JUNE 2017  VOLUME 1 ISSUE 1 JIDEG 41 Abstract: This work present two methods for determining the development of double-curved surface. The aims of… [602995]

JUNE 2017  VOLUME 1 ISSUE 1 JIDEG 41
Abstract: This work present two methods for determining the development of double-curved surface. The
aims of this paper is to show to comparative methods for determination of the sheet metal requirements for
complex roof cover shape. In first part of the paper are presented the basic sketch and information about
the roof shape and some consecrated buildings, which have a complex roof shape. The second part of the
paper shows two methods for determining the developed of the spherical roof. The graphical method is the
first method used for developing of the spherical shape. In this method it used the poly-cylindrical method
to develop the double-curved surface. The second method is accomplishing by using the dedicated CAD
software method.

Key words: roof shape, descriptive geometry, developed surface, CAD.
INTRODUCTION
Nowadays rapid development of the building with the
complex surface prior require to use the graphical
methods. From the oldest times the people has used
approximate methods to make roofs with complex
surface from different buildings. The style and shape of
the roof varies depending on the geographic location,
type of the building and the nation from that area, or
country. In the ancient time the construction of the many
buildings are accomplished according to golden ratio
proportion. Using this method, the buildings have a
beautiful aspect and a strong structure.
Since the 18th century the mathematician Gaspard
Monge published the first treaty of Descriptive
Geometry, creating a graphical representation approach
of the spatial object in orthogonal projections, solving the
many problems in an easy a precise engineering manner.

2. ROOF SHAPES
Some of the arhitectural roofs shapes it used
especially in religious buildings, witch are composed of
arcs of a circle, [7], [9] are presented below:
• Semicircle
The ends of the semicircle points that define the roof
are collinear with the center point, the shape of the arch
are presented in figure 1.

Fig. 1 Semicircle arch Fig. 2 Stilted semicircle
Stilted semicircle arch, presented in figure 2, are ends
extended below of the center point of the center arch.
These are vertical lines from the center point of the arch.
Semi-circular arch is especially characteristic of
Romanesque architecture. • Segmental arch
The extreme points of this construction are located
above of the center point arc. The curve is an arch with
an angle less than 180 degrees, as can be seen in figure 3.

Fig. 3 Segmental arch Fig. 4 Stilted segmental arch
The stilted segmental arch center is located below of
the ends. Both ends of the arc are extended by two
vertical line segments. The shape of the stilted segmental
arch is presented in figure 4.
• Horseshoe Arch
The ends of the horseshoe arch, presented in figure 5,
are situated below of the arc center. This architectural are
also called Moorish arch, it is based on the Islamic
concept. Construction of the mosques roof used of the
horseshoe arch. This arch has an angle greater than 180
degrees.

Fig. 5 Horseshoe Fig. 6 Stilted Horseshoe
The stilted horseshoe has the arch ends extended two
vertical line segment, as can be seen in figure 6.
• Pointed Arch
This shape is specifically of the gothic style. Figure 7
presents the shape of the pointed equilateral construction.
The radius of the arch has the same size with the base of METHOD TO DEVELOP THE DOUBLE-CURVED SURFACE OF THE ROOF
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Method to develop the double-curved surface of the roof
JUNE 2017  VOLUME 7 ISSUE 1 JIDEG 42 the constructions. The pointed lancet arch, present ed in
figure 8, has a radius size greater than the base l ength.

Fi g.7 Pointed equilateral Fig. 8 Pointed lancet

In figure 9 is presented the pointed obtuse shape. The
center of the arc is situated on the horizontal lin e with the
arch start point. Pointed segmental shape presented in
figure 10 has the arch center situated below of the arch
start point.

Fig. 9 Pointed obtuse Fig. 10 Pointed segmental

• Three centered arch

Construction of the three centered arch is composed
from there circle arcs, as shown in figure 11. The center
of the extremity arc is placed on the horizontal li ne.

When the shape is composed from two circle arcs,
which are drawn to the tangent segments, like as in figure
12, the shape is called quasi four-centred.

• Four-Centered Arch

This shapes are presented in figure 13 a, b. Four-
centred arch are parts of four different circle. Th is arch
style has been considered characteristic to England , but it
was common both in Flanders.

a) b)
Fig. 13 Four-Centered Arch

The arch presented in figure 13, a), b) is called T udor
arch. Architectural Tudor style was developed in
medieval England, during the Tudor period (1485-160 3). • Ogee Arch

The ogee arch consisting of a concave arc into
convex arc, with vertical ends. This ogee arches we re
feature of the English gothic architecture and it a re used
mainly for decoration, because does not have a stro ng
resistance to the action of the environmental loads . The
shapes of the ogee arch are presented in figure 14.

a) b)
Fig. 1 4 Ogee Arch
• Trefoil arch

Trefoil shape is composed from three center arc, as
can be seen in figure 15, a, b.

a) b)
Fig. 1 5 Trefoil arch

• Pointed arch trefoil

The shape pointed arch trifoliate presented in figu re
16. This shape is used mainly for decoration,
characteristic of Gothic architecture, characterize d by the
ribbed vault. A shouldered arch, presented in figur e 17, is
a style of arch used especially over a doorway. Tho se
shapes are called Corbels [1], [5]. This style is a pplied in
castle constructions, both inside and outside, or i n the
bridge constructions [3].

Fig. 1 6 Pointed arch
trifoliate Fig. 17 Shouldered arch

Among the many buildings that have spherical roof
are mosques. The oldest mosque, Quba Mosque was bui lt
in the Islamic style in 622 in Medina, Saudi Arabia [8].
In figure 18 is presented a view of this mosque, it can be
observing the semi spherical elements from the roof ,
specific of the Islamic architectural style.

