Dumitru MARIN
THE INTERSECTION OF CYLINDRICAL SURFACES
GRAPHO-ANALYTICAL STUDIES
Abstract: : This paper presents the determination of the section curve of two rotation cylinders with
intersecting axes and parallel with a projection plane, using the analytical method, as well as the
graphical method (of auxiliary spheres). Regardless of the method employed, one reaches the conclusion
that the projection of the section curve is an equilateral hyperbola, for any angle between the axes of the
two intersecting cylinders and for any diameters.
Key words: intersection curve, auxiliary sphere, symmetry axis, hyperbola, asymptote.
1. INTRODUCTION
The intersection of two rotation cylinders with
intersecting axes and unequal diameters can be solved by
using either the method of auxiliary planes or the method
of auxiliary spheres.
The method of the auxiliary spheres exhibits two
major advantages: it is fast, and can also be applied when
the rotation surfaces are represented in one projection.
The basic constraint for the application of the method
of auxiliary spheres is that the rotation cylindrical
surfaces have intersecting axes and parallel with a
projection plane.
In the following, the similarity between the analytical
and the graphical method is observed, considering the
most common practical case: the intersection of two
rotation cylindrical surfaces with intersecting axes and
parallel with the vertical projection plane.
2. THE INTERSECTION OF TWO ROTATION
CYLINDERS ANALYTICAL METHOD
Given is the rotation cylinder in Fig. 1.
The equation of the vertical cylinder with radius r and
symmetry axis OZ is:
 x 2 + y 2 = r 2

 z = z
(1)
The equation of the horizontal cylinder with radius R
and symmetry axis OX is:
 z 2 + y 2 = R 2

 x = x
(2)
The intersection of the cylinders with the equations
(1) and (2) is a space curve, having as vertical projection
an equilateral hyperbola with the equation:
z2 − x 2 = R2 − r 2
(3)
For x = ± r we obtain z = ± R , which are the
intersection points of the apparent contour generators of
the two cylinders in Fig. 1.
For x = 0 we obtain z = ± R 2 − r 2 , which are the
apexes of two hyperbola arms located on the axis OZ.
For R = r equation (3) change to:
( z − x )( z + x ) = 0
(4)
The lines in equation (4) are
asymptotae of the hyperbola arms.
For r > R , equation (3) becomes:
perpendicular
x2 − z2 = r 2 − R2 > 0
(5)
Equation (5) is an equilateral hyperbola with Ox as
symmetry axis (Fig. 2) and with the same asymptotae as
yielded by equation (4).
The intersection points of the apparent contour
generators and the apexes of the hyperbola arms are
determined analogous to the hyperbola in Fig. 1.
3. THE INTERSECTION OF TWO ROTATION
CYLINDERS.
METHOD
OF
AUXILIARY
SPHERES
An auxiliary sphere with radius RS intersects the two
cylinders yielding the circles C1 , located on the cylinder
of radius R and C2 , located on the cylinder of radius r
(Fig. 1.).
The point M, common to the circles C1 and C2 is
located on the intersection curve of the two cylinders and
its coordinates can be calculated from the equations (see
also Fig. 1):
x 2 = R s2 − R 2
(6)
z 2 = R s2 − r 2
(7)
The elimination of RS from (6) and (7) yields the
equation of the vertical projection of the intersection
curve of the two cylinders:
z 2 − x2 = R2 − r 2
(8)
Equation (8) is equivalent to equation (3) and
represent the equation of an equilateral hyperbola with
OZ as a symmetry axis.
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The Intersection of Cylindrical Surfaces Grapho-Analytical Studies
Fig. 1 Intersection of two rotation cylinders (R>r).
Fig. 2 Intersection of two rotation cylinders (R<r).
In practice cases arise where the two rotation
cylinders don`t have perpendicular axes (Fig. 3).
It can be demonstrated, that the vertical projection of
the intersection curve between the rotation cylinders in
Fig. 3 is a hyperbola.
The x coordinate of point M on the intersection curve
found through the method of auxiliary curves is:
x = R s2 − R 2
(9)
From figure 3 one can also observe that:
OA 2 = Rs2 − r 2
(10)
Fig. 3 Intersection of two rotation cylinders with
nonperpendicular axes (R>r).
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The Intersection of Cylindrical Surfaces Grapho-Analytical Studies
The equation of the line passing through point A and
perpendicular on the z1 axis of the oblique cylinder of
radius r is:
z = − xtgα +
The lines in equation (16) are the asymptote
(perpendicular to each other) of the hyperbola from
figure 3.
If x = 0 then:
OA
cos α
(11)
z2 =
or
Rs2 − r 2
z = − xtgα +
(12)
cos α
(R
2
− r2
)
cos α
2
(17)
which are the intersection points of the hyperbola.
If R < r then R 2 − r 2 < 0 and the hyperbola is as in
figure 4.
The elimination of RS from (9) and (12) and
subsequent transformations yield the geometrical locus
of the point M:
z2 =
(
)
 1 + sin 2 α x 2 − r 2 + R 2 −


