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The spherical perspective has not yet been widely introduced into design education. Its ability to serve as a meta-class model of vanishing point perspective systems, giving a teacher the opportunity to present approximations of the straight linear perspective models (with one, two or three vanishing points) all in one system, is presented in this article. The mathematical basis for a spherical grid as a curvilinear approximation to one-eyed human vision and a didactic approach for its integration into design oriented perspective freehand drawing are also discussed.
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© Faculty of Mechanical Engineering, Belgrade. All rights reserved FME Transactions (2019) 47, 343-348 343
Received: June 2018, Accepted: November 2018
Correspondence to: Matthias Kulcke, Lecturer,
HafenCity Universität Hamburg, Faculty of
Architecture, Hamburg, Germany
Matthias Kulcke
HafenCity Universität Hamburg
Faculty of Architecture
Hamburg University of Technology
Institute of Building Technology
Spherical Perspective in Design
The spherical perspective has not yet been widely introduced into design
education. Its ability to serve as a meta-class model of vanishing point
perspective systems, giving a teacher the opportunity to present
approximations of the straight linear perspective models (with one, two or
three vanishing points) all in one system, is presented in this article. The
mathematical basis for a spherical grid as a curvilinear approximation to
one-eyed human vision and a didactic approach for its integration into
design oriented perspective freehand drawing are also discussed.
Keywords: Spherical perspective; curvilinear perspective; vanishing point
perspective; design education; augmented reality; immersion.
With a simple example Bruno Ernst [1] provided an
explanation for a work of M.C. Escher who is known
for having played an important role in introducing the
use of curvilinear (in this case cylindrical) perspective
into modern art. Ernst did this by asking the reader to
picture him/herself lying underneath two parallel
infinite telegraph wires and then establishing two basic
rules for perspective spatial perception:
1. Distant objects are perceived smaller than objects
of the same size located closer to the viewer;
2. Lines or edges that are continuous and diffe–
rentiable in reality are perceived preserving these
qualities if they are in eye-focus and unobstructed by
other objects.
One consequence is, e.g. that, these rules are to be
strictly observed, buildings viewed by people standing
on street-level can only be displayed correctly on a flat
picture plane in spherical perspective (Fig. 5, left-hand
1.1 Mathematics
To construct a spherical grid, an orthogonal grid based on
horizontal and vertical straight lines in a two-dimen–
sional plane is projected onto a sphere. The resulting grid
on the sphere’s surface is then rolled out cylindrically in
the horizontal and vertical direction onto a plane parallel
to the original grid, following its orthogonal logic. To
achieve this, each projection from straight line onto the
sphere, resulting in a half circle on the sphere’s surface, is
again projected onto the sphere’s equator plane as an
ellipse. The distance p from the sphere’s center to the
points defining the contour of each ellipse is calculated;
finally the arccos of this distance represents the y-value
of the correponding points of cylindrically rolled out
spherical projection plane.
=− (2)
cos 1ba
=∧= (3)
11cos cos
−− ⋅
arccos arccos
1sin cos
1.2 A Macro to Display Sinoid Curves
Thus it becomes feasible to construct a curvilinear grid,
e.g. in AutoCAD, in this case using the CAD-software's
VBA (Visual Basic for Applications) macro progra–
mming API. VBA for AutoCAD in the used version
only allows for calculating arctan (atn) - to avoid
division by zero. Therefore arccos is a reference to an
additional function not listed here. Also 0° and 90° are
not calculated by the macro program.
