Transformation Formulas





Transformation Formulas

A one-to-one function with the set of all points in the plane as the domain and the range is called transformation.

The important formulas of Transformation  as listed below:-

1.            reflection in X-axis: P(a, b) = p’(a, – b)

2.            reflection in Y-axis : P(a, b) = p’(- a, b)

3.            reflection in Y = X[W]: P(a, b) = P’(b, a)

4.            reflection in Y = X[Z]: P(a, b) = P’(- b, a)

5.            reflection in X = h[R]: P(a, b) = P’(2h – a, b)

6.            reflection in Y = K[M]: P(a, b) = P’(a, 2k – b)

7.            positive quarter turn about O[Q]: P(a, b) = P’(- b, a)

8.            negative quarter turn about O [Q^{-1}] : P(a, b) = P'(b, - a)

9.            half turn about O[H]: P(a, b) = P’(-a, – b)

10.          translation with T = \left( \dfrac{a}{b} \right) [T]:P(x, y) = P'(a + x, b + y)

11.          enlarging at O with scale factor K[E]: P(a, b) = P’(ka, kb)

12.          enlarging at P with a scale factor K: P(a, b) = P’(ka + x, kb + y)

13.          enlargement with centre at C (x, y) and K as scale factor denoted by E (c, k); P(a, b) = P’[x + k (a – x), y + k(b – y)]

14.          the scale factor (k) \\ = \dfrac{length \, of \, the \, side \, of \, image}{length \, of \, the \, corresponding \, sides \, of \, object \, figure} \\ \\ = \dfrac{CP'}{CP}

15.          rotation P(a, b) about centre

C (x, y) by -90 = P’(b + x – y, -a + x +y)

16.          rotation P(a, b0 about centre

C(x, y) by +90 = P’ (-a + x +y, a – x + y)

17.          rotation P(a, b) about centre

C(x, y) by 180 = p’(2x – y, 2y – b)

18.          combination of transformation:

i.              if Q = positive turn about the origin & W is the reflection in the line y = x, then QW = Y where Y is the reflection in Y-axis.

ii.             Q^2 = H where H is the half turn about O.

iii.            WQ = X where X is the reflection in the X-axis

iv.           W^2 = I where I is the identity transformation.

v.            Q^{-1}Z = Y where Q^{-1} is the negative quarter turn about O.

vi.           ZQ^{-1} = X where Z is the reflection in Y = X axis.

vii.          XH = Y

viii.         YH = X

ix.           XY = H

x.            HYX = 1

 

TRANSFORMATION USING MATRICES:

 

I.             \begin{pmatrix}1 & 0 \\ 0 & -1\end{pmatrix} represents reflection in X-axis.

II.            \begin{pmatrix}1 & 0 \\ 0 & 1\end{pmatrix} represent identity transformation.

III.           \begin{pmatrix}-1 & 0 \\ 0 & 1\end{pmatrix} represents reflection in Y-axis.

IV.          \begin{pmatrix}0 & 1 \\ 1 & 0\end{pmatrix} represents reflection in the line Y = X.

V.            \begin{pmatrix}0 & -1 \\ -1 & 0\end{pmatrix} represents reflection in the line Y = -X.

VI.          \begin{pmatrix}0 & -1 \\ 1 & 0\end{pmatrix} represent positive quarter turn \left ( +90 ^\circ \right)about the origin.

VII.         \begin{pmatrix}0 & 1 \\ -1 & 0\end{pmatrix} represents negative quarter turn\left ( -90 ^\circ \right) about the origin.

VIII.        \begin{pmatrix}-1 & 0 \\ 0 & -1\end{pmatrix} represents half 180^\circ turn about origin.

IX.           \begin{pmatrix}k & 0 \\ 0 & k\end{pmatrix} represents the enlargement with centre at the origin and scale factor k.

X.            The 2 * 1 matrix \begin{pmatrix}a \\ b\end{pmatrix} represents translation through ‘a’ units along X-axis and ‘b’ units along Y-axis.

 



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