Drawing with divergent perspective, ancient and modern.pdf

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Perception, 2011, volume 40, pages 1017 ^ 1033
doi:10.1068/p6876
Drawing with divergent perspective, ancient and modern
Ian P Howard, Robert S Allison
Centre for Vision Research, York University, Toronto, Ontario M3J 1P3, Canada;
e-mail: ihoward@cvr.yorku.ca
Received 24 November 2010, in revised form 26 July 2011
Abstract.
Before methods for drawing accurately in perspective were developed in the 15th century,
many artists drew with divergent perspective. But we found that many university students draw
with divergent perspective rather than with the correct convergent perspective. These experiments
were designed to reveal why people tend to draw with divergent perspective. University students
drew a cube and isolated edges and surfaces of a cube. Their drawings were very inaccurate.
About half the students drew with divergent perspective like artists before the 15th century.
Students selected a cube from a set of tapered boxes with great accuracy and were reasonably
accurate in selecting the correct drawing of a cube from a set of tapered drawings. Each subject's
drawing was much worse than the drawing selected as accurate. An analysis of errors in drawings
of a cube and of isolated edges and surfaces of a cube revealed several factors that predispose
people to draw in divergent perspective. The way these factors intrude depends on the order in
which the edges of the cube are drawn.
1 Introduction
Methods for drawing in accurate perspective were developed in 15th-century Florence
(Richter 1970; Edgerton 1975). A correct perspective drawing of an object creates the same
image in an eye as the object when the eye is in the location from which the object was
drawn. Consider a cube viewed with one eye with one face in a frontal plane. A projec-
tion of the cube from the nodal point of the eye onto a sheet of paper parallel to the
front surface of the cube defines a drawing of the cube in one-point perspective, as
shown in figure 1. The drawing is said to be in polar projection and has the following
features: (i) The front surface is square. (ii) The receding edges converge on a vanish-
ing point on the horizon directly opposite the eye. This convergence specifies linear
perspective. (iii) Diagonals of receding horizontal surfaces converge on two distance
points on opposite sides of the vanishing point. The distance between each distance point
and the vanishing point equals the distance of the eye from the drawing. Convergence
of the diagonals specifies foreshortening in the drawings of the receding surfaces (aspect-
ratio perspective).
Distance
point 1
Horizon
(eye level)
Vanishing point
(directly opposite the eye)
Distance
point 2
Figure 1.
A drawing of a cube with one surface parallel to the drawing surface. The eye is opposite
the vanishing point at a distance from the drawing equal to the distance between the vanishing point
and either of the two distance points.
1018
I P Howard, R S Allison
In an orthographic drawing the projection lines connecting each point in the object
and the corresponding point in the drawing are parallel. An orthographic drawing
of a cube with one surface parallel to the plane of the drawing is simply a square.
An orthographic drawing of a cube with no surface parallel to the plane of the
drawing has no linear perspective (parallels in the cube are parallel in the drawing),
but the drawings of the surfaces are foreshortened (they have aspect-ratio perspective).
The surfaces are also sheared into parallelograms. The polar projection of a cube
becomes more orthographic as its distance from the projection plane increases.
A plan view, a front elevation, and a side elevation of a cube are three ortho-
graphic drawings made from three orthogonal directions. Each drawing has parallel
edges and no foreshortening, as shown in figure 16.
Before the 15th century, the receding edges of rectangular objects were often drawn
either parallel or diverging. We looked for drawings made before the 15th century of
rectangular objects, such as pedestals and tables with the front surface parallel to the
picture plane. We found 15 examples from China, Persia, ancient Greece, Pompeii,
and medieval Europe. In every case the top of the object was drawn with parallel or
near-parallel sides, and the side of the object was drawn with divergent perspective.
Figure 2 shows some examples with lines added and the convergence angles of the
tops and sides of the rectangular objects indicated in the caption.
We report here that a large proportion of university students draw with parallel or
divergent perspective, like pre-15th-century artists. None of the students we tested came
close to drawing with correct convergent perspective. One can understand drawing in
parallel perspective. The receding edges of a rectangular object are indeed parallel, and
we see them as parallel in 3-D space. Our visual system evolved to allow us to see
in 3-D, not to make 2-D drawings or to perceive the 2-D layout of the retinal image.