cos 2 α − 2 x sin α x 2 + R 2 − r 2 
1
(13)
2
Since the coefficient of x from (13) is positive, this
is the eqution of a hyperbola.
If α = 0 , then equation (13) becomes:
z2 − x 2 = R2 − r 2
(14)
which is the equation of the equilateral hyperbola from
equation (8).
If R = r then:
z2 =
1
cos α
2
(
x 2 1 + sin 2 α ± 2 sin α
)
(15)
Fig. 4 Intersection of two rotation cylinders with
nonperpendicular axes (R<r).
or:
z2 =
(1 ± sin α ) 2
cos 2 α
x2
(16)
From the Fig. 3 and 4 one can observe that a rotation
of the vertical cylinder axis with an angle α causes the
symmetry axis of the hyperbola arms to rotate with an
angle α / 2 .
Fig. 5 The variation of the form of the equilateral hyperbola.
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The Intersection of Cylindrical Surfaces Grapho-Analytical Studies
Fig. 5. shows the variation of the form of the
equilateral hyperbola for different angles between the
axes of the two cylinders, as well as the unchanged
position of the minimum point and the permanent
perpendicularity of the hyperbola’s asymptote.
4. CONCLUSIONS
The grapho-analitical studies concerning the
intersection of the rotation cylindrical surfaces presented
in this paper emphasize the following:
- the projection of the intersection curve on the plane
of the axes is an equilateral hyperbola, regardless of
the diameter of the two surfaces and of the angle
between the axes.
- the perpendicularity of the hyperbola’s asymptote can
be demonstrated analytically and verified graphically.
The grapho-analitical studies can be extended to the
case presented in the paper, as well as in the case of the
two cylinders being cyrcular oblique.
The paper is interesting from a didactical point of
view, as well as from a practical point of view (at the
unfolding of intersected surfaces, the determination of
minimum points and the symmetry axes of the
intersection curve etc.).
28 JUNE 2008  NUMBER 4 JIDEG
5. REFERENCES
[1] Aldea, S. (1979). Geometrie descriptivă. Studiul
suprafeţelor şi al corpurilor, I.P.B. Publishing House,
Bucharest.
[2] Botez, Şt. (1965) Geometrie descriptivă, Didactic and
Pedagogic Publishing House, Bucharest.
[3] Marin, D., (2003). Geometrie descriptivă. Noţiuni
teoretice şi aplicaţii, BREN Publishing House,
Bucharest.
[4] Marin, D., Adîr, V. (2000). Metoda sferelor auxiliare
cu centrul mobil la intersecţia suprafeţelor cilindroconice, GRAFICA – 2000 - The VIIth edition of the
national conference with international participation,
pp. 223-226, Craiova.
[5] Marin, D., Raicu, L. ş.a. (1998). Geometrie
descriptivă. Probleme şi aplicaţii, BREN Publishing
House, Bucharest.
[6] Postelnicu, V., Coatu, S. (1980). Mica enciclopedie
matematică – Traducere din limba germană, Editura
Tehnical Publishing House, Bucharest.
Author:
Eng.
Dumitru
MARIN,
Ph.D.,
Professor,
POLITEHNICA University of Bucharest, Departament
of Descriptive Geometry and Engineering Graphics,
E-mail: [email protected].