ne
an
an
Jam
dDepartment Physics of Man, Helmholtz Institute, Utrecht University, Padualaan 8, 3584 CH Utrecht, The Netherlands
eBehavioural Biology Group, Utrecht University, Paduala
n Resea
Universi
roadwa
2-364, C
sity, 216
eidelber
DE Soes
‘‘Monocular stereopsis” is frequently treated as an oxymoron as
titles of scientific papers referring to ‘‘paradoxical monocular stere-
opsis” illustrate (Enright, 1991). Yet experientially ‘‘visual space”
does in no way collapse into the (flat!) ‘‘visual field” if one closes
one eye. Moreover, many animals, such as cows and rabbits, have
structure of binocular visual space, which Luneburg identifies as
Lobachevky’s hyperbolic space. But even generic Riemann spaces
lack a projective structure (Berger, 2007), showing that the
Cayley–Klein spaces are quite special. The existence of a projective
structure should not be assumed as Luneburg (1947) did, but
should be treated as an empirical issue.
One of the few empirical studies is Foley’s (1964), dating from
the nineteen sixties. At the time it was assumed that readers would
be familiar with the required geometrical background. Since this
cannot be assumed for our modern readers we start the paper with
a summary review of this material.
q This work resulted from an ad lib collaboration at the occasion of the (forced)
retirement of the first author.
* Corresponding author. Address: Delft University of Technology, Faculty of
EEMCS, Mekelweg 4, 2628 CD Delft, The Netherlands. Tel.: +31 152 84145.
E-mail address: jan.koenderink@telfort.nl (J.J. Koenderink).
Acta Psychologica 134 (2010) 40–47
Contents lists availab
Acta Psych
.e l1 Address: 5083 Hanging Moss Ln., Sarasota, FL 34238, USA.Received in revised form 28 November 2009
Accepted 1 December 2009
Available online 6 January 2010
PsycINFO classication:
2323
Keywords:
Visual space
Monocular stereopsis
data apply to binocular visual space and date from the 1960s (Foley, 1964). This appears to be both the
first and the last time this basic issue was addressed empirically. Yet the question is of considerable con-
ceptual interest. Here we report on a direct empirical test of the existence of planes in monocular visual
space for a group of sixteen experienced observers. For the majority of these observers monocular visual
space lacks a projective structure, albeit in qualitatively different ways. This greatly reduces the set of via-
ble geometrical models. For example, it rules out all the classical homogeneous spaces (the Cayley–Klein
geometries) such as the familiar Luneburg model. The qualitatively different behavior of experienced
observers implies that the generic population might well be inhomogeneous with respect to the structure
of visual space.
 2009 Elsevier B.V. All rights reserved.
1. Introduction
The geometrical structure of ‘‘monocular visual space” has
rarely been the subject of formal, geometrical research, whereas
the empirical studies are also comparatively scarce. Indeed, ‘‘stere-
opsis” is usually taken to be synonymous with ‘‘binocular stereop-
sis” by many dictionaries and informative web-sites (‘‘WIKI’s”).
no, or only very minor, binocular overlap but nevertheless show
signs of possessing a well-developed visual space. Thus the topic
certainly deserves the attention of vision science.