Only architects and artists need to transfer the 3-D layout of a scene into 2-D. But
why did early artists and many people today draw in divergent perspective, particularly
when drawing the side of a rectangular object? Receding rectangular surfaces do not
diverge into the distance. One possibility is that a receding surface drawn with paral-
lel edges appears to diverge. For example, the drawing of a table in figure 2f has
parallel sides that appear to diverge. This is because a horizontal surface would have
to diverge to produce a projection on the frontal plane with parallel sides. Perhaps
some early artists drew with divergence because they copied drawings made with paral-
lel perspective. But we report here that many university students drew with divergent
perspective even when drawing an actual cube.
It is well known that, when selecting a frontal shape to match with a shape inclined
in depth, people select a frontal shape that lies between the image of the shape and
the actual shape of the inclined stimulus. Thouless (1930) called this perceptual effect
``regression to the real object''. The real object is the shape of the inclined stimulus
when viewed with the visual axis orthogonal to the stimulus. We will use the term
``regression to the orthogonal view''. We show that, when selecting a drawing that
most resembles a cube, people show regression to the orthogonal view. However,
we show that, when drawing a cube, people make errors that are far larger than the
effect of regression. The five following experiments were designed to reveal why
many people draw in divergent perspective, and why divergent perspective is more
evident in drawings of the vertical side of an object than in drawings of the top of an
object.
These experiments were approved by York University Ethics Committee in accordance
with the World Medical Association Declaration as revised in 2008.
Drawing with divergent perspective, ancient and modern
1019
(a)
(b)
(c)
(d)
(f)
(e)
Figure 2.
Examples of early divergent perspective. Added lines indicate the angles of perspective.
(a) A fresco from the grotto of Touen Houang, China, Tang dynasty (618 ^ 906). The top of the
table diverges 68, while the side diverges 128 (from Fourcade 1962). (b) Medieval Puppeteers.
(c)
Miracle of St Guido
by Jacopo da Bologna. In Pomposa Abbey, Ferrara, ca 1350. Fototeca
Berenson. The edges of the top of the table diverge 28, while the edges of the side diverge 128.
d) From
Life in the Middle Ages
by R Delort, Universe Books, New York, 1972, page 46. Side
ïdiverges 118, top converges 18. (e) From the Velislav Bible, ca 1340 (Prague University Library).
Top diverges 138, side diverges 228. (f ) Illusory divergence of parallel lines on the receding sides
of the drawing of a table.
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I P Howard, R S Allison
2 Experiment 1. Drawing a cube and a 2-D projection of a cube
This experiment was designed to reveal how accurately young adults produce a
perspective drawing of a cube and how accurately they copy a correct perspective
drawing of a cube.
2.1
Method
2.1.1
Subjects.
The subjects were eighty university students aged 18 to 22 years. None of
them had been trained to draw in perspective. They all had normal or corrected-to-
normal vision and a stereoacuity of at least 60 min of arc as tested by the stereotest
circles of the Stereo Optical Company.
2.1.2
Stimuli.
The stimuli were a 15.2 cm cube and a correct full-scale drawing of the
cube, which will be referred to as the 2-D cube. The stimuli were made from stiff black
card. The nine visible edges of each stimulus were white lines about 3 mm wide. Since
the rear edges of the cube were not visible, it appeared like a solid black cube with
white edges. The stimuli were presented one at a time in a box lined with black velvet.
They were illuminated by a dimmed tungsten lamp so that nothing other than their
white edges was visible. There were no visible shadows or shading. The near surface of
the cube and the 2-D cube was vertical and parallel to the subject's interocular axis.
The stimuli were viewed with both eyes. Each stimulus was below and to the left of a
point directly in front of the centre of the interocular axis. This point defined a vanish-
ing point that was a compromise between the distinct vanishing points of the two
eyes. The vanishing point was used to define a correct drawing of the cube in one-
point perspective. The head of each seated subject was supported on a chin-rest, with
the eyes 76 cm from the plane containing the near surface of the cube or the 2-D cube.