Fig. 11 Three centered arch Fig. 12 Quasi four-centred

Method to develop the double-curved surface of the roof
JUNE 2017  VOLUME 1 ISSUE 1 JIDEG 43

Fig. 18 Quba Mosque from Medina, Saudi Arabia [8]
Another dome construction is presented below in
figure 19, were is presented the picture of Dome of the
Rock from Jerusalem, Iterioer [6].

Fig. 19 Dome of the Rock from Ierusalem, Iterioer [4]

3. DEVELOPING METHOD
In the following paragraphs are presented two
methods to develop the spherical and cylindrical su rface.
In the first method is applied the graphical descri ptive
methods and second method are solved using CAD
modelling techniques.

3.1 Descriptive geometry method

It is known that the sphere is a double curved and is
not developable, however it can be developed using the
approximatively methods, divided the surface in sma ll
elements [2]. The most popular methods for developi ng
the spherical surface are:

• poly-cylindrical method (Gore method) – this
method is applied to approximate development
of the spherical surface dividing surface into
single-curved cylindrical surface. Precision
results is influenced by the number o that is
divided the spherical surface, if the surface is
divided into several elements the accuracy of
the results is higher.
• poly-conical method – this method it used to
approximate development of a spherical
surface by substituting the double-curved
surface into single-curved conical surface.

In figure 20 are presented the semi-sphere and
cylindrical surface development using construction by
the poly-cylindrical method. The height of the cyli nder
surface is 3 m and the radius of the semi-sphere ar e 2 m.
After solving the problem with descriptive geometry
method result an approximate length of the unfolded
cylinder equal to 12.24 m.

Fig. 20 Semi-sphere and cylindrical surface development us ing construction by the poly-cylindrical method

3.2 CAD method

Developed of the semi sphere and the cylindrical
surface it solved using the CAD dedicated software. Three-dimensional model is modelled in SolidWorks,
using the surface module. The surface is developed
using Flatten surface command. In the first step is
developed the cylinder surface, resulting a necessa ry

Method to develop the double-curved surface of the roof
JUNE 2017  VOLUME 1 ISSUE 1 JIDEG 42length of the sheet metal of 12.56 meters, as in figure
21. The spherical surface is developed in figure 22.
Fig. 21 Cylinder
development Fig. 22 Semi-sphere
development
Deformation plot of the cylinder surface and semi
spherical surface are presented in figure 23 a), b),
were can be observed the distribution of the
deformation necessary to fold the model.

a) b)
Fig.23Deformation plot of the cylinder and semi spherical
surface
In table 1 are presented the area results of the studied
surface. It can be seen as the results of the CAD method
are more accurate by solving the problem with
descriptive geometry method.
Table 1 – Comparative results
Descriptive
geometry CAD
Area of the
cylindrical surface 36.72 38.67
Area of the
spherical surface 24.88 24.91

4. CONCLUSION
In this work are presented two methods to develop
double-curved surface. Following the results, it can be
observed as the CAD method provides more accurate
results. A great advantage for using CAD software are
given by the accuracy solving of the problem. A
considerable disadvantage is represented by the necessity
of specialized training and high cost of software. Every
method used in the paper presents specific advantages.
Descriptive geometry method is suitable for educational
purpose, creating a better vision of the spatial and
orthogonal representation of the body and surface.
REFERENCES
[1] Abushadi, E., The evolution of Islamic arches till
1250, 17pg, Department: American University in Cairo. Dept. of Arab and Islamic Civilizations, April
27, 2011, http://dar.aucegypt.edu/handle/10526/4291,
Accesed: 2017-02-14.
[2] Bodea, S, Geometrie descriptiv ă, Publisher Risoprint,
ISBN 973-656-353-7, Cluj Napoca, 2006.
[3] Bjurstrom, H., Capacity assement of a bridge with
backfill- A case study of the Glomman Bridge, ISSN
1103-4297, Master of Science Thesis, Stockholm,
Sweden, 2009.
[4] Camhi, N., The Manipulation of Sacred Places: The
Role of Jerusalem’s Temple Mount in the
Construction of Identity , An essay submitted to the
faculty of Wesleyan University in partial fulfillment
of the requirements for the Degree of Bachelor of
Arts with Departmental Honors in the Art History
Program, Middletown, Ct, 2012,
http://wesscholar.wesleyan.edu/cgi/viewcontent.cgi?a
rticle=1875&context=etd_hon_theses, Accesed:
2017-02-03.
[5] Harris, C., Dictionary of architecture & construction,
Fourth Edition, DOI: 10.1036/0071452370, 2006,
https://evrosoriou.files.wordpress.com/2011/05/dictio
nary-of-architecture-and-construction.pdf, Accesed:
2017-02-10.
[6] Kaptan, K., Early Islamic Architecture and Structural
Configurations, International Journal of Architecture
and Urban Development Vol. 3, No. 2, Spring 2013,
http://ijaud.srbiau.ac.ir/article_591_5f29d7ae86d363f
7bf593124a0e84686.pdf, Accesed: 2017-02-06.
[7] Rahimi, A., Zahra, A., Etezadi, S., The methodology
of Iranian curved arches (Sagh) based on their
geometry, Journal of Applied Environmental and
Biological Sciences, TextRoad Publication, ISSN:
2090-4274, 2015.
[8] site: http://www.islamguru.com/2016/05/masjid-quba
.html, Accesed: 2017-03-02.
[9] site: http://www.catholicliturgy.com/index.cfm/ Fuse
Action/EncyclopediaArticle/Index/15/SubIndex/149/
EncyclopediaIndex/27, Accesed: 2017-01-02.
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