The few current models of monocular visual space are fre-
quently based on the classical homogeneous spaces that are the
Cayley–Klein spaces of constant curvature (Yaglom, 1979). Such
spaces have been proposed by Luneburg (1947) as describing thefDepartment of Psychology and Vanderbilt Visio
gDepartment of Psychology, Western Kentucky
hVision Laboratories, Skidmore College, 815 N B
iCSAIL, Massachusetts Institute of Technology, 3
jPsychology Department, The Ohio State Univer
kExperimental Psychology, Utrecht University, H
l TNO Defensie en Veiligheid, Kampweg 5, 3769
a r t i c l e i n f o
Article history:
Received 4 September 20090001-6918/$ - see front matter  2009 Elsevier B.V. A
doi:10.1016/j.actpsy.2009.12.002an 8, De Uithof, 3584 CH Utrecht, The Netherlands
rch Center, Vanderbilt University, TN, USA
ty, Bowling Green, KY 42101-1030, USA
y, MS 2106, Tisch Learning Center 155, Saratoga Springs, NY 12866-1632, USA
ambridge, MA 02139, USA
Lazenby Hall, 1827 Neil Avenue, Columbus, OH 43210, USA
glaan 2, 3584 CS Utrecht, The Netherlands
terberg, The Netherlands
a b s t r a c t
The issue of the existence of planes—understood as the carriers of a nexus of straight lines—in the monoc-
ular visual space of a stationary human observer has never been addressed. The most recent empiricalaDelft University of Technology, Faculty of EEMCS, Mekelweg 4, 2628 CD Delft, The Netherlands
bDepartment of Cognition and Education Sciences, Trento University, Rovereto Branch, Corso Bettini 31, 38068 Rovereto, Italy
cDelft University of Technology, Faculty of Industrial Design, Landbergstraat 15, 2628 CE Delft, The NetherlandsDoes monocular visual space contain pla
Jan J. Koenderink a,d,*, Liliana Albertazzi b, Andrea J. v
Astrid M.L. Kappers d, Joe S. Lappin f,1, J. Farley Norm
Flip Phillips h, Sylvia C. Pont c, Whitman A. Richards i,
journal homepage: wwwll rights reserved.s?q
Doorn c, Raymond van Ee d, Wim A. van de Grind e,
g, A.H.J. (Stijn) Oomes a, Susan P. te Pas k,
es T. Todd j, Frans A.J. Verstraten k, Sjoerd de Vries l
le at ScienceDirect
ologica
sevier .com/ locate /actpsy

The primitive elements of a three-dimensional projective space
are ‘‘points”, ‘‘lines” and ‘‘planes”. Any plane is by itself a two-
dimensional projective space. Axiomatically the role of ‘‘points”
and ‘‘lines” in planar or ‘‘points” and ‘‘planes” in spatial (3D) pro-
jective geometry are mutually interchangeable (the so-called prop-
erty of duality), thus these formal entities are quite distinct from
the visual qualities usually associated with them (e.g., the ‘‘extend-
edness” of lines as compared to points). Basic axioms of incidence
have that—in the projective plane—any two distinct points define a
unique line and any two distinct lines define a unique point,
whereas—in a three-dimensional projective space—any three dis-
tinct points define a unique plane and any three distinct planes de-
fine a unique point. Moreover, any two distinct planes define a
unique line, a plane and a line define a distinct point, and so forth.
In cases of interest to vision lines carry a continuous sequence of
points and planes contain a continuous nexus of criss-crossing
lines (thus we disregard finite geometries here). In particular, if
one considers the nexus of all lines connecting any vertex to any
point on its opposite side in a triangle, any two lines from different
vertices meet in a point of the interior, thus the nexus is planar.
This is the property addressed in this study.
In a generic Riemannian space such planar webs of geodesics do
not exist (Berger, 2007; Buseman, 1955). The required projective
the issue of the existence of a projective structure, as exemplified by
the existence of ‘‘planes” (as coherent webs of geodesics), is up to
empirical verification.
One particularly convenient set of axioms (at least in the con-
text of this paper) that define a ‘‘projective space” as a synthetic
geometry is the following (Bennett, 1995):
Definition A projective space is an ordered pair ðP; LÞ, where P
is a nonempty set whose elements are called points andL is a non-
empty collection of subsets of P called lines subject to the follow-
ing axioms:
[PS1] Given any two distinct points P and Q, there is one and
only one line (called ‘ðP;QÞ) containing them.
[PS2] (The Pasch Axiom) If A, B, C, and D are distinct points such
that there is a point E in ‘ðA;BÞ \ ‘ðC;DÞ, then there is a point F
in ‘ðA;CÞ \ ‘ðB;DÞ (see Fig. 1).
[PS3] Each line contains at least three points; not all points are
collinear.
