1. Perspective

2. Piero della Francesca. Ideal City (1475). Oil on canvas.

3. What is Linear Perspective?
a system for representing three-dimensional space on a two-dimensional flat surface
developed in Florence in the early 15th century by Filippo Brunelleschi and Leon Batista Alberti

4. drawing machines

5. The Invention of Linear Perspective

6. Development of Linear Perspective
Filippo Brunelleschi ( ) was the sculptor and architect who demonstrated the principles of perspective through mathematics
In 1415, Brunelleschi painted his picture of the Baptistery on the surface of a small mirror, right on top of its own reflection. 
The Baptistery in Florence

7. Brunelleschi’s “Peep Show”
To demonstrate the fact that his painting was indeed an exact replica that could fool the eye, Brunelleschi drilled a small hole in the mirror and then stood directly in front of the Baptistery, looking through the peephole to see the real building. 
He then held up a second, clean mirror in front of his painted panel.  The second mirror blocked the view of the real building, but now reflected his painted version on the original mirror.     

8. Brunelleschi’s “Peep Show”
By holding up the panel and pressing the hole to one eye while holding a mirror with the other hand, the viewer could see the painting’s reflection.
A viewer standing in the cathedral doorway could check the painted illusion against the real view.

9. Leon Battista Alberti (1404-72)
architect and writer who was the first to formulate rules that artists could follow
imagined the picture surface as an “open window” through which a painted world is seen
showed how a perspective “checkerboard pavement” is created within the picture space—in which the receding parallel lines represent the visual rays connecting the spectator’s eye to a spot in the distance

10. Leon Batista Alberti
based his system on the height of the human figure, being 3 braccia tall
drew a rectangular picture area, imagined as an open window
divided the ground line into scaled braccia
fixed the central vanishing point by drawing a vertical line three braccia high
from the center of the ground line
drew diagonals—orthogonals—joining the ground line to the vanishing point

11. Application of Linear Perspective
Brunelleschi devised the method of perspective for architectural purposes.
He is said by Manetti to have made a ground plan for the Church of Santo Spirito on the basis of which he produced a perspective drawing to show his clients how it would look after it was built.
We can compare this drawing with a modern photo of the actual church.

12. Application of Linear Perspective
Brunelleschi's proposal drawing of the  interior of Santo Spirito (1543)
Interior of Brunelleschi's Santo Spirito

13. Early Approaches
Prior to the Renaissance, artists were less concerned with the illusion of reality and more concerned with the content and symbolism of their work.

14. Ancient Egyptian only the front planes of objects are shown
figures assembled from separate views
depth suggested by overlapping forms

15. Byzantine/Medieval
Duccio di Buoninsegna. The Virgin and Child with Saints (c.1315).
Egg tempera on poplar.
the gold backgrounds suggest majesty and internal light
denies any sense of depth, allowing figures to inhabit a spiritual space
figures, however, modeled in light and shade

16. Defining Space Before the Renaissance
The size of each element in the image related much more to its importance, rather than it's placement in a space.

17. Judging by the Eye
Giotto introduced a new kind of realism by creating convincing spatial arrangements.
He angles the building, removing side walls to reveal the cubic interior.
Giotto. Birth of the Virgin Mary (1320s).
Fresco.

18. Judging by the Eye
The sense of depth is partly achieved in this painting by the diminishing size of the floor tiles.
The receding lines of the floor converge toward a single focus (vanishing point), although the other receding diagonals converge toward points higher up in the picture.
Ambrogio Lorenzetti. The Presentation
in the Temple (1342). Tempera on wood.

19. Judging by the Eye
Dutch master van Eyck created highly convincing interiors and landscapes by relying on his own observations rather than theoretical rules.
In this painting, an intimate interior is created by the sloping lines of the boarded floor and beamed ceiling, the relative size of objects, and the use of light.
Jan van Eyck. The Arnolfini Marriage (1434).
Oil on oak.

20. A convex mirror, like the one in the
painting, may have been used by
van Eyck as a compositional aid.

21. The Renaissance

22. Masaccio: a “hole in the wall”
Ten years after the invention of linear perspective, Masaccio applied the new method of mathematical perspective even more spectacularly.
This fresco’s painted architectural framework is so carefully constructed that it could almost have been translated from an architectural plan.
Masaccio. The Trinity (1427). Fresco.

23. A “hole in the wall”
The barrel vaulted ceiling is incredible in its complex, mathematical use of perspective.
These lines eventually
meet at a distance point
on the horizon line.

24. Paolo Uccello: Playful Measurement
explored the geometry of nature and objects
studied how to solve perspective problems
combined scientific probing with the love of pattern and splendor
Chalice

25. Perspective: Eye Levels

26. Normal Viewpoint

27. Paolo Uccello. The Hunt in the Forest (1460s). Oil on canvas.

28. High Viewpoint

29. Low Viewpoint

30. Bramantino. Adoration of the Kings (1498). Oil on panel.

32. Piero della Francesca: Divine Measurement
Piero dell Francesca. The Flagellation (1460). Tempera on wood.

33. Piero della Francesca: Divine Measurement
perspective so logical and precise that scholars have been able to reconstruct the room as if it were real architecture
architectural proportions and dimensions reflect the divine order of things

34. Andrea Mantegna: Eyewitness Art
perspective of the oculus constructed according to the upward gaze of an observer standing directly below
Andrea Mantegna. From The Painted
Room ( ). Fresco.

35. Andrea Mantegna: Eyewitness Art
Christ shown in a dramatically foreshortened pose
heavy head propped on a pillow so that his features can be seen
feet projected out of the picture so viewer can focus on gaping wounds
Andrea Mantegna. Lamentation Over the
Dead Christ (1480). Distemper on canvas.

36. Perspective Foreshortening
Foreshortening is based on studies from life as well as perspective principles.
Figure is encased in a gridlike “box” divided into equal units.
When box is laid on the ground, the units diminish as they recede.

37. Leonardo da Vinci’s Explorations
The Last Supper (1497). Fresco.

38. Leonardo da Vinci’s Explorations
The Last Supper (1497). Fresco.

39. Leonardo da Vinci’s Explorations
The Last Supper (1497). Fresco.

40. Leonardo da Vinci’s Explorations
To seem like an extension of the refectory,
the vanishing point—which is on Christ’s right
eye—should be lower, at the spectator’s level.
Instead, it is about 15 feet above the floor
level.
The strong pull of the perspective, however,
“lifts” the viewer from the floor to the correct
viewpoint—a strangely spiritual phenomenon.
Convent of Santa Maria delle Grazie, Milan

41. Perspective, Human Anatomy, and Proportions
Albrecht Dürer
Perspective, Human Anatomy, and Proportions

42. Albrecht Dürer He traveled to Italy to learn about perspective.
Besides his great studies of perspective, he was also interested in human anatomy and proportion.
Melancholia (1514). Copper engraving.

43. Dürer’s Perspective Aid
frame consisted of wooden
stand
grid threads probably made of
silk
The Draftsman’s Net

44. Dürer’s Perspective Aid
The grid is used by the artist to copy the outline of the model’s form onto a squared drawing surface
An eyepiece, fixing the artist’s viewpoint, is positioned at a specific distance—twice the frame’s height—from the device.
The closer the net is placed to the object, the more foreshortened the perspective.