Projective geometry has its roots also in Perspective, “perspectiva” being the Latin word that the philosopher and Roman senator Severinus Boethius (c. 475–526) chose as a translation for the Greek Optiké, the title that Euclid gave to his work on the geometry of vision. This chapter begins by discussing the first basic rules of “practical perspective”, in order to get better pictorial images of real scenes, which were given by the Italian architects and painters of the fifteenth century: Filippo Brunelleschi, Leon Battista Alberti, and Piero della Francesca. Their ideas were essentially based on Euclid’s Optics, and their constructions made use of specific instruments. Then, since 1570s, perspective grew up gradually as an autonomous geometrical discipline. This was essentially due the Italian Giovanni Battista Benedetti and Guidobaldo del Monte, who fixed mathematical basis of the new discipline. Their works are illustrated and commented in the second part of the chapter. The mathematical perspective they found, led artists to represent parallel straight lines as concurrent at “vanishing points” in the “plane of the picture”, and this undoubtedly forwarded the idea that parallel straight lines have to be considered as converging in a point an infinite distance away, whose “perspective image” is a “vanishing point” on the canvas. It is also seen how mathematical perspective led to reconsider conics as images of a circle under central projection; that is, as curves obtained from the circle by the fundamental operations of projection and section. The final part of the chapter is devoted to see how the ideas of the above mentioned mathematicians were received and disseminated through Europe, and contributed to open the way toward what it can be called the “dawn” of Projective geometry.

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Perspective in the Renaissance

  • Andrea Del Centina,
  • Alessandro Gimigliano

摘要

Projective geometry has its roots also in Perspective, “perspectiva” being the Latin word that the philosopher and Roman senator Severinus Boethius (c. 475–526) chose as a translation for the Greek Optiké, the title that Euclid gave to his work on the geometry of vision. This chapter begins by discussing the first basic rules of “practical perspective”, in order to get better pictorial images of real scenes, which were given by the Italian architects and painters of the fifteenth century: Filippo Brunelleschi, Leon Battista Alberti, and Piero della Francesca. Their ideas were essentially based on Euclid’s Optics, and their constructions made use of specific instruments. Then, since 1570s, perspective grew up gradually as an autonomous geometrical discipline. This was essentially due the Italian Giovanni Battista Benedetti and Guidobaldo del Monte, who fixed mathematical basis of the new discipline. Their works are illustrated and commented in the second part of the chapter. The mathematical perspective they found, led artists to represent parallel straight lines as concurrent at “vanishing points” in the “plane of the picture”, and this undoubtedly forwarded the idea that parallel straight lines have to be considered as converging in a point an infinite distance away, whose “perspective image” is a “vanishing point” on the canvas. It is also seen how mathematical perspective led to reconsider conics as images of a circle under central projection; that is, as curves obtained from the circle by the fundamental operations of projection and section. The final part of the chapter is devoted to see how the ideas of the above mentioned mathematicians were received and disseminated through Europe, and contributed to open the way toward what it can be called the “dawn” of Projective geometry.