Define Reflection transformation in 2D computer graphics. Derive or state the…
Define Reflection transformation in 2D computer graphics. Derive or state the transformation matrix for reflection about the line y = x. Also, calculate the reflected coordinates of a triangle with vertices P(2, 1), Q(5, 1), and R(3, 4) using this line as the axis of reflection.
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Definition of Reflection
Reflection is a geometric transformation that creates a mirror image of an object. The object is flipped 180o across a specific line known as the Axis of Reflection. While the size and shape of the object remain identical (congruent), its orientation is reversed.
Reflection about the line y = x
When an object is reflected across the diagonal line y = x, the x-coordinate and y-coordinate of every point are swapped.
Mathematical Rule: (x, y) => (y, x)
Transformation Matrix (Homogeneous Coordinates):
To represent this in a 3 * 3 format for computer graphics pipelines:
Numerical Calculation
We apply the transformation matrix or the swap rule to the triangle vertices:
Vertex P(2, 1):
x' = y = 1
y' = x = 2
New Point: P'(1, 2)
Vertex Q(5, 1):
x' = y = 1
y' = x = 5
New Point: Q'(1, 5)
Vertex R(3, 4):
x' = y = 4
y' = x = 3
New Point: R'(4, 3)
Final Result: The coordinates of the reflected triangle are P'(1, 2), Q'(1, 5), and R'(4, 3).