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This article presents a geometric exploration of the reconstruction of a model of the three-dimensional space that Leonardo da Vinci had before him or imagined when he created The Last Supper. The purpose is to reveal the geometrical principles inherent in this work of art, as well as to propose an invert geometrical method, from 2D to 3D space reconstruction.

This exploration begins by describing the method used to construct a perspective drawing, in which the role of three elements is highlighted: the base line of the picture plane, on which the perspective drawing is made, the height of the horizon and how it is depicted on the picture plane—which is connected to the height of the artist’s station point and, generally speaking, to the work of art’s viewing height—and, lastly, the distance from the picture plane at which the artist positions the viewer in order to better observe the work of art. These three concepts, namely the picture plane’s base line, horizon and viewing distance, play a decisive role in the work of art. Once the geometric meanings of these concepts have been presented, using reverse geometric perspective processes we conduct a geometric abstraction on da Vinci’s painting in order to determine its base line, horizon line and the viewing distance the artist has set. The article ends with a presentation of the three-dimensional space model resulting from the preceding process and an analysis of whether the perspective image of the model, from the viewpoint set by the artist, coincides with the actual work of art.

A perspective drawing is the central projection of an object or space on a plane (

To define the horizon line [

on horizontal line α also meets the picture plane at 1, since that is already on the picture plane. Thus, point 1’s perspective will be itself. We therefore need the perspective of one more point to determine the perspective of straight line α. Let us say that we choose to construct the perspective of a point on α that is an infinite distance away from the viewer, in other words, α’s point at infinity. For this purpose, let us consider the visual ray that connects the observer’s viewpoint with the point at infinity. This means that the ray will be parallel to α and will intersect the picture plane at a point we refer to as α’s vanishing point Φ1. Line α’s vanishing point is the perspective of the point along its length that is an infinite distance away from the viewer. If we then join Φ1 with 1, we will have the perspective image on the picture plane of α from 1 to infinity. Let us now consider the horizontal plane that is at the observer’s eye level. The line formed where this plane intersects the picture plane is called the horizon line. All the vanishing points of the horizontal lines in physical space are located along this line. In classical perspective drawing, horizontal lines have vanishing points along the horizon line, and straight lines that are parallel in space have the same vanishing point in perspective. This is what happens in physical space. In drawing, in other words on paper, we consider the picture plane to have been rotated on its base and collapsed on the horizontal plane. Thus, on paper we have the base line, the horizon line and the projection of station point O’. On paper we do not have the actual viewpoint, that is, O. It is, however, important to remember that the distance between the horizon line and the base line represents the viewing height and the distance between O’ and the base line represents how far the observer is standing from the painting.

in a straight line, we meet perspective (α) of α at point (A), which is the perspective of point A.

In

Line α intersects the picture plane at point M, which is a distance Z from point 1 of line a’ on the picture plane. The vanishing point of α will be Φ1, which is the same as the vanishing point of α’, since α and α’ are parallel in space. Thus, the perspective of α will be line (α), in other words Φ1(Μ). Given that we have the position of O’, but not of O, in order to find point M we can say that from point 1, where the projection α’ of α intersects the picture plane at its base line, we can extend a line equal to α’s height Z, perpendicular to the base line. This perpendicular line ends at M, which, if joined with vanishing point Φ1, gives us the perspective of α. We can therefore conclude that the height of a horizontal

line, or more commonly its distance from the ground plane, is shown in perspective in its actual size, as the distance from the point where the line’s projection intersects the base line and from there in a perpendicular direction to the perspective of the said line. The further we move away from the picture plane and the observer, the smaller the height of the straight line generally becomes in perspective.

According to the rules of classical perspective [

1 | Cube behind picture plane | |||
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2 | Cube in front of picture plane | |||

3 | Cube intersected by picture plane | |||

4 | Cube’s front face flush with picture plane |

In order to reconstruct a model of the three-dimensional space represented by this painting (

To begin with, the painting’s qualitative elements, such as colour and shading, are removed, and only its structural elements are redrawn as shown in

equal perspective parts, given our second assumption that the ceiling consists of equal square panels (

As we continue with our study, to facilitate our drawing, so that the ground plan of the reconstructed space does not overlap its perspective, we will move the base line and projections of the vanishing points a bit further up our paper (

actual size since it is located on the picture plane. The perspective drawing allows us to directly deduce the room’s width and height, which are depicted in their actual size, as seen in the corresponding case with the cube. We then use the visual ray technique mentioned early on in the article to pinpoint the position of the room’s back wall on the ground plan: From the perspective drawing, we project the front wall’s left border, as well as the back wall’s left border, onto and perpendicular to the base line (

the real viewing distance and viewing height.

This study suggests a geometrical method from 2D to 3D imaginary space [

between Art and Architecture, using a pure geometrical synthetic tool. After all, geometry has always been a very important base knowledge in the education of architects.

I would like to thank Professor Anthi Maria Kourniati (School of Architecture, National Technical University of Athens, Greece) for her advice on the geometrical methods.

The authors declare no conflicts of interest regarding the publication of this paper.

Kourniatis, N. and Architect, N.T.U.A. (2019) Leonardo da Vinci’s The Last Supper: Reconstruction of the Room Using Reverse Geometric Perspective Processes. Journal of Applied Mathematics and Physics, 7, 1941-1957. https://doi.org/10.4236/jamp.2019.79134