Axiom PS2 (see Fig. 1) is due to Moritz Pasch (1843–1930) (but
in a different context) and was first used by Ostwald Veblen
(1880–1960) as a clever way to avoid postulating the existence
of planes (Bennett, 1995). Instead, planes are defined as flats:
poi
ne d
posite s
to the o
J.J. Koenderink et al. / Acta Psychologica 134 (2010) 40–47 41structure occurs only in the classical homogeneous spaces, the
Cayley–Klein spaces, in which the curvature is constant. Examples
of such spaces are not only the familiar Euclidean space, but also all
the spaces considered by Luneburg (elliptic and hyperbolic geom-
etries), including the one he singled out as descriptive of ‘‘visual
space”. Although Luneburg considered mainly binocular stereopsis,
a close analysis of his arguments reveals that these arguments ap-
ply equally to monocular visual space. Luneburg considers ‘‘free
movements of objects” (Helmholtz, 1868) obvious, this indeed lim-
its the possibilities to spaces of constant curvature. It is his key
assumption from which most of his formalism follows. But the
existence of such a group of congruences is quite independent of
the binocularity issue. It is easy enough to come up with very
reasonable models of the geometry of visual space that fail to be
Cayley–Klein geometries and for which no planar webs of geode-
sics exist. In order to illustrate this we discuss an instance of such
a model in the Appendix. The upshot of this discussion is that
Fig. 1. The ‘‘Pasch Axiom”: Given four points fA; B;C;Dg, let the lines determined by the
fA; Cg and fB;Dg possess a common point F (say). In this figure we also indicated the li
determine a flat triangular area in the sense that lines that connect a vertex to the op
point ‘‘inside the triangle” (here D), in other words, the lines connecting the vertices
experiment, shown at left. Intuitively, it is easy to conceive of EC and FB as slightly curved (‘‘o
at right). Of course this would imply the non-existence of the triangular ‘‘plane”! This show
this!).fA; Bg and fC;Dg possess a common point E, then the lines determined by the points
mined by the points fE; Fg in order to show that the points fA; E; Fg are meant to
ide (like the lines determined by the points fE;Cg and fF; Bg) meet in a common
pposite sides ‘‘mesh” to form a planar nexus. This is the condition tested in thents
eterDefinition Let M be a point of ðP; LÞ not in line ‘. The plane a
determined by M and ‘ is given by
a ¼
[
A2‘
‘ðM;AÞ ¼ P : ‘ðP;MÞ \ ‘–£f g [ Mf g: ð1Þ
We will denote this plane by ðM; ‘Þ. Notice that a plane is defined as
the closure of a fan of lines that connect the points of a line with a
point not on that line, a very intuitive notion (Fig. 2).
One proves that the line defined by two points in a plane lies in
that plane and that two lines intersect if and only if they are copla-
nar. One also shows that the point M that appears in the definition
is nothing special: Any point T (say) in the plane a not on the line ‘
may be substituted for M and one still obtains the same plane. One
also proves that any three non-collinear points in a projective
space lie on a unique plane. Thus the ‘‘planes” defined in this man-
ner indeed possess all the familiar properties.
The definition of a plane as a ‘‘flat” can be generalized to higher
dimensional flats. This allows the definition of dimension. For in-ut of the plane of the triangle”) such as not to meet in a common point D (figure
s that the Pasch axiom really has non-trivial content (Euclid should have caught

stance, three-dimensional space is a flat obtained from a plane and
a point not on it. One proves the familiar properties, e.g., two dis-
tinct planes intersect in a line, and so forth. The crucial fact for our
stead one checks the Pasch Axiom instead, which is a much easier
task. This is what Foley (1964) did and it is also our paradigm.
As said above, the only attempt to check for the existence of
planes in a visual space known to us is the most remarkable paper
Fig. 2. Given a point M and a line ‘ (M not on ‘), one constructs the fan of lines that
connect M with ‘. The fan describes a plane a.
Fig. 4. A really bad case as ‘‘planes” go. The line has two distinct points in common
with the ‘‘plane”, yet it fails to lie fully in it. In such a case one concludes that in this
space ‘‘planes do not exist”. In Foly’s terms the space ‘‘fails the Desarguasian
property”. The Pasch Axiom (Axiom PS2; Fig. 1) is obviously violated in such a
geometry.
42 J.J. Koenderink et al. / Acta Psychologica 134 (2010) 40–47experiment is that Desargues’ Theorem for projective space (two
triangles that are ‘‘point perspective” are also ‘‘line perspective”
and vice versa; see Fig. 3) can be proven for dimensions three and
higher, but not in planar geometry. For planar geometry it needs
to be included in the axioms, albeit perhaps in some roundabout way.
Thus the only non-Desarguesian projective spaces are necessar-
ily two-dimensional, that is to say planes, the Moulton plane per-
haps being the most familiar example (Moulton, 1902). Any
projective space which points are not all coplanar is Desarguesian.
Thus any three-dimensional space in which the theorem of Desar-
gues fails (that does not have the ‘‘Desarguasian property”) cannot
be a projective space. Conversely, if all planes are Desarguesian,
then they can be embedded into a three-dimensional projective
geometry. This illustrates the conceptual load of the ‘‘Desarguesian
property”.
There are various ways to check the existence of planes. In prac-
tice one looks for a method that is easily implemented psycho-
physically. In this paper we check Pasch’s Axiom (Axiom PS2).