Figure 3a shows the cube drawn in one-point perspective relative to the vanishing
point. It also shows relevant distances, the angle of convergence of a correct drawing
of the cube, and labels of the relevant edges. Cross-fusion of the images in figure 3b
creates an impression of the cube seen by the subjects.
Eye level
Convergence of
edges A and C
25 cm
268
Vanishing point
76 cm
to subject's
head
Median
plane
Edge D
Edge A
Edge E
B
e
dg
458
E
Black surround
25 cm
Front surface in
subject's frontal
plane
Edge C
(a)
15.2 cm
(b)
Figure 3.
(a) A 2-D projection of the cube
used in all the experiments. The point midway
between the subject's eyes was 76 cm directly
in front of the vanishing point. Subjects drew
the cube on a sheet of paper to the right
of the cube at a viewing distance of 25 cm.
The front face of the cube was pre-drawn.
Subjects drew the remaining five edges (A, B,
C, D, and E). (b) Cross fusion of the two
images creates an impression of the cube seen
by the subjects.
Drawing with divergent perspective, ancient and modern
1021
2.1.3
Procedure.
All eighty subjects drew the cube. The first forty subjects drew with a
pencil on a vertical sheet of white paper just to the right of the cube at a viewing
distance of 25 cm. All the other drawings in all experiments were done with a white
crayon on black paper. Subjects looked at the cube with both eyes and then rotated
the head on the chin-rest 258 so as to look squarely at the sheet of paper. They could
look back and forth between the cube and drawing as often as they wished. The sheet
of paper contained a 7.4 cm square in such a position that the vanishing point of
a correct drawing of the cube based on the square was in its correct location on the
subject's visual horizon and median plane. Thus, the image of the cube and a correct
drawing of the cube produced the same images in the eyes. The pre-drawn square
standardised the size and location of all the drawings.
A subset of sixteen students also drew the 2-D cube. Half of them drew the cube
first and half of them drew the 2-D cube first.
2.2
Results and discussion
Figure 4 shows the two extreme examples of the drawings of the cube and the
mean drawing derived from the eighty subjects. Table 1 shows the mean errors derived
from drawings of the cube produced by the eighty subjects. For edges A, B, and C
a positive angular error indicates that the drawn edge was rotated towards the vertical
(counterclockwise) from its position in a correct drawing. For edges D and E a positive
linear error indicates how far, in millimetres, the edge was displaced out from its
correct position. Table 1 also shows the mean divergence errors for each pair of reced-
ing edges. A positive divergence error indicates how far the angle between a pair of
edges diverges from the correct angle of convergence. In a correct drawing, edges A
D
A
17 mm
2X428
28
12X88
B
5.2 mm
E
À5X88
C
(a)
(b)
(c)
Figure 4.
Drawings of a cube. The front surface was pre-drawn. Dashed lines indicate the accurate
drawings relative to a vanishing point opposite the eye. (a) The best drawing. (b) The mean of
the drawings produced by the eighty students. The error for each edge is with respect to the
accurate drawing. (c) The worst drawing. Edges A and C diverge 968 relative to the true value
and diverge 708 relative to parallel. Note how this subject drew edges D and E curved in order
to connect them to edge B while approximating a right angle between them.
Table 1.
Mean errors of eighty subjects in drawing the edges of a 3-D cube. A positive angular
error indicates that the edge was drawn more vertical than in the correct drawing. Divergence errors
A ^ B and B ^ C are the angles between the drawings of edges A and B, and B and C with respect
to the correct angle of convergence of 138. A divergence error of 138 indicates that the edges
were drawn parallel. Divergence error A ^ C signifies the angle between edges A and C with
respect to the true angle of convergence of 268. A convergence error of 268 indicates that the edges
were drawn parallel. All pairs of edges were drawn more divergently than the correct value.
Item
Angular errorsa8
edge
A
Means
SD
SEM
24.19
11.38
1.27
edge
B
edge
C
Divergence errorsa8
edges
A±B
11.36
9.38
1.05
edges
B±C
18.59
9.31
1.04
edges
A±C
29.95
14.26
1.59
Linear errorsamm
edge
D
16.93
10.07
1.59
edge
E
5.2
6.25
2.59
12.83
À5.76
7.75 10.54
0.87
1.18

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