Private Sub
Dim pt1(0 To 2) As Double
Dim pt2(0 To 2) As Double
Dim dblX1 As Double
Dim dblX2 As Double
Dim intZaehl1 As Integer
Dim intZaehl2 As Integer
Dim pi As Double
pi = 3.14159265358979
dblX1 = pi / 2 - 0 / 360 * 2 * pi
dblX2 = pi / 2 - 1 / 360 * 2 * pi
For intZaehl1 = 1 To 89
For intZaehl2 = 1 To 89
344 VOL. 47, No 2, 2019 FME Transactions
pt1(0) = (pi / 2) / 90 * (intZaehl2 - 1)
pt1(1) = arccos(Sqr((Cos((pi / 2) / 90 * (intZaehl1)))
^ 2 / (1 - ((Sin((pi / 2) / 90 * (intZaehl1))) ^ 2) *
Cos(dblX1) ^ 2)))
pt2(0) = (pi / 2) / 90 * (intZaehl2)
pt2(1) = arccos(Sqr((Cos((pi / 2) / 90 * intZaehl1)) ^
2 / (1 - ((Sin((pi / 2) / 90 * intZaehl1)) ^ 2) *
Cos(dblX2) ^ 2)))
ThisDrawing.ModelSpace.AddLine pt2, pt1
dblX1 = pi / 2 + ((intZaehl2 - 1) / 360 * 2 * pi)
dblX2 = pi / 2 + ((intZaehl2) / 360 * 2 * pi)
Next intZaehl2
Next intZaehl1
End Sub
Putting this macro to use the following partial (π/2 *
π/2) representation of horizontal lines in a spherical grid
can be produced - only one third of the calculated
curves are shown here (Fig. 1). This calculated quadrant
can be mirrored and assembled to form a full (π * π)
spherical grid (Fig. 2).
Figure 1. Top right quadrant of the spherically projected
and rolled out horizontal lines (Kulcke 2018)
1.3 Changes in the Grid according to View Point
In contrast to straight-line vanishing point perspective
systems the curvature of lines is changed according to
eye-focus (see e.g. [2]). This explains why different
spherical systems can be found in literature.
Systems showing straight lines going through a
central vanishing point (Fig. 2) show actually a very
special case; when exactly this vanishing point is in eye-
focus. And, geometrically speaking, it never is, since it
is defined only by the lines pointing to it (for an artistic
discussion on this issue see [3]). As soon as the eye
chooses one of the lines rather than the point itself, the
system should change accordingly (Fig. 3 and Fig. 4).
Figure 2. Curvilinear system with exact center focus
(Kulcke 2018)
Figure 3. Change of curvature due to line focus, horizontal
direction (Kulcke 2018)
Figure 4. Change of curvature due to line focus, vertical
direction (Kulcke 2018)
FME Transactions VOL. 47, No 2, 2019 345
To raise awareness with the students considering this
phenomenon and to also stay within the explanatory
logic of Ernst's example, the spherical grid introduced to
the students (Fig. 5) uses sinoid curves representing the
third dimension going toward the systems center. Also
the curves are simplified to sinus curves so they are
easier to draw in freehand.
Figure 5. Spherical perspective (using simplified sinoid
curves) as an explanatory model (Kulcke 2007)
An interest for using curvilinear perspective or distor–
tions of orthogonal architectural concepts, e.g. hinting at
a wide-angle (fish-eye) distortion, is seen frequently in
contemporary architecture, as in building structures like
Norman Fosters "Chesa Futura" [4]. These designs are
not merely free form but can be perceived as a reference
to the use of distortion via a curvilinear perspective grid.
One of the reasons for the current use of such concepts
might be that architects and builders have only recently
significantly increased their ability to handle them in
design and construction processes by using CAD-
Software (like Rhino/Grasshopper) in combination with
CAM (computer aided manufacturing).
However, the spherical perspective (as the
curvilinear perspective par excellance) has not been
widely introduced into design education; literature on
applied perspective construction is often confined to
straight linear perspective (e.g. [5], [6]). And if
spherical perspective is mentioned, then it is usually
presented as an "outside the box" model (e.g. [7], [8],
[9]) rather than as what it should be viewed as: A meta-
class model of perspective representation, giving a
teacher the opportunity to present approximations of
straight linear perspective models (with one, two or
three vanishing points) all in one system (Fig. 5).
2.1 The Didactic Approach
Thus it becomes easier to explain to students in which
cases to use a certain straight linear model and also to
show the limits of each of these models. The author has
used spherical perspective as an explanatory system to
teach students the basics of freehand-drawn perspective
within the subject "Prozesse der Innenraumgestaltung"
("the interior design process") at the Hamburg
University of Technology since 2007 and in the course
"Methoden der Darstellung" ("presentation methods")
since 2011. The free use of the spherical grid (Fig. 7)
and the benefits as well as the limits of the straight
linear perspective models are told in several sets of
lecturing, practical exercises and corrections, leading to
an understanding of the basic principles of perspective
and their application in design processes.
