Figure 1 - uploaded by Luisa Simonutti
Content may be subject to copyright.
Giotto, La cacciata dei diavoli da Arezzo, scene from "Storie di San Francesco", (1295-1299), fresco, Basilica Superiore di Assisi.
Source publication
We show how the birth of perspective painting in the Italian Renaissance led to a new way of interpreting space that resulted in the creation of projective geometry. Unlike other works on this subject, we explicitly show how the craft of the painters implied the introduction of new points and lines (points and lines at infinity) and their projectiv...
Contexts in source publication
Context 1
... are the equations of three straight lines in π that, again, are not only nonparallel, but that all meet at the point U = (1/m, 1, 0). Again, the point U does not coU ntribute with a pixel coming from the ceiling, and anyway belongs to the painting π. ...
Context 2
... interest was slowly shifting away from the ascetic body of the teachers of the medieval scholastics, and was turning to three-dimensional figures, the divine Maestà inside gothic churches, or the suggestive backgrounds of battles where the powerful soldiers and the vigor of the horses could find an effective representation. A philosophical development was forcing the painters towards a new understanding of their art as evidenced in the work of artists such as Paolo Uccello (figure 13), Mantegna (figure 11), Masaccio, and the Giambellino (Giovanni Bellini) ( figure 12). Among them Leon Battista Alberti (who wrote in 1435 the treatise De pictura praestantissima [1], where he offers a practical guide to perspective drawing) and the great painter and mathematician Piero della Francesca, who built on his knowledge of Euclid and Alberti, to write (towards the end of the XV century) De prospectiva pingendi [15], probably the ultimate text on prospective in painting. ...
Context 3
... Step 1. Design the projections of the "orthogonal" straight lines of the floor on the painting π. This step can be done formally as explained in Section 3. It can be practically performed as follows: it is enough to join each intersection of a straight line of the floor with the basis of the painting with the vanishing point ( figure 14). Step 2. Design the heights of the projections of the "parallel" straight lines of the floor on the painting π. ...
Context 4
... distance of the point of view of the painter from the vanishing point has to intervene in this step. Consider the set painter-painting-floor seen from someone on the right, staying on the plane of the painting π. Figure 15 shows how to construct these heights. ...
Context 5
... 3. Put together steps 1 and 2, and design the projection of the entire square-tile ground floor on the painting π. As shown in figure 16, it is enough to add to the painting obtained in ...
Context 6
... one can then use the projection of the side of a square tile parallel to the painting as a unit to give the measures of any object that is placed in the painting precisely on this side (figure 17). Figure 17. Leon Battista Alberti, Of Painting in three books, 'Book II', in [2]. ...
Context 7
... now show how to use this representation to calculate the distance of the eye of the painter from the painting ( figure 18). Note that a horizontal straight line L exiting from the eye and making an angle of 45 degrees with the plane of the painting is parallel to one of the diagonals of the square tiles. ...
Context 8
... by means of the side-of-tile-meter one can find the desired distance. Figure 18. Leon Battista Alberti, Of Painting in three books, 'Book I', in [2]. ...
Context 9
... this is the case, and if one knows both the distance D of the point of view O from the plane of the painting π, and the height H of the point of view from the floor, then all the vertical figures and objects that are standing on the floor can be well placed in 3D. This is made clear by the following Thales-style 14 drawings (figures 19, 20): Figure 19 Figure 20 14 By this we mean a drawing that utilizes a theorem that is often referred to as Thales' Theorem, namely an important result in elementary geometry about the ratios of different line segments that arise if two intersecting lines are intercepted by two parallel lines. ...
Context 10
... us now see how this was done in a specific case [16], for Piero della Francesca's Flagellazione ( figure 21), a first example -we should say the example -of the mathematically well constructed theory of perspective contained in his De prospectiva pingendi 15 . Figure 21. ...