This is an apt choice, because in case it fails visual space lacks a
projective structure due to the fact that lines fail to fully coincide
with planes that have at least two distinct points in common with
them. Because of the geometrical facts mentioned above axioms
PS1. . .3 imply Desargues Theorem, this is why the (sparse) litera-
ture on the topic uses the ‘‘Desarguesian property” (meaning that
Desargues’ theorem holds) as a key word. It does in no way imply
the actual empirical check of the Desargues Theorem though, in-Fig. 3. Two major theorems of ‘‘standard” (or perhaps ‘‘intended”) projective plane geo
‘‘Pappus configuration”. In the Desargues configuration one starts with two triangles tha
they are also ‘‘line perspective” (the sides meet in collinear points on the single black lin
and gray) and notices that the cross-wise intersections of points taken from each triple ha
that in projective plane geometry the Pappus property implies the Desarguesian propert
drawn in many ways that would at first sight appear almost unrecognizable.by Foley (1964). What Foley means by the ‘‘Desarguesian property”
is that any line that meets a plane in two distinct points lies fully in
that plane (see Fig. 4). This addresses the very existence of planes,
in Foley’s case in a binocular visual space. Here we study the exis-
tence of planes in a monocular visual space.
Foley worked with luminous points in a dark room, at finite dis-
tance and observed with both eyes. Foley’s aim was to empirically
check the axiom of Blank’s axiomatization of visual space (Blank,
1953) that boils down to the existence of planes. The outcome of
this attempt was unfortunately indecisive. In his abstract Foley
concludes:
‘‘. . .that the visual spaces of a significant proportion of the observ-
ers [2 out of 6 observers] are Desarguesian; those of others may be
non-Desarguesian”.
In this paper we report on an attempt to settle the issue for the
case of monocular visual space. It is a priori likely that one might
find very significant interindividual differences, as we have
encountered them in previous investigations (Norman, Crabtree,
Clayton, & Norman, 2005).
2. Essential rationale for the experiment
We implement a purely monocular visual space based upon the
single cue of apparent size. This is done by considering a space that
is empty except for a number of white, Lambertian spheres ofmetry: left, an example of the ‘‘Desargues configuration”;right, an example of the
t are ‘point perspective” (the sides concur in the single black point) and notices that
e). In the Pappus configuration one starts with two triples of collinear points (black
ppen to be collinear (the white points on the black line). In 1905 Hessenberg proved
y, thus the Pappus is the stronger condition. Notice that these configurations can be

one defined by a pair of points that are separated in the visual field
and the other a visual ray, specified by a third, single point. The
three points are collinear in the visual field. The ‘‘lines” in a visual
space are mental entities produced by the observer. The task is to
indicate the intersection of these lines by moving the third point
along the visual ray (purely in depth) so as to make it appear col-
linear with the point pair in visual space.
To continue the construction: let themidpoints of the arcs AB, BC
and CA (in the visual field) be, respectively, c, a and b (see Fig. 5).
The arcs cC, aA and bB intersect in the visual field in the common
point P. In visual space the point P is actually represented by three
different (‘‘parallel”) points at distinct distances from the eye, one
on the arc cC, one on the arc aA and one on the arc bB. In the exper-
iment a single session is composed of the following subtasks (notice
that arcs are oriented, for instance, the arc AC is the same geometri-
cal locus as the arc CA, but in the opposite orientation):
1. find the point b on the arc CA;
2. find the point b on the arc AC;
3. find the point c on the arc AB;
4. find the point c on the arc BA;
5. find the point P on the arc bB;
6. find the point P on the arc Bb;
7. find the point P0 on the arc cC;
8. find the point P0 on the arc Cc.
(See Fig. 6.) Notice that subtasks #1 and #2 are identical except
for the orientation. In the experimental paradigm the points A, C
and b are presented on a horizontal line, the arcs Ab and bC at
the correct angular sizes (see Fig. 7). The orientation difference
then implies a left–right mirror image. This also applies to subtasks
sychidentical diameter. The size cue is enabled by instructing the ob-
server that the spheres are indeed of equal sizes. This appears to
be natural enough to our observers (equal size appears to be the
default assumption of the visual system in cases of the complete
absence of prior information), though we evidently have to rely
on their applying this prior knowledge to the stimulus configura-
tions used in the experiment.