2.2 Freehand Drawing in the CAD Era
The author also intends to raise the awareness with the
students that a general understanding of CAD doesn't
have to be limited to 2D and 3D technical drawing and
modeling on the computer. The refinement of sketches
e.g. with image enhancing software may also serve as a
feedback regarding the status of their designs and often
lead to the creation of alternative concepts and sketches.
Therefore, in contrast to the craft of drafting with a ruler
and similar devices, freehand drawing hasn't been
replaced to the same extent by digital tools and
techniques; it has, on the contrary, even gained
importance since image enhancing software can easily
enrich a quick sketch with color and contrast.
2.3 Course Structure
The course "Methoden der Darstellung" contains
freehand drawing instruction on free sketching, parallel
projections, vanishing point perspective systems as well
as layout and design exercises. Due to the articles’s
subject only the part on vanishing point perspectives
will be explained in further detail.
Figure 6. Spherical perspective in freehand drawing, grid
construction (Kulcke 2011)
In four sessions curvilinear and straight line vanishing
point perspectives are explained to the participants. The
first session introduces Ernst's example and thus a
346 VOL. 47, No 2, 2019 FME Transactions
spherical grid is successively discussed in its
representation of all three dimensions. In this order the
lessons on vanishing point perspective systems start with
the seemingly most complicated, but actually with the
least abstract system of the four, since it is closest to
actual human vision. After the introduction the grid is
constructed in a freehand drawing exercise and simple
objects are drawn into the system (Fig. 6 and Fig. 7).
Figure 7. Spherical perspective in freehand drawing, filling
in objects (Kulcke 2011)
Figure 8. Student work (Grocholl 2016)
Figure 9. Student work (Volkmann 2013)
The freehand drawing approach exempts the stu–
dents from worrying too much about actual dimensions,
even if they draw more specific objects (Fig. 9). This
allows them to focus on questions of edge constellations
and directions.
2.4 Explaining Straight Line Perspective as a Fol-
Figure 10. From curvilinear to straight line perspective 1
(Kulcke 2014)
Figure 11. From curvilinear to straight line perspective 2
(Kulcke 2014)
The results are promising, almost every student
manages to comply with the basic rules of perspective
drawing within the one and a half hours of the first
FME Transactions VOL. 47, No 2, 2019 347
session. In the following lessons straight linear pers–
pective systems, with one, two or three vanishing
points, are successively introduced and explained as
simplifications in reference to the spherical grid.
In the age of the geometrical CAD System that
creates images in vanishing point perspectives on the
basis of 3D models, it is important to adapt and refine
didactic strategies in the academic realm when teaching
students about the essence of vanishing point perspec–
tive representations. Using spherical perspective as an
explanatory model may serve as a foundation regarding
such approaches.
While it still remains to be proven, that the described
didactic approach is superior to others in regards to un-
derstanding straight line vanishing point perspective,
there are further reasons to teach students about cur-
vilinear systems.
An awareness and deeper knowledge of spherical
perspective could prove to be crucial for advanced aug–
mented reality (AR) applications and the design and pla–
nning processes using them. Since the human eye
perceives straight edges next to an edge in focus, due to the
curvature of its lens and other geometrical factors, in
spherical distortion (e.g. [2]), the simulated objects that
complete an urban situation, for example as simulated
image parts on an optical see-through display seen within
the already existing surroundings, will be more con–
vincing if the simulated part is itself distorted accor–dingly.
Virtual reality (VR) software usually uses a static
representation of curvature, which is still a crutch, beca–
use eye movement constantly changes the focal point and
therefore the vanishing point of the system and thus the
relationship between object and edge curvature as well as
constellation (see also [2] and [10]).
VR glasses and the accompanying applications keep
the vanishing point at the center of the calculated image
(as if the eye would stare all the time in a right angle on
the image), thereby potentially weakening immersive
illu–sion as soon as eye movement without head
movement occurs.
The application of curvilinear grids to willingly distort
an object or a building should not be reduced to simply
displaying a prerogative of the designer or to create a
superficial effect. Consciously integrating the category
of spherical distortion into the design process might
actually lead to better designs, especially in but not
limited to the urban design context. Thus designers and
architects may reconnect modern practise to design
techniques already known and used, e.g. for the shaping
of columns in ancient Greek architecture.
