7.C] Consider an image point [2, 2]. Perform the following operations and show the results:
a) Translate the image right by 3 units.
b) Scale the image in both x-axis and y-axis by 3 units.
c) Rotate the image in x-axis by 45°.
Answer:
Matrix Operations on Point [2, 2]
1. Initial Point:
The given point is [2, 2], represented as a column vector:
⎛ 2 ⎞
⎜ 2 ⎟
⎝ ⎠
2. Translation by 3 Units to the Right:
Translation Matrix:
⎛ 1 0 3 ⎞
⎜ 0 1 0 ⎟
⎝ 0 0 1 ⎠
Point in Homogeneous Coordinates:
⎛ 2 ⎞
⎜ 2 ⎟
⎝ 1 ⎠
Translated Point:
⎛ 1 0 3 ⎞ ⎛ 2 ⎞ ⎛ 5 ⎞
⎜ 0 1 0 ⎟ × ⎜ 2 ⎟ = ⎜ 2 ⎟
⎝ 0 0 1 ⎠ ⎝ 1 ⎠ ⎝ 1 ⎠
Resulting Point: [5, 2]
3. Scaling by 3 Units in x and y Directions:
Scaling Matrix:
⎛ 3 0 0 ⎞
⎜ 0 3 0 ⎟
⎝ 0 0 1 ⎠
Point in Homogeneous Coordinates:
⎛ 2 ⎞
⎜ 2 ⎟
⎝ 1 ⎠
Scaled Point:
⎛ 3 0 0 ⎞ ⎛ 2 ⎞ ⎛ 6 ⎞
⎜ 0 3 0 ⎟ × ⎜ 2 ⎟ = ⎜ 6 ⎟
⎝ 0 0 1 ⎠ ⎝ 1 ⎠ ⎝ 1 ⎠
Resulting Point: [6, 6]
4. Rotation by 45° Around the Origin:
Why \(\frac{1}{\sqrt{2}}\) is used:
The trigonometric values for 45° are:
- \(\cos(45^\circ) = \frac{\sqrt{2}}{2}\)
- \(\sin(45^\circ) = \frac{\sqrt{2}}{2}\)
To simplify calculations, these values are often expressed as \(\frac{1}{\sqrt{2}}\) because \(\frac{\sqrt{2}}{2}\) is mathematically equivalent to \(\frac{1}{\sqrt{2}}\).
Rotation Matrix:
⎛ 1/√2 -1/√2 0 ⎞
⎜ 1/√2 1/√2 0 ⎟
⎝ 0 0 1 ⎠
Point in Homogeneous Coordinates:
⎛ 2 ⎞
⎜ 2 ⎟
⎝ 1 ⎠
Rotated Point:
⎛ 1/√2 -1/√2 0 ⎞ ⎛ 2 ⎞ ⎛ 0 ⎞
⎜ 1/√2 1/√2 0 ⎟ × ⎜ 2 ⎟ = ⎜ 2√2 ⎟
⎝ 0 0 1 ⎠ ⎝ 1 ⎠ ⎝ 1 ⎠
Simplifying:
⎛ 0 ⎞
⎜ 2.83 ⎟
⎝ 1 ⎠
Resulting Point: [0, 2.83]