Context 11
... us now see how this was done in a specific case [16], for Piero della Francesca's Flagellazione ( figure 21), a first example -we should say the example -of the mathematically well constructed theory of perspective contained in his De prospectiva pingendi 15 . Figure 21. Piero della Francesca, Flagellazione di Cristo, (1444-1470), tempera on wood, Galleria Nazionale delle Marche, Urbino. ...
Context 12
... are the equations of three straight lines in π that, again, are not only nonparallel, but that all meet at the point U = (1/m, 1, 0). Again, the point U does not coU ntribute with a pixel coming from the ceiling, and anyway belongs to the painting π. ...
Context 13
... interest was slowly shifting away from the ascetic body of the teachers of the medieval scholastics, and was turning to three-dimensional figures, the divine Maestà inside gothic churches, or the suggestive backgrounds of battles where the powerful soldiers and the vigor of the horses could find an effective representation. A philosophical development was forcing the painters towards a new understanding of their art as evidenced in the work of artists such as Paolo Uccello (figure 13), Mantegna (figure 11), Masaccio, and the Giambellino (Giovanni Bellini) ( figure 12). Among them Leon Battista Alberti (who wrote in 1435 the treatise De pictura praestantissima [1], where he offers a practical guide to perspective drawing) and the great painter and mathematician Piero della Francesca, who built on his knowledge of Euclid and Alberti, to write (towards the end of the XV century) De prospectiva pingendi [15], probably the ultimate text on prospective in painting. ...
Context 14
... Step 1. Design the projections of the "orthogonal" straight lines of the floor on the painting π. This step can be done formally as explained in Section 3. It can be practically performed as follows: it is enough to join each intersection of a straight line of the floor with the basis of the painting with the vanishing point ( figure 14). Step 2. Design the heights of the projections of the "parallel" straight lines of the floor on the painting π. ...
Context 15
... distance of the point of view of the painter from the vanishing point has to intervene in this step. Consider the set painter-painting-floor seen from someone on the right, staying on the plane of the painting π. Figure 15 shows how to construct these heights. ...
Context 16
... 3. Put together steps 1 and 2, and design the projection of the entire square-tile ground floor on the painting π. As shown in figure 16, it is enough to add to the painting obtained in ...
Context 17
... one can then use the projection of the side of a square tile parallel to the painting as a unit to give the measures of any object that is placed in the painting precisely on this side (figure 17). Figure 17. Leon Battista Alberti, Of Painting in three books, 'Book II', in [2]. ...
Context 18
... now show how to use this representation to calculate the distance of the eye of the painter from the painting ( figure 18). Note that a horizontal straight line L exiting from the eye and making an angle of 45 degrees with the plane of the painting is parallel to one of the diagonals of the square tiles. ...
Context 19
... by means of the side-of-tile-meter one can find the desired distance. Figure 18. Leon Battista Alberti, Of Painting in three books, 'Book I', in [2]. ...
Context 20
... this is the case, and if one knows both the distance D of the point of view O from the plane of the painting π, and the height H of the point of view from the floor, then all the vertical figures and objects that are standing on the floor can be well placed in 3D. This is made clear by the following Thales-style 14 drawings (figures 19, 20): Figure 19 Figure 20 14 By this we mean a drawing that utilizes a theorem that is often referred to as Thales' Theorem, namely an important result in elementary geometry about the ratios of different line segments that arise if two intersecting lines are intercepted by two parallel lines. ...
Context 21
... us now see how this was done in a specific case [16], for Piero della Francesca's Flagellazione ( figure 21), a first example -we should say the example -of the mathematically well constructed theory of perspective contained in his De prospectiva pingendi 15 . Figure 21. ...
Context 22
... us now see how this was done in a specific case [16], for Piero della Francesca's Flagellazione ( figure 21), a first example -we should say the example -of the mathematically well constructed theory of perspective contained in his De prospectiva pingendi 15 . Figure 21. Piero della Francesca, Flagellazione di Cristo, (1444-1470), tempera on wood, Galleria Nazionale delle Marche, Urbino. ...
