International Journal of Mathematics and Physical Sciences Research ISSN 2348-5736 (Online)
Vol. 2, Issue 2, pp: (1-17), Month: October 2014 - March 2015, Available at: www.researchpublish.com
“HCR’s Infinite-series”
Analysis of Oblique Frustum of a Right
Circular Cone
Mr Harish Chandra Rajpoot
Madan Mohan Malaviya University of Technology, Gorakhpur-273010 (UP) India
Abstract: Three series had been derived by the author, using double-integration in polar co-ordinates, binomial
expansion and
-functions for determining the volume, surface-area & perimeter of elliptical-section of
oblique frustum of a right circular cone as there had not been any mathematical formula for determining the same
due to some limitations. All these three series are in form of discrete summation of infinite terms which converge
into finite values hence these were also named as HCR’s convergence series. These are extremely useful in case
studies & practical computations.
Keywords: HCR’s Convergence-series, oblique frustum, binomial expansion,
I.
-functions.
INTRODUCTION
When a right circular cone is thoroughly cut by a plane inclined at a certain angle with the axis of a right circular cone, an
oblique frustum with elliptical section & apex point is generated So far there are no mathematical formulae for
determining the volume & surface area of oblique frustum & perimeter, major axis, minor axis & eccentricity of elliptical
section. This article derives mathematical formulae for all the parameters.
II.
VOLUME OF OBLIQUE FRUSTUM
Let there be a right circular cone with apex angle
Now, it is thoroughly cut by a plane (as shown by the extended line
AB in fig 1.) inclined at an angle with axis OZ of the cone & lying at a normal distance h from the apex point O
Fig 1: Oblique frustum of right cone with elliptical section
Page | 1
Research Publish Journals
International Journal of Mathematics and Physical Sciences Research ISSN 2348-5736 (Online)
Vol. 2, Issue 2, pp: (1-17), Month: October 2014 - March 2015, Available at: www.researchpublish.com
Now, consider any parametric point (
) on the conical surface
Equation of conical surface: in Cartesian co-ordinates
(
)
In Polar co-ordinates
(
)
(
)
Equation of cutting plane: by intercept form
⇒
Equation of elliptical boundary: Ellipse is the curve of intersection of conical surface & cutting plane. Hence on
) in the equation of plane, we have the following equation in polar
substituting the co-ordinates of parametric point (
co-ordinates
*(
)
+
*
+
⇒
(
)
( )
Now, let‟s consider an imaginary cylindrical surface ABDC completely enclosing the frustum of solid cone (as shown in
the fig 2. below). It is easier to calculate the volume of void space enclosed by the frustum of cone, cylindrical surface &
XY-plane.
Fig 2: Imaginary cylindrical surface enclosing the frustum of cone
Let‟s consider an elementary surface area
on the conical surface (see fig 1.)
*(
Taking the projection
of elementary area
+
)
on XY-plane
(
⇒
)
Now, volume of elementary vertical strip
⇒
(
)
Page | 2
Research Publish Journals
International Journal of Mathematics and Physical Sciences Research ISSN 2348-5736 (Online)
Vol. 2, Issue 2, pp: (1-17), Month: October 2014 - March 2015, Available at: www.researchpublish.com
Hence, using double integration with proper limits, total volume ( ) of the enclosed void space
⇒
∫
∫
∫
∫
*∫
∫
* +
(
+
)
∫
( ), we have
∫
(
(
)
)
∫
(
(
)
) .
.
/
(
∫
.
∫
/
(
.
(
/
)
)
/
)
Since, the frustum has finite elliptical section, hence we have a condition
⇒
|(
)
|
Now, using Binomial Expansion
(
(
)
)
(
)(
)
( | |
)
We have,
⇒
∫
(
∫
[
(
)
)
(
(
*∫
(
)
)
)
(
)
(
)
]
∫
∫
∫
∫
∫
+
Page | 3
Research Publish Journals
International Journal of Mathematics and Physical Sciences Research ISSN 2348-5736 (Online)
Vol. 2, Issue 2, pp: (1-17), Month: October 2014 - March 2015, Available at: www.researchpublish.com
(Multiplying & diving each term by 2 in above series)
*
∫
∫
∫
∫
∫
( )
∫
(
∫
∫
)
( )
∫
∫
+
( )
(
)
( )
∫
∫
∫
Now, we have
[
∫
∫
∫
]
[
∫
[
∫
∫
Since, we know from
∫
∫
]
∫
]
-functions
.