In architectural and urban design the inclusion of
public and/or customer opinion at an early state into the
planning process becomes an increasingly important
factor, especially if participatory planning processes are
put into practice. The use of VR and AR technologies, as
described under 3, on building sites, in learning envi–
ronments and excavation sites (combining the excavation
with 3D models of reconstructed building elements) that
are open to the public is also a tempting possibility, if not
already in use. In the light of these developments aug–
mented reality and its immersive perfection, through a
better understanding of and awareness for spherical
perspective, and its implementation into 3D-simulation
CAD-software, also taking stereo vision (see [11]) and
last but not least eye movement into consideration, are
bound to become more important. To integrate especially
the latter into modern CAD and visualization systems a
deeper and widespread understanding of spherical pers–
pective and its dynamic grid changes, due to eye move–
ment are a critical factor for its designers and developers.
[1] Ernst, B.: Der Zauberspiegel des M.C. Escher,
TACO Verlagsgesellschaft und Agentur mbH,
Berlin, 1986.
[2] Vero, R.: Understanding Perspective, Van Nost–
rand Reinhold Company a division of Litton
Educational Publishing, New York, 1980.
[3] Kandinsky, W.: Punkt und Linie zu Fläche -
Beitrag zur Analyse der malerischen Elemente,
Benteli-Verlag, Bern-Bümpelitz, 1959.
[4] Foster + Partners, Kuchel Architects: Project Chesa
Futura, St Moritz, Switzerland, https://www.foster–, 2000– 2004.
[5] Schaarwächter, G: Perspektive für Architekten,
Verlag Gerd Hatje, Stuttgart, 1964.
[6] Weiner, G.: Perspektive - Konstruktionsaufgaben
mit einer Einführung in die allgemeinen Grund–
lagen, Fachbuchverlag GmbH, Leipzig, 1950.
[7] Mauldin, J. H.: Perspective Design, Van Nostrand
Reinhold Company Inc, New York, 1985.
[8] Raynes, J.: Handbuch Künstlerische Perspektive,
Knaur Ratgeber Verlage, München, 2005.
[9] Smith, R.: Perspektive: Zeichnen und Malen,
Ravensburger Buchverlag, Ravensburg, 1996.
[10] Klee P.: Das bildnerische Denken, Schwabe & Co
Verlag, Basel/Stuttgart, 1964.
[11] Cyganek, B., Siebert J.P.: An Introduction to 3D
Computer Vision Techniques and Algorithms, John
Wiley and Sons Ltd, Chichester, West Sussex, 2009.
М. Кулчке
Сферна перспектива још увек није у већој мери
уведена у дизајнерско образовање. Могућност њене
примене као модела мета-класе система перспективе
нестајуће тачке, која даје предавачу прилику да
представи апроксимацију модела перспективе праве
линије (применом једне, две или три нестајућих
тачака) у оквиру истог система је приказана у овом
раду. Такође је разматрана и математичка основа
348 VOL. 47, No 2, 2019 FME Transactions
сферне мреже са криволинијском апроксимацијом
једнооког људског вида и дидактичким приступом
њеној интеграцији у концепт дизајна слободном
руком у оријентисаној перспективи.
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A digital cooperation methodology based in the scientific definition of cubical perspective is presented. The research gathered new methods for cubical perspectives drawing with their final VR fruition through digital techniques. The developments belong to the disciplinary field of drawing and are reinforced with definitions from the mathematical one. The utility of the approach was tested through cases study. Among the outputs, the investigation defined the “hybrid immersive model” (HIM), a model that joins traditional drawing with digital immersive techniques. As well, the research developed new theories and methods for cubical perspective, starting to fill an existing gap in the field of drawing. To such an end, it organized two methods with systematic definitions for drawing cubical perspectives by using simple tools, both from direct observation and from associated floorplan and section. Specifically, the second method transferred several properties already in use by spherical perspectives (e.g., the use of exactly two vanishing points per line) within the compact and linear setup of the cubical map. Four cases study were conducted, highlighting the advantages of a holistic vision in immersive modality, where both traditional and innovative techniques are a correlated way of thinking and not antagonists anymore. The applications developed prototypes using the new methods for cubical drawing, covering both survey and project cultures, and showing the impact of the procedure for the study, analysis and spread of architectural data. Hence, the research used representation as a way of knowledge and study verified in the architectural composition plane. The operator highlighted and interpreted relations among extracted/abstracted elements using the hybrid model. Although, such a personal analysis resulted methodologically comparable in the plane of representation, revealing the procedure as a useful tool for the philological reading, the critical analysis of the spatial geometry’ relationships and the definition of modular laws.
