JUNE 2012 VOLUME 7ISSUE 1 JIDEG 4 1Abstract: To avoid the routine of buildings composed of classical geometric bodies, some methods used in [602576]

JUNE 2012 VOLUME 7ISSUE 1 JIDEG 4 1Abstract: To avoid the routine of buildings composed of classical geometric bodies, some methods used in
contemporary architecture consist in: the use of intersected solids, the cutting with planes in different
positions, as well as the recomposing of volumes.
In this paper, after a theoretical introduction regarding the intersections of planes, from both analytically
and graphically point of view, the authors approach applications of these in the achievement of some
elements in the civil engineering field.
Key words: architectural styles, intersections of planes, facades, awnings, roofs, geodesic dome.
1 .INTRODUCTION
Starting with the XXthcentury the trends in
architecture have undergone a real metamorphosis
caused by the necessity to build in a fast, ratioanal and
efficiently rate. Modern architecture is characterized by
vastness and diversity -Fig. 1.
Thus, in a first stage the internationalist style was
contoured Fig. 1a which was characterized by the
construction of tall buildings skyscrapers of
parallelepipedic form with flat and bright surfaces,
suggesting order and regularity. One of the prominent
architects of this style was Le Corbusier who established
as a priority the importance of the functionality features
and has combined in an efficent way the available
materials, respectively steel, glass and concrete.
T h e n i t f o l l o w e d t h e D a d a i s m a n d C u b i s m
movements Fig. 1b which represented attempts to
rebuild the form through decomposition and
recomposition of the geometric solids.
In parallel, the futurism current was contoured Fig.
1c. It manifested as a rejection of the historical currents
that, one can admit, are veritable architectural jewels, but
no longer corresponded with the idea of speed,
movement, dynamism in the most general way. It was
inspired by the military architecture.
Once with the unprecedented technological
development in the last decades of the XXthcentury, it
appeared a post modern style, the so called high tech
architecture named also structured or technological
expressionism, characterized by simplicity and elegance.
Thus, by applying various methods to the classical
geometric volumes as the using of cuts intersections of
planes or some simple curved surfaces, result buildings
with their own personality, specific to sophisticated
urban areas, denoting the courage and valorizing the
imagination of architects. Taken to the edge of the
possible, from a structural viewpoint, this current turns
into deconstructivism Fig. 1d, which by displacement
and distortion of solids suggests a controlled chaos, ie a
non-Euclidean volumetry composition.
The achievement of these types of construction was
made possible by the discovery of advanced materials –
as high-strength glass, carbon fiber and other composite
materials, of the high performance technologies, and alsodue to the possibility of computer modeling of the
structures.
a. b .
c. d .
Fig. 1Modern and postmodern stylebuildings [4, 5 ]
In this paper the authors aim an analysis regarding the
applications of the intersections of planes in the civil
engineering field, as a way out method from the
conformism of simplistic volumes specific to modern and
postmodern architecture trends.
2. BASIC ABOUT THE PLANE AND
INTERSECTION OF THE PLANES
As it is known from geometry, the plane can be
defined by its elements -three non-collinear points, a line
and an exterior point, two parallel or intersecting lines,
respectively by two of its trac es.The most direct
applications in civil engineering result from theFirstname1 SURNAME1, Firstname2 SURNAME2 1talic)APPLYING THE THEORY OF THE PLANE IN THE CIVIL ENGINEERING FIELD
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Applying the theory of the plane in the civil engineering field
JUNE 2012VOLUME 7ISSUE 1 JIDEG 42definition of the plane by its elements, namely defining
the plane as a plate shape. If the plate is a different
polygon than a triangle, will put the condition that the
other vertices tobelong to a straight line of the plane that
was defined by three explicit vertices/points.
The intersection of two planes is a line, obtained
either if certain conditions are fulfilled -when the plane
is analytically defined, either using the points of
intersection of the edges of a plate with the other plate –
when the plane is graphically defined.
2.1.Analytical determination of the intersection of
planes
A plane can be defined mathematically in several
ways. For the current applications in civil engineer ing are
of interest the forms that will be further presented.
Cartesian implicit equation of the plane [1]:
 0 :  DCzByAxP (1)
respecting the condition 0222CBA
where,
A, B, C = directors parameters of the plane (components
ofa normal vector to the plane)
D = parameter depending on the position o f the plane
relative to the axe s origin.
The equation of plane defined by three non –
collinear points M 1(x1,y1, z1), M2(x2,y2, z2), M3(x3,y3, z3),
is expressed in the form [1]:
 0
1111
:
333222111
zyxzyxzyxzyx
P (2)
where (x i, yi, zi), i=1,2,3 represent the point scoordinates
through which the plane passes.
If the points through which the plane will pass
represent exactly the intersections with the coordinate
axes, respectively A(a,0,0), B(0 ,b,0), and C(0,0,c) –
called in descriptive geometry vanishing points, the
equations (2) became:
01cz
by
ax(3)
The coordinates of the previous vanishing points can
be expressed as:
ADa;BDb;CDc; (4)
if.0D
Being two planes [P 1] and [P 2] defined by the implicit
form:
 0 :1 1 1 11  DzCyBxAP (5)
 0 :2 2 2 22  DzCyBxAP (6)these are intersecting (secant) about a straightline, ifis
satisfiedthe condition that the rank of the matrix to be
two [1]:
2
222111