∫
/ .
/
.
( )
√
/
Hence, we have
0
.
.
/ .
/
.
.
.
.
.
.
/
/ .
.
0
/
. / . /
( )
/
/ .
.
/
/
/
/
1
/
/
.
. / . /
( )
/
.
. / . /
( )
/
1
Page | 4
Research Publish Journals
International Journal of Mathematics and Physical Sciences Research ISSN 2348-5736 (Online)
Vol. 2, Issue 2, pp: (1-17), Month: October 2014 - March 2015, Available at: www.researchpublish.com
0
√
.
√
/
√
.
√
/
√
.
√
/
1
0
.
[
.
(
[
)
[
(
(
1
(
)
)
(
)
]
)
]
( )
series
(
)
/
)
(
(
)
.
(
is called Factor of volume or HCR’s
(
/
)
(
⇒
Where,
/
)
)
(
(
)
)
(
)
]
In generalised form
∑*
∑*
⇒
(
(
(
)
)
(
,
(
)
(
∑ *(
Now, the volume (
)
)(
)
(
)
)
(
)
-
(
,
(
),
)
+
)
(
(
)
-
)
-
+
+
(
)
) of imaginary cylinder enclosing the frustum of solid cone (see fig 2)
, (
) (
)-
⇒
(
(
)
)
Page | 5
Research Publish Journals
International Journal of Mathematics and Physical Sciences Research ISSN 2348-5736 (Online)
Vol. 2, Issue 2, pp: (1-17), Month: October 2014 - March 2015, Available at: www.researchpublish.com
(
)
(
)
Now, on setting the values, we get the volume of cylinder
* (
(
⇒
) (
)
(
)
(
)
(
(
(
)
)
(
)
)+
)
)
(
Hence, the volume (V) of the oblique frustum of right cone
(
(
)
)
(
)
Above is the required expression for calculating the volume of oblique frustum.
series can be simplified as follows
III.
SURFACE AREA OF OBLIQUE FRUSTUM
We know that the elementary area
conical surface (see fig 1.)
*(
+
)
Hence, total surface area (S) of frustum of cone (using double integration with proper limits)
∫
∫
∫
∫
*∫
∫
* +
+
∫
( ), we have
∫
(
(
)
(
) .
)
∫
(
)
.
/
/
(
∫
.
.
/
)
/
Page | 6
Research Publish Journals
International Journal of Mathematics and Physical Sciences Research ISSN 2348-5736 (Online)
Vol. 2, Issue 2, pp: (1-17), Month: October 2014 - March 2015, Available at: www.researchpublish.com
∫
(
(
|(
)
)
)
|
(
(
)
)
We have,
⇒
(
∫
∫
(
[
)
)
(
(
(
)
(
)
*∫
)
)
∫
∫
∫
∫
∫
+
∫
∫
∫
∫
+
( )
∫
(
∫
∫
)
( )
∫
∫
)
]
∫
*
(
( )
(
)
( )
∫
∫
∫
Now, we have
[
∫
∫
∫
]
Page | 7
Research Publish Journals
International Journal of Mathematics and Physical Sciences Research ISSN 2348-5736 (Online)
Vol. 2, Issue 2, pp: (1-17), Month: October 2014 - March 2015, Available at: www.researchpublish.com
[
Using
∫
∫
∫
]
-functions
.
∫
/ .
/
.
( )
√
/
Hence, we have
0
.
.
.
/ .
.
.
/
/ .
/
.
0
.
0
.
. / . /
( )
√
√
/
/
.
.
/ .
.
/
/
/
/
1
/
/
. / . /
.
/
( )
√
.
/
√
. / . /
.
/
( )
/
1
√
.
√
/
1
0
.
/
.
[
(
[
)
(
⇒
/
(
)
)
(
(
.
/
(
)
1
)
(
]
)
)
]
( )
Above is the required expression for calculating the surface area of oblique frustum.
Where,
is called Factor of surface area or HCR’s
(
)
(
series
)
(
)
In generalised form
∑*
(
)
(
)
,
(
)
(
)
-
+
Page | 8
Research Publish Journals
International Journal of Mathematics and Physical Sciences Research ISSN 2348-5736 (Online)
Vol. 2, Issue 2, pp: (1-17), Month: October 2014 - March 2015, Available at: www.researchpublish.com
∑*
⇒
(
)
(
(
)
)
(
,
(
),
∑ *(
)
)
(
(
)
)
-
-
+
+
(
)
series can be simplified as follows
IV.