In this setting the size cue tends to function immediately and
transparently, that is to say, if one varies the relative size of objects
many observers experience a relative distance variation, rather than
size changes. There is a definite spectrum here though, several
observers noticed combined distance and size variations, whereas
one observer experienced all objects close to the frontoparallel
plane with mostly size variations. The observers thus varied be-
tween the extremes of (intuitively and automatically) ascribing
most of the variation to depth and to ascribing most of the varia-
tion to size. This is no doubt an important factor in the interpreta-
tion of the results.
We assume that the visual field as well as visual space is invari-
ant with respect to rotations about the vantage point. This should
apply at least approximately, although a variety of objections could
easily be raised. We use this assumption so as to be able to com-
bine observations obtained in a canonical configuration. In the
experiment we consider various configurations of three spheres
that are coplanar with the vantage point. In the canonical configu-
ration we use the horizontal plane, with the center sphere in the
primary (frontal) direction. Moreover, we repeat any configuration
in the left–right reversed configuration and average over the re-
sults. These conventions serve to force isotropy of the results. It
thus increases the confidence one might have in the results by fac-
toring out possible complications due to horizontal–vertical–obli-
que differences and so forth. Although all these restrictions could
be removed, this would involve a major effort and would render
the results less decisive.
Theoretically, any violations of Pasch’s Axiom (Axiom PS2) are
expected to vanish for small configurations and monotonically
grow with their size. Thus we design the experiment for a triangu-
lar configuration in the visual field of the maximum feasible size.
The limit is set by the extent of the visual field and the difficulties
many observers experience with extremely eccentric targets. We
decided on a spherical triangle in the visual field (see Fig. 5 left)
ABC with three equal sides of an arclength equal to a quarter great
circle (a right angle, that is p2 radians or 90). As a consequence the
interior angles of the triangle are also three right angles. We place
the spheres in the directions of the vertices of this spherical trian-
gle, but at a (very) different distance from the origin. Of the result-
ing triangle ABC in space (see Fig. 5 right) the distance from the
origin (that is the vantage point) to B is twice the distance to A
and the distance to C is even four times the distance to A.
At this point a short digression is in order because of a likely
confusion: Notice that we aim to address the projective structure
(if it exists) in a visual space, yet in the above we use metrical
descriptions in terms of Euclidean distances and angles. It might
be thought that this introduces methodological inconsistencies.
However, one has to distinguish sharply between physical space
in which we describe the structure of the stimulus and visual space
which is the arena of descriptions of the responses. These spaces
have different ontologies, physical space may simply be taken to
be ‘‘the space we move in”, whereas visual space is a mental entity.
Keeping these distinctions in mind at all times avoids possible con-
fusion considering consistency. In some discussions we use a third
type of space, namely a formal, hypothetical model of visual space.
This model is not Euclidean, but a space with a simple Riemannian
J.J. Koenderink et al. / Acta Pmetric. In this model we have a nexus of well-defined geodesics
which allows us to predict entities in visual space. Of course such
predictions have only hypothetical value, they are up for empiricalcheck. The key conceptual fact is that the psychophysical task is of a
purely projective nature. The observer is presented with two lines,
Fig. 5. left: A spherical triangle ABC with lines connecting the vertices to the
midpoints of the opposite sides. The common intersection at P is to be thought of as
potentially three different points, one on aA, one on bB and one on cC. The triangle is
in the visual field, thus the three points might be at different depths in visual space.
In the configuration used in the experiment ABC is an equilateral spherical triangle
with sides p2 (such a triangle is also equiangular with interior angles
p
2). In the
triangle APc the side Ac is p=4 (because half the side AB), the angle AcP ¼ p=2 (for
half of p, the straight line AcB), the angle cAP ¼ p=4 (for half of CAB ¼ p=2) and the
angle APc ¼ p=3 (because the six triangles meeting at P are congruent and the
interior angles at P add up to 2p). This suffices to solve for the remaining sides using
the conventional rules for the right angled spherical triangle. right: The configu-
ration in space. Notice that OB is twice OA and OC is four times OA. All the curved
lines are geodesics (shortest connecting ‘‘lines”) according to the model discussed in
the appendix. The direction Ob bisects the angle AOC and the direction Oc bisects
the angle AOB. The geodesics u and v evidently lack a common point, thus
(visually!) violating the Pasch axiom.
ologica 134 (2010) 40–47 43#3, #4, subtasks #5, #6 and subtasks #7, #8.
Notice that subtasks #1 and #2 have to be completed before
subtasks #5 and #6, and that subtasks #3 and #4 have to be com-

pleted before subtasks #7 and #8. The depth of b is found as the
mean of the depths found in subtasks #1 and #2 and the depth
of c as the mean of the depths found in subtasks #3 and #4. Thus
there exist constraints on the order in which the subtasks can be
performed. Subject to these constraints the order was randomized
for each full task.
