Non Homogeneous Transformation

Non Homogeneous Transformation

T.Srinivasarao

Asst.Professor

University College of Engg. Science & Technology Adikavi Nannaya University

Andhra Pradesh India

Dr.V.Mallipriya Asst.Professor Dept.Of Math.

Adikavi Nannaya University Andhra Pradesh

India

Abstract: A linear transformation

A:V F W F maps 0V 0W that helps to confirm that the range is the subspace of the

vector space W (F) and the null space is the subspace of V (F).

So, if the range space is a plane or a k dimensional hyperplane in the n dimensional space, then the null space is a line or an n k dimensional subspace. Further, they both have the point of intersection at the origin resulting in the orthogonal complement spaces of each other.

Since the row space of the matrix of A is range space and the column null space is the null space of A, ifV F W F , then the

row space and the column null space become the orthogonal complement subspaces ofV F . So, it can be followed that a linear transformation can be mapped to the Euclidean space or a plane or a line to the zero space or a line or a plane as its orthogonal complement spaces respectively.

In the present discussion, restricting the discussion to 3 dimensions, we wish to find a transformation that do not map the spaces through the origin and that identifies any point in the space at a fixed distance from the given point.

INTRODUCTION:

The set of points at a fixed distance from a fixed point make a sphere in n space. In a Euclidean space, the standard basis vectors are the unit vectors along the co-ordinate axes and the representation of an arbitrary point in the Euclidean space is (x, y,

z) which when transformed into the spherical co-ordinate system, the angles between the co-ordinate axes are considered to be right angles and so, the angle of measure is considered from the positive part of x axis at the origin to the projection of the terminal ray is denoted by and the angle between the projection and the terminal ray at the origin is denoted by . For each

unique pair , and , we get a unique point identified by the transformation from the given point.

If the fixed point in the space is Pa,b,c and Qx, y, z is any point in the space, then the fixed distance

x a2 , 0 2

and 0

is the radius of the sphere satisfying the conditions

tan1 y and cos1 z

x

x

Chapter 1:

Definition: 1.1: A: 3

3 given by

Ax, y, z cos sin, sin sin, cos is a transformation for a fixed triad

, , that can transform any point in the Euclidean space to any other point required depending on the unique triad.

x , , cos sin

A y , , sin sin

z , , cos

cos sin

sin sin

sin

cos

cos cos

sin cos 0

cos

0 sin

0

cos

sin

0 sin

0 cos cos sin

sin cos 0 0 1 0 0 sin sin

0 0 1 cos 0 sin 0 cos

Remark 1.2: for each fixed value of , taking 0 2 , a horizontal circle is formed which identifies the locus of the circle while increasing by a small angle, the new circle is formed and the family of these circles will form the entire sphere upon which every point can be identified with the help of the transformation A.

To identify the transformation that uses the spherical co-ordinate system, conveniently, shift the rectangular frame of reference

OXYZ to PX YZsuch that OX & PX ,OY & PY ,OZ & PZ are respectively parallel. The new rectangular frame will

identify the point Qx, y, z from Pa,b,c and all those points which are at the constant distance PQ .

Definition: 1.3: a transformation A: 3 3 given by

Ax, , , y , , , z , , a cos sin,b sin sin, c cos

that does not pass through origin for a,b,c 0,0,0 .

This transformation can be called a non homogeneous transformation.

The matrix form of this linear transformation will be

x , , a cos sin

A y , , b sin sin

z , , c cos

a

cos

sin

0 sin

0 cos

a cos sin

b sin cos 0 0 1 0 0 b sin sin

c 0 0 1 cos 0 sin 0 c cos

Chapter 2:

Definition 2.1: the shape of polygon P is translation invariant or simply a polygon P is translation invariant under a transformation if

2

2

Ax, k , , y , k , , z , k , Ax , , , y , , , z , ,

For a fixed value of , the transformation A acts on a circle. Further, for the fixed value of , the transformation rotates

n

the n roots of unity 2k ,1 k n & n 3 by treating them as the vertices, they form a regular n gon joining them

k n

in the order of rotation, which is translation invariant under the transformation A.

Definition 2.2: a group (G, ) is a translation group if the operation is a metric or a geometric operation on it.

Theorem 2.3 : a group formed by 2k / 1 k n & n 3 under additional of angles is a translation group.

k n

Clearly addition of angles is a metric on a circle.

For each fixed value of , 0 , the non homogeneous transformation defines a circle (a circle horizontal if it is

2k

considered to be in the Euclidean space) upon which for a particular value of k

,1 k n , the translation group under

n

k j

2k 2 j 2k j mod 2 ,1 k, j n & n 3

n n n

2k

2 j

2k

2 j

See that 0 2 & 0 2 result in 0 mod 2 2

n n n n

But, if the resultant angle is equal to 2 , then we consider it as 0 or 2 represent the vertex of the n gon upon the positive part of the x – axis.

This verifies that is closed under .

is the member of such that

2k 2n k mod 2

which verifies the identity and inverse

2 k nk n n 2

axioms.

Note: the translation is geometrically invariant. However, this is the source of discussion for the permutation groups with instances of dihedral group.

REFERENCES:

1. Conway, John Horton; Delgado Friedrichs, Olaf; Huson, Daniel H; Thurston Willam P.(2001), On three dimensional space groups, Beltrage zur Algebra und Geometrie, 42(2): 475-507, arXiv:math.MG/9911185,MR 1865535

2. Coornaert, M; Delzant.T; Papadopoulos,A.(1990), Geometrie et theorie des groups[Geometry and Group Theory],

3. Serre, Jean Pierre(1977), Linear representations of finite groups, Berlin, New York. Springer Verlag. ISBN 978 0 691 08017 8. MR 0347778

4. Wussing, Hans(2007), The Genesis of Abstract Group Concept: A contribution to the History of the Origin of Abstract Group Theory, New York:

   Dover Publications, ISBN 978 0 486 45868 7

5. Arfken, George. Mathematical Mthods for Physicists, Academic Press. ISBN 0123846544

6. Keith, Sandra, Visualizing Linear Algebra using Maple.Upper Saddle River, NJ: Prentice Hall, 2001

7. Schreier, O. And Sperner, E., Introdution to Modern Algebra and Matrix Theory, 2nd Ed., Chelsea Publishing Co., New York, 1955

8. Handbook of Finite Translation Planes, Normal L.Johnson, Vikram Jha, Mauro Biliotti, Chapman & Hall/CRC, Taylr & Francis Group, University of Leece, Italy, ISBN 13: 978 58488 605 1