PERIMETER OF ELLIPTICAL SECTION OF OBLIQUE FRUSTUM
Now, consider any parametric point (
) on the periphery of elliptical section (see fig 3. below)
Now, the elementary perimeter
(
)
(
)
Hence, the total perimeter (P) of the elliptical section (using integration with proper limits)
∫
∫
∫
Fig 3: Perimeter of elliptical section of oblique frustum
( ), we have
∫
(
(
)
)
Page | 9
Research Publish Journals
International Journal of Mathematics and Physical Sciences Research ISSN 2348-5736 (Online)
Vol. 2, Issue 2, pp: (1-17), Month: October 2014 - March 2015, Available at: www.researchpublish.com
∫
(
).
(
)
∫
.
/
/
(
∫
.
(
.
/
(
)
)
|(
)
|
(
(
)
/
(
)
)
)
(
| |
)
We have,
⇒
,
∫
(
∫
(
*∫
)
)
(
)
(
∫
)
(
(
∫
+
∫
∫
∫
∫
+
( )
(
( )
∫
∫
-
)
∫
∫
∫
∫
)
∫
∫
∫
)
∫
∫
*
(
(
)
)
( )
( )
∫
∫
∫
Page | 10
Research Publish Journals
International Journal of Mathematics and Physical Sciences Research ISSN 2348-5736 (Online)
Vol. 2, Issue 2, pp: (1-17), Month: October 2014 - March 2015, Available at: www.researchpublish.com
Now, we have
[
∫
[
Now, using
∫
∫
∫
∫
]
∫
]
-functions
.
∫
/ .
/
.
( )
√
/
Hence, we have
0
.
.
/ .
/
.
/
.
.
/ .
/
/
.
/
.
.
/
/ .
/
.
/
/
1
0
.
0
.
. / . /
( )
√
√
/
. / . /
.
/
( )
√
.
/
√
. / . /
.
/
( )
/
1
√
.
√
/
1
0
.
[
[
/
(
.
)
(
(
)
(
⇒
/
)
)
(
.
(
(
/
1
)
]
)
]
)
(
)
Above is the required expression for calculating the perimeter of elliptical section of oblique frustum.
Where,
is called Factor of surface area or HCR’s
(
)
(
)
series
(
)
Page | 11
Research Publish Journals
International Journal of Mathematics and Physical Sciences Research ISSN 2348-5736 (Online)
Vol. 2, Issue 2, pp: (1-17), Month: October 2014 - March 2015, Available at: www.researchpublish.com
In generalised form
(
∑ *,
⇒
)
(
∑ *,
(
)
)
(
-
+
)
-
+
(
)
Above series can be simplified as follows
V.
CONCLUSION
Thus, all the results can concluded from equations III, IV, V, VI, VII & VIII as follows
Let a right circular cone with apex angle
be thoroughly cut by a plane inclined at an angle
& lying at a normal distance h from the apex point O (as shown in the fig 4 below)
with axis OO‟ of the cone
Fig. 4
Now, consider the oblique frustum with elliptical section & apex point „O‟ (see above fig 4)
The following parameters can be determined as tabulated below
Volume (V) of
frustum
(
(
)
)
where
∑ *(
Surface Area (S)
of frustum
)(
(
),
(
)
(
)
-
+
)
Page | 12
Research Publish Journals
International Journal of Mathematics and Physical Sciences Research ISSN 2348-5736 (Online)
Vol. 2, Issue 2, pp: (1-17), Month: October 2014 - March 2015, Available at: www.researchpublish.com
where
∑ *(
Perimeter (P) of
Elliptical section
(
),
(
)
(
)
-
+
)
where
(
∑ *,
)
(
)
-
+
Major Axis (2a)
(
)
Minor Axis (2b)
)
√(
Eccentricity (e)
Conditions
for
known quantities
(
)
where
VI.
IMPORTANT DEDUCTIONS
1.
All the series converge to a finite value for the given values of
is as follows
2.
It is obvious that the rate of convergence of
3.
Although, the order of convergence rates of
depends on the values of
a.
If
⇒
converge to finite values with less number of terms i.e. terms of higher power can be
neglected with insignificant errors in the results
b.