The result of the experiment is the ratio of the mean of the
depths of P (obtained from subtasks #5 and #6) to that of the depth
of P0 (obtained from subtasks #7 and #8).
As mentioned above, all subtasks are presented in the canonical
configuration, a horizontal linear presentation. Fig. 7 gives an
impression of a typical situation. The observer is permitted to shift
the center sphere in depth (by changing its size) and is required to
make it collinear (in visual space!) with the two outer spheres.
Since this task has to be performed in visual space, it is far from
being trivial.
A full session yields a direct test of the validity of Pasch’s Axiom
(Axiom PS2) in a monocular visual space. Of course the structure of
the session implies that various settings are dependent upon each
other (for instance the result of subtask #5 evidently depends upon
the result of subtask #1). However, repeated sessions are indepen-
dent of each other. Therefore we use the mean of a number of
sessions in order to increase the accuracy of the estimate of any
possible violation. Such a repetition also allows us to find a mea-
sure of the significance of the result.
Our prior experience with the structure of monocular visual
space for both restricted (Koenderink & van Doorn, 2008) and very
wide (Koenderink, van Doorn, & Todd, 2009) visual fields suggests
at all believe this to be relevant. After the conclusion of the exper-
Fig. 7. A typical stimulus configuration. This is a view of three equal sized spheres
in the horizontal plane through the eye (the ‘‘horizon”) at different distances from
the observer. The spheres have white Lambertian surfaces and are illuminated by a
uniform, parallel beam from behind the observer. As is evident from the shading,
the center sphere is illuminated frontally. It is perhaps less obvious that the outer
spheres are illuminated in exactly the same way, to many observers it appears like
they were illuminated from different directions. The reason is that the field of view
is very large as can be judged from the black line segment which indicates the
correct viewing distance. From the correct vantage point the perspective deforma-
tions vanish, but (to most observers) the shading patterns still look somewhat
unexpected. The observer has control over the distance of the center sphere and has
the task to place it collinear with the outer spheres in visual space.
SP
JT
ST
AK
FP
SV
JL
WG
LA
WR
JK
FV
RE
SO
FN
AD
0.7 0.8 0.9 1.0 1.1 1.2 1.3
Fig. 8. Results for all observers. The boxes show the 25% and 75% quartiles as wel as
44 J.J. Koenderink et al. / Acta Psychologica 134 (2010) 40–47that large interindividual differences are likely to occur. This
expectation, coupled with the fact that observers find spatial tasks
involving very wide visual fields difficult, almost forced us to find
an occasion where a relatively large number (16) of very experi-
enced visual observers were available. All except three of the
observers were initially kept unaware of the structure and aim of
the experiment. This seemed at least prudent, although we do not
C
A
β
BA
B
β
P
C
γ
P'
γ
Fig. 6. From left to right, top to bottom, these are subtasks #1, #2 (symmetrical
pair), subtasks #3, #4 (symmetrical pair), subtasks #5, #6 (symmetrical pair),
subtasks #7, #8 (symmetrical pair). The actual appearance as a stimulus is shown in
Fig. 7.iment the observers (all well-known visual scientists) switched
their role and assumed their responsibilities as that of a coauthor.
3. Methods
Stimuli were presented on a large plasma display (conventional
96 cm diagonal TV monitor) driven by a macintosh powerbook
computer. The display was fitted with a frame designed to aid
the observer in using the preferred eye and keeping it at approxi-
mately the same location. In order to be able to do the task at all
the observer frequently had to perform head movements. The
eye position was kept within a 2-cm diameter area by way of a
‘‘key hole” aperture. The frame provided an additional aperture
so as to be able to leave room for the nose at the screen-side of
the key hole. The available field of view was 90 in the vertical
and appreciably larger (about 120) in the horizontal direction.
The stimuli were programmed in OpenGL via the Cocoa object-
C environment under the macintosh OS X Leopard operatingthe medians for the ratio of depths. In the planar case the value would be unity, as
indicated by the black vertical line. The model prediction is indicated by the gray
vertical line at right.