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We analyze in detail a specific image of curvilinear perspective developed by Andre Barre and Albert Flocon in 1968 which depicts a square room from its interior by a 360° total perspective. Since its publication, their work has become an important point of reference, but they did not say how it was done. We discuss how this specific case of a pseudo-spherical projection for a hypercube was achieved by hybridized using ellipse curves in one of the spatial directions and a different spherical projection in the other direction. For this analysis we develop the mathematical expressions and study how the ellipsoidal and spherical transformations are hybridized into one solution. Barre and Flocon attempted to use both projections in an angular curvilinear grid as a practical drawing method. One of the main interests of having access to a 360° curvilinear grid arises its ability to act as a drawing guide. We provide an example of its use to demonstrate its practicality.
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Cubical perspective is maybe the youngest alternative among the options for immersive drawing. Its theoretical definition, as every immersive perspective, has its pros and cons. Furthermore, its utility has not been fully explored, e.g. for early conceptual design. In fact, while its theory is on its way to being solved with different methods, there is still a doubtful judgement about the practice: is cubical perspective a way to think? A tool? Can we do something different with it, or are we just playing with technology? Let us try to give some answers to these questions by presenting some experiences held in university classrooms. We are presenting the experience of architecture, engineering and product design students that have applied cubical perspective to the creation of immersive panoramas subsequently assembled into a virtual tour. This article, more than closing a discussion, aims at opening the debate about the real applications and utility of cubical perspective and push for experimentation.
Computer vision encompasses the construction of integrated vision systems and the application of vision to problems of real-world importance. The process of creating 3D models is still rather difficult, requiring mechanical measurement of the camera positions or manual alignment of partial 3D views of a scene. However using algorithms, it is possible to take a collection of stereo-pair images of a scene and then automatically produce a photo-realistic, geometrically accurate digital 3D model. This book provides a comprehensive introduction to the methods, theories and algorithms of 3D computer vision. Almost every theoretical issue is underpinned with practical implementation or a working algorithm using pseudo-code and complete code written in C++ and MatLab®. There is the additional clarification of an accompanying website with downloadable software, case studies and exercises. Organised in three parts, Cyganek and Siebert give a brief history of vision research, and subsequently: present basic low-level image processing operations for image matching, including a separate chapter on image matching algorithms; explain scale-space vision, as well as space reconstruction and multiview integration; demonstrate a variety of practical applications for 3D surface imaging and analysis; provide concise appendices on topics such as the basics of projective geometry and tensor calculus for image processing, distortion and noise in images plus image warping procedures. An Introduction to 3D Computer Vision Algorithms and Techniques is a valuable reference for practitioners and programmers working in 3D computer vision, image processing and analysis as well as computer visualisation. It would also be of interest to advanced students and researchers in the fields of engineering, computer science, clinical photography, robotics, graphics and mathematics.
Der Zauberspiegel des M.C. Escher, TACO Verlagsgesellschaft und Agentur mbH
  • B Ernst
Ernst, B.: Der Zauberspiegel des M.C. Escher, TACO Verlagsgesellschaft und Agentur mbH, Berlin, 1986.
Understanding Perspective, Van Nostrand Reinhold Company a division of Litton Educational Publishing
  • R Vero
Vero, R.: Understanding Perspective, Van Nostrand Reinhold Company a division of Litton Educational Publishing, New York, 1980.
Perspektive - Konstruktionsaufgaben mit einer Einführung in die allgemeinen Grundlagen
  • G Weiner
Weiner, G.: Perspektive -Konstruktionsaufgaben mit einer Einführung in die allgemeinen Grundlagen, Fachbuchverlag GmbH, Leipzig, 1950.
Handbuch Künstlerische Perspektive
  • J Raynes
Raynes, J.: Handbuch Künstlerische Perspektive, Knaur Ratgeber Verlage, München, 2005.
Perspektive: Zeichnen und Malen
  • R Smith
Smith, R.: Perspektive: Zeichnen und Malen, Ravensburger Buchverlag, Ravensburg, 1996.