If
⇒
converge to finite values with more number of terms i.e. neglecting the terms of
higher power may cause some error. In this case, the values of
must be determined by taking more
number of terms until higher power terms become significantly negligible.
If a right circular cone with apex angle
be thoroughly cut by a plane normal to the axis of the cone & lying at a
normal distance h from the apex point then by setting
in the above results, we have the following results for
the frustum generated
Page | 13
Research Publish Journals
International Journal of Mathematics and Physical Sciences Research ISSN 2348-5736 (Online)
Vol. 2, Issue 2, pp: (1-17), Month: October 2014 - March 2015, Available at: www.researchpublish.com
a.
Volume of frustum
(
(
)
(
(
)
(
)
(
)
(
)
(
)(
)
(
)
(
)
)
(
)
)
(
)
(
)
Major axis of generated section
(
(
)
(
)
)
(
)
Minor axis of generated section
)
√(
√(
f.
)
)
(
e.
(
Perimeter of section generated
(
d.
)
)
Surface area of frustum
(
c.
(
)
(
b.
)
)
(
)
(
)
Eccentricity of generated section
It is clear from the above results that when a right circular cone is thoroughly cut with a plane normal to the axis of cone
generates a frustum which is itself a right cone with circular section. Hence, all the mathematical results derived by the
author are verified.
Page | 14
Research Publish Journals
International Journal of Mathematics and Physical Sciences Research ISSN 2348-5736 (Online)
Vol. 2, Issue 2, pp: (1-17), Month: October 2014 - March 2015, Available at: www.researchpublish.com
Thus, the above results are very useful for analysis of oblique frustum in determining the volume, surface area &
perimeter of elliptical section. All the derived results can be easily verified by the experimental results. These results can
be used in practical applications, case studies & other academic purposes.
VII.
ILLUSTRATIVE EXAMPLE
Consider a right circular cone, with apex angle
with the axis & lying at a normal distance
is thoroughly cut by a smooth plane inclined at an angle
from the apex point.
In this case,
hence the rate of convergence of
terms of higher power can be neglected with no significant error in the results
Let‟s first calculate the values of
as follows
(
(
)
series is much higher i.e.
(
)
(
(
)
)
)
(
(
)
(
)
)
⇒
(
)
(
(
)
)
(
(
)
)
(
)
(
)
(
)
(
)
⇒
(
)
(
(
)
)
(
(
)
)
⇒
Page | 15
Research Publish Journals
International Journal of Mathematics and Physical Sciences Research ISSN 2348-5736 (Online)
Vol. 2, Issue 2, pp: (1-17), Month: October 2014 - March 2015, Available at: www.researchpublish.com
Now, on setting the corresponding values of
a.
Volume of frustum
(
(
)
(
)
)
)
)
)
(
)
)
Major axis of generated section
(
(
)
(
)
)
(
)
Minor axis of generated section
)
√(
(
√(
f.
(
Perimeter of section generated
(
e.
(
)
(
d.
)
(
)
Surface area of frustum
(
c.
(
)
(
b.
we get the following results
)
)
Eccentricity of generated section
Thus, all the mathematical results obtained above can be verified by the experimental results. The symbols & names used
above are arbitrary given by the author Mr Harish Chandra Rajpoot (B Tech, Mech. Engg.).
ACKNOWLEDGEMENTS
Author is extremely thankful to all his teacher & professors who inspired & guided him for the research work in the field
of Applied Mathematics. He also thanks to his parents & his elder brother who have been showering their blessing on
him. Finally, he thanks to Almighty God who kept him physically fit during the completion of his research work.
Page | 16
Research Publish Journals
International Journal of Mathematics and Physical Sciences Research ISSN 2348-5736 (Online)
Vol. 2, Issue 2, pp: (1-17), Month: October 2014 - March 2015, Available at: www.researchpublish.com
REFERENCES
The work of the author is original. It is an outcome of deep studies & experiments carried out by the author on the
analysis of oblique frustum of right circular cone.
[1]
Some results are directly taken from “Advanced Geometry”, a book of research articles, by the author published
with Notion Press, India (www.notionpress.com)
[2]
Previous paper entitled “HCR‟s Rank Formula” was subjected to the peer review by Dr K. Srinivasa Rao, Senior
professor, Distinguished Professor (DST-Ramanujan Prof.), director of Ramanujan Academy of Maths Talent, Luz
Church Road, Chennai, India.
Page | 17
Research Publish Journals