CBACBA
rang (7)
or, expressed in other way, if the following condition is
accomplish:
21
21
21
21
DD
CC
BB
AA (8)
2.2.The graphical/descriptive determin ation of the
intersection of planes
As specified previously, in civil engineering
applications, the representation of planes in the form of
plates are frequent used. In Fig.2 it is represented the
intersection between two triangular plates [ABC] and
[MNP].
a.
b.
Fig. 2Intersection of twooblique triangular plates

Applying the theory of the plane in the civil engineering field
JUNE 2012 VOLUME 7ISSUE 1 JIDEG 43The intersection line (IJ) is obtained in this case using
the auxiliary vertical projecting planes [R 1] and [R 2],
passing through the sides (AB) and (BC) of the plate
[ABC] [3]. One est ablished the visibility of the plates in
each projection. In Fig .2b is the 3D representation of the
intersection of the plates. But in applications to various
elements of construction -such as cuts on the facades,
awnings, decorations, are kept only the visible part of the
plates bounded by the intersection line.
Similarly, in Fig.3 it is represented the intersection
between two quadrilateral oblique plates. One observe
that for establish the projections of the forth point of the
plate it is necessary to impose the condition that the last
point to be lying in the plane defined by the three known
vertices. This can be solved with the help of diagonals of
the quadrilateral from the known projection, respecting
the affinity condition [3].
Fig. 3Intersection of oblique quadrilateral plates
In many applications, at least one of the plates is in
particular position relative to the projection planes. In
this case, the intersetion yields directly. In Fig. 4a is
represented the intersection between an oblique plate and
a horizontal plate, respectively in Fig. 4b is represented
the intersection between an oblique plate and a frontal
plate.
3.APPLICATIONS OF THE INTERSECTION OF
THE PLANES IN CIVIL ENGINEERING
FIELD
Even if from analytical or descriptiv e point of view
the intersection of planes do not raise problems, we meet
frequently in construction applications of this subject,
which is the reason why the authors have focused to
detail some specific aspects of this field. It must be
mentioned that in terms of structure, they generate, in
some cases, additional calculus, not always easy.
a.
b.
Fig. 4Intersection between an oblique plate and plates in
particular positions.
As was specified in the introduction, in modern
architecture -which respo nds to the needs of
contemporary people to be pragmatic and efficient, one
of the solutions for avoiding the ordinary building
appearance, is to use intersections of planes or polyhedra
cuttingbyplanes.
Among the most known applications, are the
buildings having truss roofs, which, depending on the
complexity of the outline shape, consist of a number of
plane faces called slopes with equal or unequal
inclination that intersect after rafters, valleys or ridges
[2]. In Fig. 5a was represented the solving of a roof
having equal slopes and a polygonal contour as
horizontal projection, using projections with elevations.
In Fig.5b is given the 3D representation of the roof.

Applying the theory of the plane in the civil engineering field
JUNE 2012VOLUME 7ISSUE 1 JIDEG 44As it is known, the intersection edges of equal
inclination slopes are projected on the horizontal plane
about the bisecting lines of the interior angles of the
slopes traces on the eaves plane, which in particular are
represented by the contour line.
a.
b.
Fig. 5Solving a roof with slopes of equal inclinations
In the case of the roofs having slopes with unequal
inclination, it is necessary to use the steepest line ( the
lineof maximum inclination) of the sides, which is
perpendicular to the horizontal trace of the plane,
respectively on each of the horizontal line of the given
plane. For determining the intersection edges of the
sides, one draws slope planes having the steepest line
perpendicular to each side of the polygon contour, that is
the perimeter projection. Using the horizontal lines of the
planes one obtained the intersect ios of the slopes.
There are situations when a nonconformist approach
is desired in order to achieve the covering of a buildin g–
see Fig.6.Thus,in Fig.6ait is represented the to p view
of the roof respectively, inFig.6ba picture of the roof
composed of a joining of planes having different
inclinations.
Building envelopes have an important aesthetic role.
In some situations one meet different plane joints to
decorate windows, loggias and balconies or their
intrados, and often to the a chivement of the facades
(Fig.7). The special form of the facade is in some cases
related to the interior distribution of useful space indoors,but in other cases represent merely curtain walls or false
glass surfaces.
In all cases, the aim is to obtain aesthetic and vis ual
effects, to avoid the monotony and regularity of flat
surfaces.
Fig. 7 shows the folowing representative buildings:
-Fig. 7a and 7d –commercial buildings on Orchard
Road in Singapore;
-Fig.7b–Polyvalent Hall in Cluj Napoca;
-Fig.7c–China Financial Information Center in
Shanghai;
-Fig. 7e-Health Department building in Bilbao , Spain;
-Fig. 7f-production and office building in the
industrial zone of Munich, Germany.
a.
b.
Fig. 6Roof with slopes having arbitrary inclination [8]
There are frequent situations where the space
achieved by applying these types of facades are used to
provide natural ventilation, respectively for directing air
flow, depending on the season.
Another frequently application is that of the awnings
composed of intersecting planes used to cover or closed
open spaces -see Fig.8as it follows:
-Fig.8a–Park Pavilion, Cuerca, Spain [9];
-Fig. 8b-Compass Stadium, Huston, Texas, USA [6].