Appendix A. Prediction based on a simple dilation–rotation
invariant metric
The predictions of any model of monocular visual space should
be invariant with respect to arbitrary rotations–dilations about the
ego center. The reason is that neither absolute distance nor abso-
lute direction is optically specified. In order to be able to arrive
at a prediction one needs a metric. An obvious, simple model is
thus a Riemannian space with the required symmetry. Let the ego-
center be taken at the origin of the conventional Cartesian space R3
with coordinates fx; y; zg. The required ‘‘line element” (or metric) is
ds2 ¼ 2
C
dx2 þ dy2 þ dz2
x2 þ y2 þ z2
 
; ð2Þ
where C denotes a constant. The metric is (by construction) invariant
against arbitrary rotation–dilations. One easily checks that it defines
a curved space with constant positive scalar curvature equal to C.
A transformation into the conventional polar coordinates
f.; #;ug (. the distance from the egocenter, # the elevation with
respect to the horizon and u the azimuth) yields
sychologica 134 (2010) 40–47proportion of the observers are Desarguesian; whereas those of
others may be non-Desarguesian. Of course Foley’s conclusion ap-
plies to binocular visual space, whereas ours applies to monocular
visual space.
There can be no doubt that the monocular visual space of many
observers does not allow the definition of ‘‘visual planes”, since
Pasch’s Axiom (Axiom PS2) is (frequently very) significantly vio-
lated. This suggests that it might be prudent to refrain from refer-
ence to ‘‘visual planes” in any case. This is not necessarily
problematic since infinitesimal planar elements can always be de-
fined locally which is enough to be able to make sense of the local
curvatures of smooth surfaces. It is typical for general Riemannian
spaces to have violations of Pasch’s Axiom (Axiom PS2), the classi-
cal homogeneous Cayley–Klein spaces being very special, or singu-
lar, in this respect. The lack of planes by no means excludes the
existence of surfaces, which is what really matters in vision. The
main impact of our finding is as a constraint on conceptual geo-
metrical models of the structure of visual space.
The violations of the Desarguesian property occur in two qual-
itatively different ways in that the ratios either exceed or fall short
of unity. This might well indicate true qualitative differences with-
in the normal population. In earlier experiments (Koenderink et al.,
2009) we have found that such variations exist and are surprisingly
large. It will no doubt be rewarding, although very cumbersome, to
study this in greater detail, for many more observers and with a
considerable battery of different tests.
There can be no doubt that all observers—quite independent of
their psychophysical results—have no problems with their opti-
cally guided behavior in the physical world. This might seem sur-
prising given the fact that their visual experiences may differ
appreciably. Here one has to remember that even very different
‘‘user interfaces” (e.g., a UNIX command line interface and the
Windows GUI on PC’s) might very well prove to serve equally well
in various tasks (e.g., deleting or duplicating files in the example)
as also argued by Hofmann (in press).
Is it possible that the observers used a purely visual field (two-
dimensional) based strategy? Suppose the observer would perform
a linear interpolation of angular sizes, using angular distance in the
visual field to do the interpolation. For instance the objects at dis-
tances one and two would yield angular sizes proportional to one
and one-half. At the midpoint the interpolated size would be
three-quarters, yielding (using the size cue) a predicted distance
of four-thirds (that is 1:33 . . .). Continuing this type of calculation
one arrives at a ratio of 1:119 . . ., very close to the prediction
(1:12934 . . .) of the model discussed in the Appendix (the gray line
shown in Fig. 8). We cannot exclude that some (at least four) of the
observers used this strategy. This is in accord with the fact that
variations were sometimes experienced as depth sometimes as
size variations. However, one should remember that in the final
analysis any strategy is based upon (two-dimensional) visual field
structure. Whether an observer ‘‘performs the task in visual space”
instead of ‘‘performing it in the visual field” is a fundamentally
undecidable issue. Pictorial ‘‘depth” is a quale that has no psycho-
physical equivalent.
Thus far we have found no likely model that would predict vio-
lations less than unity. This means little though, since it is very dif-
ficult to come up with such models in a principled manner. Given
any Riemann metric it is a mere matter of calculation to find the
predicted violation, but given a violation there exists no principled
method to construct a Riemann metric that would predict it. Given
the data such violations no doubt are a bona fide category.
Acknowledgement
46 J.J. Koenderink et al. / Acta PHans Kolijn helped us to adapt a wide screen plasma display to
requirements of the experiment.ds2 ¼ 2
C
dn2 þ d#2 þ cos2 #du2
 
; ð3Þ
where n ¼ logð.=.Þ (with . an arbitrary unit length). Apparently the
space is the product of the 2-sphere and a line. In the polar coordi-
nate system the Riemann curvature tensor has two distinct, inde-
pendent components (others follow from the antisymmetry under
exchange of the last two indices), namely R2332 ¼  cos2 # and
R3232 ¼ 1.
