 Open Access
 Authors : T. Srinivasarao , K. Revathi
 Paper ID : IJERTV9IS020221
 Volume & Issue : Volume 09, Issue 02 (February 2020)
 Published (First Online): 26022020
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
Coaxial Hyperplanes
T. Srinivasarao
Asst. Professor, Dept of Math University College of Science & Technology
Adikavi Nannaya University AP, India

Revathi
Asst. Professor, Dept of Math University College of S & T Adikavi Nannaya University AP, India
Abstract: For Euclidean geometry, the basis vectors are the source of inspiration for mutually perpendicular/orthogonal concepts that rules over the world in the form of tangent, normal and binormal or the rectangular coordinate system. So, the Pythagoras theorem is valid in 3 dimensions. But when the 4th and higher dimensional space does not obey Pythagoras and thus the higher ideas like norm , ortho normal basis and more are required. So, the plane in a Euclidean geometry becomes a Hyperplane in the n dimensional space for n > 3. Pivoting in the linear programming or the construction of orthonormal basis using Gram Schmidts process and the pth norm are some of the methods that scales to the required heights. In the astronomical discussions, the planets in the solar system will come as close as possible and the distance required to go round and reach back is an idea or the thought source in which the shortest distance between two hyperplanes is presented.

INTRODUCTION:
Hyper Plane:
In an n dimensional linear space, how many hyper planes would be there, of n 1 dimensional those form a family of symmetric subspaces?
Introduction: planes through origin only will be subspaces. But the collection of parallel planes cannot form the collection of subspaces while one and only one plane will have the origin on it. For instance,
a1 x1 a2 x2 … an1xn1 an xn c
will be a hyperplane in n dimensional space Vn F for c is any constant and when c = 0,
(1.1)
the set of vectors x1 , x2 ,…, xn that satisfy the condition (1.1) will form the subspace showing a1 0, a2 0, an1 0, an 0 whenever c 0. In the case of c = 0, if ai 01 i n , then x1, x2 ,…, xn1will be the basis of the subspace which is a hyperplane through the origin.
See that j 1×1 2 x2 … n xn 0 , k ,1 k n, k i
(1.2)
are the hyper planes that contain the
Xi axis as the axis of intersection or common axis or the axis of rotation for different
j
j
non zero values of k and
n
j 1
n 1/ n 1
(1.3)
Definition: a family of hyperplanes through a common line in Vn F is a coaxial family of hyperplanes.
(1.4)
There are infinitely many coaxial families of hyperplanes through origin. A class of coaxial families will have coordinate axes as the axes of symmetry and other families will have lines through origin as axes of symmetry.
Definition: a non homogeneous coaxial family of hyperplanes are the family that have the line of symmetry not through the origin. (1.5)
There will be a unique plane through origin in the family of non homogeneous coaxial family of hyperplanes.
(1.6)
The
Xi – axis can be represented by
X1 0, X 2 0,…, Xi1 0, Xi1 0,…, Xn 0
(1.7)

Line of Intersection of Hyper Planes:

Two hyper planes r 1 x1 2 x2 … n xn 0 and
s 1 x1 2 x2 … n xn 0 are clearly having the common line Xi
and so are not coincident if and only if
k m
k m
for at least one distinct pair k, m among 1, 2, …,i 1, i +1, …,n.
(2.1)
Any line not coincident with any coordinate axis will be of the axis of rotation for the family of the hyper planes
j 1 x1 2 x2 … i1 xi1 i xi i1 xi1 … n xn 0
If a line is not through origin in Vn F , then it can be parametrized as
x1 1t 1 , x2 2t 2 ,…, xn nt n
Two lines 1 defined by x1 1t 1 , x2 2t 2 ,…, xn nt n and 2 defined by
(2.2)
(2.3)
x t , x
t
,…, x
t
that are not intersecting if k
bk
for at least one k such that1 k n .
1 1 1 2 2 2
n n n
k k
Definition: If 1
and 2
are the lines on two different members of a family of hyperplanes that do not intersect, then they are
skew hyper lines on hyperplanes. (2.4)
Theorem: Two skew hyper lines in different coaxial families will be on a unique pair of parallel hyperplanes and so, the distance between the planes will be the shortest distance between the lines.
Proof: there is a unique position in which the line 1 is in the plane
1 1 x1 2 x2 … i1 xi1 i1 xi1 … n xn 0; and the line 2 is in
(2.5)
x x
… x
x … x
c , c 0 such that k m
for all 1 k,m n
2 1 1 2 2
i1 i1
i1 i1 n n
k m
Since
1 and
2 are the skew hyper lines, at one instance, these lines will be as close as possible and the next moment, the
distance between them will be more and keep increasing between any two points, each is on either the lines.
In other words, there is a unique pair of points the given lines.
p1 1 and
p2 2
such that
p2 p1 n will be the shortest distance between
It can be followed that the line of minimum distance between 1 and 2 is 3 p1 p2 such that 3 is normal to both 1 and 2 .
So, the line 3 is through the feet of the normal that are the points of intersection of the lines namely
p1 1 and
p2 1
Incidentally
1 1
and
2 2
where
1 is in one co axial system and
2 is in the other co axial system such that
1 & 2 will be parallel.
The distance between the feet of the normal skew hyper line is the distance between the parallel planes 1 and 2 . The minimum distance possible between the planes 1 & 2 the length of the skew hyper line of shortest distance is
n
j 1
c
n 1/ n
n 1/ n
j j
,1 i n
(2.6)
Note: if both co axial systems are not through origin, then the shortest distance between the skew hyper lines
d1 d2
n
n 1/ n
(2.7)
j 1
j j
REFERENCES

Charles w.Curtis(1968) Linear Algebra, page 62, Allyn&Bacon, Boston

Victor V.Prasolov & VM Tikhomirov(1997, 2001) Geometry, page 22, Vol. 200 in Transactions of Mathematical Monographs, American Mathematical Society

Heinrich Guggenheimer(1977) Applicable Geometry, page 7, Krieger, Huntington ISBN 0882753681

Berger, Marcel(1987), Geometry 1, Berlin; Springer, ISBN 3540 116583

Brannan, David A; Esplen, Matthew F; Gray, Jeremy J(1999), Geometry, Cambridege University Press, ISBN 9780521441773