Applying the theory of the plane in the civil engineering field
JUNE 2012 VOLUME 7ISSUE 1 JIDEG 45
a. b. c.
d. e. f.
Fig. 7Facade details [9,10]
a.
b.
Fig.8Types of awnings [6,9]
4.REASERCH DIRECTIONS
In cases where there are special aesthetic
requirements, the covering can be designed in the form
of geodesic domes or parts of them.What are and how are obtained these domes? We
know that any surface can be approximated by a
polygonal spatial network, taking on that surface indirect
points and resulting vertices, edges and triangular or
polygonal faces. The approximation of a surface with a
polyhedron is more accurate as the polyhedron has
smaller edges. Otherwise the two -dimensional
equipartition in space lead, as particular cases, to the five
regular or Platon’s polyhedra -tetrahedron, hexahedron,
octahedron, icosahedron and dodecahedron, respec tively
those 13 semiregular or Arhimede’s polyhedra inscribed
in a sphere.
To achieve large opening reticular domes, it is
necessary to multiply the faces and vertices of these
polyhedra, such as t oresult as many equal edges (struts)
and identical solid angles (knodes) [2]. This
multiplication, which is performed according to certain
laws, involve that the polyhedron to follow a sphere
through a series of quasi -regular polyhedra having
increasingly smaller edges and faces, closer in size to the
surface of the sphere or, in generally to the curved
surface which supports. In the context of the topic of the
paper, the reticular surfaces can be treated as a
succession of joined planes (polygons), lying on a
surface.
The most stable geodesic surfaces are the o mni-
triangular structures that provides a solid structure with
outstanding resistance to the natural factors: earthquakes
up to 8.5 degrees on the Richter scale, wind speeds
below 320 km/h, due to the aerodynamic shape. But
according to the load to which t he building is subjected
one can also find domes having pentagons or hexagons
regular faces.

Applying the theory of the plane in the civil engineering field
JUNE 2012 VOLUME 7ISSUE 1 JIDEG 4 6Geodesic domes are successfully used in the
construction of recreational facilities, exhibition
pavilions, greenhouses, and recently even in the
construction of houses.Besides the previously described
advantages, others should be mention, that in the context
of sustainable development of constructions, makes them
attractive: their shape maximizes the light effect, allows
optimum distribution of sound and heat, providing
thermal comfort. This types of constructions require
approximately 30% lower energy consumption than a
conventional parallelipipedic building.
a.
b .
Fig.1 3Geodesic dome constructions [7]
In Fig. 13a are shown the green houses from Jibou’s
Botanical Garden and Fig. 13b shows an ecological
housein the form of geodesic dome.
5 .CONCLUSIONS
Contemporary architecture reflects the dynamics that
characterizes modern society. Thus, spectacular shapes
and meticulous decorative details of styles preceding the
XXthcentury were replaced with simple volumes derived
f r o mclassical geometric solids.One of the methods used to achieve a particular
appearance consists in the intersection with other plane
or spatial geometric solids.
Present and future buildings are and will be designed
i n o r d e r t o p u r s u e o t h e r t w o challenges of the
contemporary world society:
-energy efficiency;
-responsibility towards the environment.
REFERENCES
[ 1 ]Dumitraș, D., E., Geometrie analitică si diferențială
Analytical and differential Geometry , Editura
Digital Data Cluj, ISBN 973-82010-0-4, Cluj-
Napoca, 2001.
[ 2 ]Iancău, V, Zetea, E., ș.a., Reprezentări geometrice ș i
desentehnic Geometrical representation a n d
technical drawing , EDP, București, 1982.
[ 3 ]Mârza, C., Corsiuc, G., Drăgan, D., Geometrie
descriptivă –T e o r i eși aplicații Descriptive
Geometry –Theory and application , Editura UT
Press, ISBN 978- 606-737-182-6, Cluj Napoca, 2016.
[ 4 ]Turcanu, C., R., Elemente de Arhitectura
Architectural elements ,
www.cursarhitectura.wordpress.com.
[ 5 ]https://ro.wikipedia.org/wiki/Istoria_arhitecturii
[ 6 ]https://commons.wikimedia.org/wiki/
[ 7 ]http://ecocase.ro/case-dom/
[ 8 ]https://www.dezeen.com/2013/12/27/office-
extension-faceted-copper-roof-emrys-architects/
[ 9 ]http://weburbanist.com/2013/11/11/origami-inspired-
architecture-14-geometric-structures/3/
[10]http://www.archdaily.com/536964/textilmacher-
tillicharchitektur/
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