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1. Combined Transformations
When an object undergoes two transformations one after another, it is called a combined (or composite) transformation.
ā ļø Crucial Order Rule:
If a question asks for the combined transformation T1 followed by T2, it is written notationally as T2 o T1.
You must apply T1 first to the object point to find the intermediate image, and then apply T2 second to that intermediate image to get the final answer.
2. The Unit Matrix & Special Matrices
Before computing vector multiplications, you must recognize these foundational geometric matrix constants:
The 2x2 Identity Unit Matrix ( I )
The unit matrix acts like the number "1" in normal multiplication. Multiplying any vector by the identity matrix leaves the point completely unchanged (Identity Transformation).
I = [ 1 0 ; 0 1 ]
The 2x2 Zero / Null Matrix ( O )
A transformation matrix where all entry objects are zero. It completely collapses any shape down onto the origin point (0,0).
O = [ 0 0 ; 0 0 ]
3. Advanced Custom Centers (h, k) ā Highly Tested!
When the center is not the origin (0,0), apply these specialized coordinate shifts:
A. Rotation Rules about Center (h, k)
- š Rotation of +90° (Anticlockwise):
P(x, y) → P'(-y + h + k, x - h + k)
- š Rotation of -90° (Clockwise):
P(x, y) → P'(y - k + h, -x + h + k)
- š Rotation of 180° (Half Turn):
P(x, y) → P'(2h - x, 2k - y)
B. Enlargement about Center (h, k) with Scale Factor 'k_s'
To stretch or shrink an object relative to a point outside the origin:
P(x, y) → P'( k_s(x - h) + h, k_s(y - k) + k )
4. How to Find the Matrix of Transformation
When you need to find an unknown 2x2 matrix that transforms a set of points, use the standard unit columns Eā(1, 0) and Eā(0, 1) as your foundation.
The Unit Vector Method (Eā and Eā):
Let the required transformation matrix be T = [ a b ; c d ]
- Find the image of Eā(1, 0) under the given rule. Let its image be Eā'(a, c). This forms the first column.
- Find the image of Eā(0, 1) under the given rule. Let its image be Eā'(b, d). This forms the second column.
- Combine them together: T = [ a b ; c d ]
5. Transformation Matrix Quick Reference Table (Origin Rules)
| Transformation Type |
Standard Coordinate Rule |
2x2 Matrix Notation |
| Reflection in x-axis |
P(x, y) → P'(x, -y) |
[ 1 0 ; 0 -1 ] |
| Reflection in y-axis |
P(x, y) → P'(-x, y) |
[ -1 0 ; 0 1 ] |
| Reflection in line y = x |
P(x, y) → P'(y, x) |
[ 0 1 ; 1 0 ] |
| Reflection in line y = -x |
P(x, y) → P'(-y, -x) |
[ 0 -1 ; -1 0 ] |
| Rotation +90° (Anticlockwise) |
P(x, y) → P'(-y, x) |
[ 0 -1 ; 1 0 ] |
| Rotation -90° (Clockwise) |
P(x, y) → P'(y, -x) |
[ 0 1 ; -1 0 ] |
| Rotation 180° (Half turn) |
P(x, y) → P'(-x, -y) |
[ -1 0 ; 0 -1 ] |
| Enlargement [(0,0), k] |
P(x, y) → P'(kx, ky) |
[ k 0 ; 0 k ] |
6. Inversion Geometry (Inversion Circle)
Finding the inverse of a point P(x, y) relative to a circle with center (h, k) and radius r:
Case A: Center at Origin (0,0)
x' = (r² · x) / (x² + y²)
y' = (r² · y) / (x² + y²)
Case B: Advanced Center at (h,k)
x' = h + [ r² · (x - h) ] / [ (x - h)² + (y - k)² ]
y' = k + [ r² · (y - k) ] / [ (x - h)² + (y - k)² ]