&DUPHQ0Æ5=$'HOLD'5 *$1
TECHNICAL APPLICATIONS OF THE ANTI-PARALLEL SECTIONS
IN CURVED SURFACES
Abstract: This paper presents some parts that can be met in the joining of the sectors of mains found in
the air ventilation and conditioning equipment, as applications of the anti-parallel sections in cylindercone surfaces. In this way, the joints obtained are very tight. Starting from geometrical observations
related to transformations and inversions, some mathematical relationships of help in solving and
practically building the parts have been established.
Key words: Geometrical transformation, inversion. inversion pole, anti-parallel section, joint, frustum,
development.
1. INTRODUCTORY NOTIONS
The anti-parallel sections are circular shaped sections
cut in oblique circular cylinders and cones whose planes
are skew versus the director curves, mainly they are not
parallel to them [1]. The geometrical properties of these
sections have their technical applicability in the joining
together of the mains sectors with various positions and
sizes and in building some connecting parts for
equipment made of metal sheet or by cutting from
cylinder or cone sectors. The joints obtained in this
manner are tight and simple to carry out.
Further details will be given on the mathematical
conditions to be met by the cylinder and cone section
planes in order to find anti-parallel sections.
Figure 1 shows the geometrical conditions in which
the section with a vertical projecting plane in a frontal
oblique cylinder is a circle. It is noticed that the cutting
planes are symmetrical to the director circle plane versus
a plane [R] that is normal to the generatrices. The angle
between the frontal axis and the horizontal plane [H] is
equal to the angle between the axis and the vertical trace
of the cutting plane [2, 3].
Fig. 1 Anti-parallel section in oblique cylinder.
According to the theorem:
“The inverse of a circle whose inversion pole is not
situated on the circle is a circle”, it yields that section
[1234] represents the inversion geometrical transform of
the director circle of diameter (AB) of the cylinder,
related to pole Px = Qx, whose inversion modulus is [1]:
(1)
µ = (Px b ’ )(Px a ’ )= (Px 3’ )(Px 1’ )
In Figure 2, one can see an anti-parallel section with a
vertical projecting plane in an oblique cone whose
sagittal plane is contained in a frontal plane. It is
noticeable that, together with the vertical outline
generatrix (SB), the plane [P] forms an angle equal to the
angle formed by the opposite generatrix (SA) with the
director circle plane [4] According to the theorem stated
before, section [1234] is circular, as it represents the
inversion geometrical transform of the director circle
cone with reference to the inversion pole S (the cone
vertex) [1]. The inversion modulus will be:
(2)
µ = (s’1’)(s’a’) = (s ’3’)(s’b’)
We know that two circles situated in different planes,
inverse to each other, are situated on the same sphere.
Fig. 2 Anti-parallel section in oblique cone.
DECEMBER 2006  VOLUME 1  NUMBER 2 JIDEG 29
Technical Applications of the Anti-Parallel Sections in Curved Surfaces
The sphere of centre (Ψ, Ψ’) was pointed to, on
which are situated the circles representing the basis
contained in [H] and the circle for the anti-parallel
section [2]. In both cases, the true size of the section was
solved by coincidence on [H].
Fig. 3 Anti-parallel section in oblique cone – particular case.
One observes that 2’c’ is anti-parallel
∆Px1’d’, so that:
⇒ ∆Px c’2’≈ ∆Px d ’1’
P 2’ P c’ 2’c’
⇒ x = x =
Px d ’ Px 1’ 1’d ’
to 1’d’ in
(5)
Fig. 4 Intersection between two oblique cylinders.
Figure 3 represents an anti-parallel section in an
oblique cone, whose sagittal plane is also in a frontal
plane. The cutting plane is in this case a plane parallel to
a profile plane so that the true size of the anti-parallel
section originates in the profile projection [2]. The angle
between the generatrix (SA) and the director circle
should be equal to the one between the generatrix (SB)
and the vertical trace P’.
2. APPLICATIONS OF THE ANTI-PARALLEL
SECTION FOR THE JOINING OF SOME
CYLINDER-CONE SHAPED SECTORS
Now, we are going to describe some practically
potential situations whose solutions can be easily met
using the above-mentioned remarks.
Figure 4 shows the solution of two frontal cylinders
whose bases are contained in [H]. As one can notice, b’1’
is anti-parallel to ω1’ω2’ and also parallel to it at the same
time,
⇒ ∠Px 1’b’= ∠Pxω1 ’ω 3 ’⇒ ∆d ’Px 2’= isosceles
⇒ α + α + β + β = 180 ⇒ α + β = 90
(3)
∆b’Px 1’= isosceles
⇒ b’Px = Px c’

∆c’Px 1’= isosceles
( 4),
i.e., their common inversion pole is placed at the middle
of the distance between the generatrices, and cylinder
axes respectively.
Thus, to obtain the intersection of two cylinders with
reference to a circle, we notice that [2]:
- the axes of the two cylinders are perpendicular;
- the angle conditions required in the figure are observed.
In Figure 5, a frontal cylindrical sector and a taper
sector reached from an oblique cone with a sagittal plane
in a frontal plane was achieved.
30 DECEMBER 2006  VOLUME 1  NUMBER 2 JIDEG
Fig. 5 Intersection between a cylinder and a cone.
Also, one observes that b’2’ is both parallel and
antiparallel to a’1’⇒∆b’Px2’= isosceles.
