the development of the layout of the "square conical vault" (actually, a cantilevered vault), in the treatise on stonework by Tomás Vicente Tosca, published in 1727. This author's work reflectsthe tradition of the local area (which comes mainly from the brilliant episode of the 15th century in Valencia), but it is also part of the pre-illustrated cultural trends of the time. The traces that appear are the result of combining ancient graphic habits with the development, to a certain extent, already scientific, of geometry and mathematics.All this leads, in the case of the "conical-square vault", to the development of a fundamentally descriptive trace, result of a mentality that is more analytical than practical, but that does not manage to emancipate itself from the old building tradition, as this study makes evident, and also the detected graphic habits.The research is also based on other examples of a documentary nature, such as the cantilevered domes of Vandelvira, Philibert de l'Orme, Gelabert, and on a built example; the squinch of the palace of the Generalitat de Valencia.</p></div><div data-canvas-width="145.94326164079826"> </div>

No abstract