Leonardo da Vinci’s The Last Supper : Reconstruction of the Room Using Reverse Geometric Perspective Processes

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.


Introduction
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

Geometric Interpretation of Base Line, Horizon Line and Viewing Distance in a Perspective Drawing
A perspective drawing is the central projection of an object or space on a plane ( Figure 1) [1]. When we talk about the centre of projection, we mean the observer's viewpoint, whereas central projection refers to where the visual rays intersect the aforementioned plane. Visual rays emanate from the centre of projection and end at all the key points of an object, intersecting the plane on the way.
The object's perspective image, namely its central projection from the particular centre of projection, is created on this plane. The intersected plane is called the picture plane and is usually vertical, in other words perpendicular to the ground plane on which the viewer stands. The vertical picture plane meets the ground plane along a straight line called the base line. When we speak about the distance between the centre of projection, from now on referred to as the station point or viewpoint, and the picture plane, we mean how far away the viewer (or artist) stands from the plane. Up to now, we have defined the base line of a picture plane and the viewer's distance from this plane. Next we will explain the meaning of "horizon line".

Perspective of a Horizontal Line and the Horizon Line
To define the horizon line [1], we ought to mention the existence, in the construction of a perspective image, of a horizontal line belonging on the horizontal plane on which the observer stands ( Figure 2). As mentioned before, the perspective image of an object on a plane known as a picture plane is the intersection of the visual rays, which connect the viewpoint and the object, with this plane.
To determine the perspective of a straight line on the picture plane, which will also be a straight line, all we have to do is determine the perspectives of two of its points. In Figure 2 we observe that the visual ray corresponding to point 1  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 con-

Perspective Image of Horizontal Line α Located a Distance Z from the Ground Plane
In Figure 5 we see the perspective of horizontal line α on the picture plane. Line α' is the projection of α on the ground plane. Let α's height be Z. Based on what we have said about finding the perspective of a horizontal line, to find the perspective of α we can proceed as follows: 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

Perspective Image of a Frontal Cube. Exploration of the Relationship between Perspective and the Position of an Object in Relation to the Picture Plane
According to the rules of classical perspective [2] [3], the cube in Figure 6 is projected from point O onto the picture plane, resulting in its perspective image. Table 1 shows various perspectives of the same cube, placed at different distances from the picture plane. In all cases, the distance between viewpoint O and the picture plane is fixed. What changes is the cube's position in relation to the picture plane. We observe that when the cube is placed behind the picture plane, all its edges are portrayed smaller than they actually are. When it is placed between the picture plane and the viewpoint, all its edges are portrayed bigger than they really are on the picture plane. When the cube is positioned in such a way that it is intersected by the picture plane, then the part behind the picture plane is portrayed smaller and the part in front of the picture plane bigger. Therefore, in this latter case, the cross-section obtained when the cube is cut by the picture plane, in other words the square belonging on the picture plane, is portrayed in its actual size. This is also made apparent in the last case shown, where the cube has been positioned so that its front face is flush with the picture plane. In perspective, this face is shown in its actual size, whereas the rest of the cube is portrayed smaller.

Case Study of Leonardo da Vinci's The Last Supper
In order to reconstruct a model of the three-dimensional space represented by this painting (Figure 7), we must first assume that the artist has realistically portrayed the interior of a room, and has opened the room up to the viewer by removing the front wall. Let us therefore say that the room's wall, which would have been positioned under the front beam depicted by the artist in the painting's foreground, has been removed. The picture plane of this particular work of art has therefore been positioned where the missing wall would have been. The second assumption concerns the room's ceiling panels. These panels were considered to be perspective drawings of similar, horizontal, square elements, and have been dealt with as such. Following these two assumptions, we will go on to find the picture plane's base line and horizon line and, lastly, the projection of the viewpoint. In other words, we will find at what distance from the work of art, and at what height, the artist positioned the observer's eye. We will then reconstruct the spatial model and draw the perspective of the said model from the same viewpoint as that indicated by the artist. If the vanishing points and relative distances between objects in the resulting model's perspective image coincide with Journal of Applied Mathematics and Physics the vanishing points and relative distances between the objects in the artist's painting, this will first of all mean that we have correctly determined the artist's desired viewing distance and viewing height. It will also mean that the study's initial assumptions are correct.
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 Figure 8.
It was initially observed that the perspectives of the room's horizontal lines all converge toward a single vanishing point, which coincides with Christ's right eye, at the centre of the table (Figure 9). We can thus pinpoint the position of vanishing point Φ1 of all the horizontal lines in the room. These lines are also perpendicular to the picture plane, in other words perpendicular to the missing front wall (Figure 10). Having determined the vanishing point, we can also tell where the horizon line lies, since the vanishing point is on this line ( Figure 11).
Furthermore, based on our first assumption, namely that the picture plane is positioned where the wall opened up to the viewer would stand, we can deduce that the base line is located as shown in Figure 11. In Figure 12         equal perspective parts, given our second assumption that the ceiling consists of equal square panels ( Figure 13). Figure 14 also shows the horizontal beams, thus providing a perspective of all the squares on the ceiling. Since these squares are horizontal, their diagonals will also be horizontal lines. Consequently, the vanishing points of lines Φ2 and Φ3 will also be along the horizon line. The projections of Φ2 and Φ3 on the base line will be Φ2' and Φ3' (Figure 14/Figure 15). A square's two diagonals form a 90˚ angle between them. Consequently, if we had the position of the projection of viewpoint O', to find Φ2' and Φ3' we would draw parallel lines from O' towards the two diagonals. These parallel lines would also meet at a 90˚ angle. The locus of the points on the plane that meet section Φ2'Φ3' at a 90˚ angle is a semicircle whose diameter is Φ2'Φ3', as shown in Figure 15. However, O' also runs perpendicular to the picture plane's base line from Φ1'. The position of O' is thus the intersection of the two loci, as seen in Figure 16. Journal of Applied Mathematics and Physics line and projections of the vanishing points a bit further up our paper ( Figure   17). Figure 18 shows how the general outline of the room's ground plan is reconstructed [1]. The first thing we position on the ground plan is the missing front wall, which, based on the first assumption we made in our study, is in its          the picture plane is, based on the first assumption made in the study. As a result, we observe that the perspective drawing of the room and the artist's painting coincide to a great extent. There is correspondence between both the receding lines and the relative distances between the depicted objects. This means that the initial assumptions made in the study are correct and also that our approach to the room that the artist had before him or imagined when he created this work of art is relatively precise.

The Results
This study suggests a geometrical method from 2D to 3D imaginary space [4].
This approach could open a discussion in the field of Art concerning the relation between real and imaginary space that could possibly generate a connection