Monday, 24 September 2012

What is transformations

In the previous post we have discussed about word problem solver free and In today's session we are going to discuss about What is transformations.
If we move any shape in any direction then after every rotation we get its original position is said to be transformation. Now we will see some properties based on Transformations. Properties of transformation are given below.
1.    Projective properties
2.    Affine properties
3.    Metric properties
4.    Affine geometry
5.    Projective geometry
Now we will see theorem based on transformation. Theorem of transformation is given below:
Theorem of ceva for concurrent lines;
Theorem of Menelaus for collinear points;
Theorem of pappus – pascal;
And theorem of Desargues;
Now we will discuss the concept of coordinate transformation.
As we know that the Cartesian coordinates system is fully defined by its origin and vectors along x-axis and y-axis.
Let we take first origin ‘O’ and unit vector OX1 and OX2 and we take a point ‘N’ which has coordinates (a, b) relative to that coordinate system.
Then we have new coordinates with the origin O’ and unit vectors OX1 and OX2 and point ‘N’ has coordinates (a', b') that is related to the new coordinate system.
Then the transformation formula between (a, b) and (a', b')
ON= OO' + O'M
 N = O' + a’ O'N1' + b' O'N2'
N = O' + a'(N1' - O') + b' (P2' - O')
With coordinates this will becomes
(a, b) = (ao, bo) + a'((s1, t1) - (ao, bo)) + b'((s2, t2) - (ao, bo))
a = bo + (s1 - ao) a' + (s2 - ao) a'
b = bo + (s1 - bo) a' + (t2 - bo) a'
 With matrix notation this becomes
[a]   [(s1 - ao) (s2 - ao)] [a']   [ao]
=   +[a]   [(t1 - ao) (t2 - ao)] [a']   [ao]
This is all about transformation.
Now we will see How to Find Volume of a Cylinder. Volume of cylinder is given as:
Volume = pi * r2 * h. icse syllabus can be downloaded from many websites.  

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