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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

E-mail: matthias@kulcke.de

doi:10.5937/fmet1902343K

Matthias Kulcke

Lecturer

HafenCity Universität Hamburg

Faculty of Architecture

and

Hamburg University of Technology

Institute of Building Technology

Germany

Spherical Perspective in Design

Education

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.

1. SPHERICAL PERSPECTIVE AS AN APPROXI–

MATION TO HUMAN VISION

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

object).

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

22

1cos

b

e

a

ϕ

⎛⎞

−⋅

⎜⎟

⎝⎠

(1)

2

1cose

α

=− (2)

cos 1ba

α

=∧= (3)

()

2

22

cos

11cos cos

p

α

αα

=

−− ⋅

(4)

2

22

cos

arccos arccos

1sin cos

p

α

αϕ

=

−⋅

(5)

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

Linsenraster_CommandButton_1_Click()

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)

2. SPHERICAL PERSPECTIVE AS AN APPROXI–

MATION TO HUMAN VISION

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-

low-Up

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.

3. THE SPHERICAL PERSPECTIVE IN THE CON–

TEXT OF AUGMENTED REALITY

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.

4. CONCLUSION AND OUTLOOK

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.

REFERENCES

[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–

andpartners.com/projects/chesa-futura/, 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.

СФЕРНА ПЕРСПЕКТИВА У ОКВИРУ

ДИЗАЈНЕРСКОГ ОБРАЗОВАЊА

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.