One notes: Pxb’= Px2’ = x
ω1’ω2’ = k
In these conditions, one obtains from (5), after a
calculus, an expression for finding the position of the
cutting plane function the given constants:
(k − R1 )2 − R2 2
x=
(6)
2k
In Figures 6 and 7 the connections between two
cones’ frustums obtained from the intersection of two
oblique cones with reference to vertical projecting
planes, respectively profile planes are presented. We
must notice that, in order to achieve anti-parallel planes,
the angle conditions indicated in the figures must be
satisfied [2].
Technical Applications of the Anti-Parallel Sections in Curved Surfaces
In Figure 6, one observes that c’ 1’ is anti-parallel to
2’ d’ ,
⇒ ∆Px 2’d ’≈ ∆Px c’1’
Px c’ Px 1’ 1’c’
=
=
⇒ Px c’⋅Px d ’= Px 2’⋅Px 1’
Px 2’ Px d ’ 2’d ’
Similarly, b’ 1’ is antiparallel to a’ 2’ ,
⇒ ∆Px 2’a ’≈ ∆Px 1’b’
(7 )
P b’ P 1’ 1’b’
⇒ x = x =
⇒ Px b’⋅Px a’= Px 2’⋅Px 1’
Px 2’ Px a ’ 2’a’
From (7) and (8) ⇒
(8)
⇒
Px c ’⋅Px d ’= Px b’⋅Px a ’
In Figure 8, the joining of two equal radii cylinder
sectors by means of an oblique cylinder is represented
[3].
One observes in this case that the trapezium a’ b’ 3’ 1’ =
isosceles ⇒ ∆b’ Px3’ ≈∆Px1’ a’ .
If we notice Pxb’ =x, it results:
⇒
x
b’3’
=
x + 2 R1 a’1’
(11)
(9)
One notes : Pxb’ =x
ω1’ ω2’ =k
Similarly, one finds the position of the cutting plane:
(k − R1 )2 + R2 2
x=
(10).
2k
Often met situations in practical conditions are those
in which cylinder sectors with concurrent axes at skew
angles must be connected together.
Fig. 6 Intersection between two oblique cones.
Fig. 8 The joint between two cylinders of equal radii.
In Figure 9, the two sectors have different diameters.
In such a case, the joining is made with the help of a
cone frustum [3] whose sections versus the normal
directors to the mains’ axes are circles. If the angle
condition shown in the figure is met, the diameter
sections (AB) and (PQ) are anti-parallel. In Figure 9 one
observs that the quadrilateral a’ b’ 3’ 1’ is inscriptible, so
that we can write Ptolemeu relation, wich help us to the
practical determination of the piece:
a’b’⋅1’3’+ a’1’⋅b’3’= a’3’⋅1’b’
Fig. 7 Intersection between two oblique cones – particular case.
(12)
In the figure, there was represented the development
of the frustum connecting the two mains. The true size
of the support generatrices was determined using the
revolution around a vertical axis passing through the
cone vertex.
DECEMBER 2006  VOLUME 1  NUMBER 2 JIDEG 31
Technical Applications of the Anti-Parallel Sections in Curved Surfaces
The connection of cylinders having the same size is
made through an oblique circular cylinder, while the
connection of two cylinders of different diameters is
obtained by means of a conical oblique surface.
The paper has presented some possible situations that
can be met in the engineering practice. In the field of
civil engineering, the incidence of such situations is quite
high with air ventilation and conditioning equipment,
which has a high development in our country too. The
solutions proposed to building customers must fulfill
both technical and aesthetic requirements, which are
more and more demanding. Consequently, the
combination between science and art is ever more
obvious.
4. REFERENCES
Fig. 9 The joint between two cylinders of different radii.
3. CONCLUSIONS
In practice, we can find various cases in which mains’
sectors of different diameters must be connected
together. In order to achieve a tight joint, a convenient
solution would be the one through which at the mains’
intersecion are obtained circular sections that
geometrically represent cylinders. This can be achieved
with the help of some mathematical knowledge related to
geometrical transformations and inversions. Thus, by
intersecting some cylinders with cutting planes fulfilling
specific conditions, anti-parallel sections can be obtained.
32 DECEMBER 2006  VOLUME 1  NUMBER 2 JIDEG
[1] Botez, M., St., (1965). Geometrie descriptiva, E.D.P.,
Bucuresti.
[2] Olariu, F., Gogu, M., Marza, C., The use of antiparallel section in solving connection pieces,
Proceedings
of
the
International
Conference
Constructions 2003 (3), 15-16 Oct., 1993, Cluj-Napoca.
[3] Oprea, G., Georgescu, C., Applications of the antiparallel sections to the joint of the right circular
cylinders, Proceedings of the 5th Symposium in
Descriptive Geometry, Design, Engineering and
Computer Graphics, 17-19 June 1996, Timisoara.
[4] Tanasescu, A., (1975). Geometrie descriptiva,
perspectiva si axonometrie, E.D.P., Bucuresti.
Authors:
Eng.
Carmen
MÂRZA,
Ph.D.,
lecturer,
TECHNICAL University of Cluj-Napoca Romania, email [email protected];
Eng. Delia DR GAN, Ph.D., lecturer, TECHNICAL
University
of
Cluj-Napoca
Romania,
e-mail:
[email protected]