Since the metric is centrally symmetric the geodesics have to be
planar curves, confined to planes through the egocenter. Taking a
plane u ¼ constant one has
ds2 ¼ 2
C
ðdn2 þ d#2Þ; ð4Þ
which has the structure of the Euclidean metric. Thus one sees
immediately that the geodesics are logarithmic spirals of the form
. ¼ .0 ekð##0Þ; ð5Þ
thus there is no need to integrate the geodesic equations explicitly3.
This geometry is sufficient to find a prediction for the experi-
mental results. We set . ¼ 1 for convenience, the actual value
being irrelevant to the problem. Consider a spherical triangle with
3 Although the symmetry argument appears compelling to us, a referee of the
manuscript expressed doubts. In case the reader has similar doubts, here is a formal
derivation: The geodesic equations (using the summation convention)
d2xk
dt2
þ Cklm
dxl
dt
dxm
dt
¼ 0; ð6Þ
in polar coordinates are (the mutually independent, non-zero Christoffel symbols
(others follow from the symmetry under exchange of the last two indices) are
C111 ¼ 1=., C
2
33 ¼ sin# cos# and C
3
32 ¼  tan#):
€. ¼
_.2
. ; ð7Þ
€# ¼  sin# cos # _u2; ð8Þ
€u ¼ 2 tan# _# _u; ð9Þ
where the dots indicate derivatives with respect to arc length. Given an initial point,
and an initial direction, select the coordinates system such that the coordinates of
the point are f.0;0;0g and the direction fsinl; cosl;0gðl slope of the geodesic).
This is always possible because of the spherical symmetry. Then Eq. (9) becomes
simply €u ¼ 0, thus one has u ¼ 0 throughout, that is to say, the geodesic is a planar
curve in the plane u ¼ 0. Eq. (8)becomes €# ¼ 0, with the immediate solution
# ¼ coslr, where r denotes the geodesic distance from the initial point. The
remaining Eq. (7), that is . €. _.2 ¼ 0 is solved by .ðrÞ ¼ C1 expC2r. Solving for
the constants of integration C1;2 one obtains .ðrÞ ¼ .0 exp sinlr. Elimation of the
geodesic distance r from this parametric representation of the geodesic one finds
.ð#Þ ¼ .0 exp tanl#, which is the form (5) again. Thus the geodesics are indeed pla-
nar logarithmic spirals. Notice that selecting an appropriate orientation of the polar
coordinate frame effectively formalizes the symmetry argument used above.

vertices in the directions e1 ¼ f1;0;0g, e2 ¼ f0;1;0g and
e3 ¼ f0;0;1g. This triangle is equilateral with sides p=2 and also
equiangular with interior angles p=2. The three bisectors of the an-
gles (and thus also of the opposite sides) meet in a single point and
divide each other in arcs with sines in the ratio 1:
ffiffiffi
2
p
, as application
of the sine rule in one of the six interior triangles shows. Next we
introduce unequal depths at the vertices of a spatial triangle ABC
with A ¼ e1, B ¼ 2e2 and C ¼ 4e3. The depths at the midpoints of
the sides are simply the geometrical means of the depths at the
vertices, thus the depth at the midpoint of CA is 2 and that at the
midpoint of AB is
ffiffiffi
2
p
. The depth in the direction of the center of
the triangle on the geodesic that connects the vertex C to the mid-
point of the side AB is (by a linear interpolation in the fn; #g-plane)
22
3arccos 1ffiffi
3
p
p ¼ 2:12541 . . . ; ð10Þ
whereas the depth in the direction of the center of the triangle on
the geodesic that connects the vertex A to the midpoint of the side
BC is (again by linear interpolation in the fn; #g-plane)
8
arccos 1ffiffi
3
p
p ¼ 1:88199 . . . : ð11Þ
Thus the two points are seen in the same direction but have differ-
ent depths, the depths being in the ratio (this is where the arbitrary
unit .0 cancels out)
22
6arccos 1ffiffi
3
p
p ¼ 1:12934 . . . : ð12Þ
This is the prediction of the model.
Notice that even such a very simple conceptual model, based on
nothing more than invariance against arbitrary rotations (eye
movements) and dilations (mental scale adjustments, since no
scale is optically specified) yields a metric that violates Pasch’s Ax-
iom (Axiom PS2).
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