Shape A can be transformed to shape B by a rotation 90° clockwise about (-1, 1) followed by a translation by vector
(
a
b
)
S
e
e
h
o
w
a
n
n
o
y
i
n
g
v
e
r
t
i
c
a
l
t
e
x
t
i
s
?
To transform shape A to shape B by a rotation of 90° clockwise about (-1, 1), followed by a translation by vector (a, b), follow these steps:
1. Start with shape A.
2. Plot the point (-1, 1) on a coordinate plane.
3. Rotate shape A 90° clockwise about the point (-1, 1).
- To do this, draw lines connecting each point in shape A to the point (-1, 1).
- Measure the angle between each line and the x-axis, and subtract 90° to get the new angle.
- Use this new angle to find the new coordinates for each point after the rotation.
4. Translate the rotated shape by the vector (a, b).
- To do this, add the values of (a, b) to the x and y coordinates of each point in the rotated shape.
- This will move the shape horizontally by 'a' units and vertically by 'b' units.
5. The result is shape B, which has been transformed from shape A by a rotation of 90° clockwise about (-1, 1), followed by a translation by vector (a, b).
To determine the transformation matrix for the given transformation, we need to consider each component separately - rotation and translation.
1. Rotation 90° clockwise about (-1, 1):
- First, we need to translate the rotation center (-1, 1) to the origin. This can be done by subtracting (-1) from the x-coordinate and subtracting 1 from the y-coordinate of each point.
- Next, we rotate the translated shape by 90° clockwise. For a 2D shape, this can be achieved by multiplying each point's coordinates by the rotation matrix:
[cos(θ) -sin(θ)]
[sin(θ) cos(θ)]
In our case, θ = -90° (clockwise), so the rotation matrix becomes:
[0 -1]
[1 0]
- After the rotation, we need to translate the shape back to its original position. To do this, we add (-1) to the x-coordinate and add 1 to the y-coordinate of each point.
2. Translation by vector (a, b):
- We simply add a to the x-coordinate and b to the y-coordinate of each point.
So, the final transformation matrix can be obtained by combining the translation matrix and rotation matrix:
[T] = [1 0 a]
[0 1 b]
[R] = [0 -1]
[1 0]
[T*R*T_inv] = [1 0 a][0 -1][-1 0][0 -1 1][1 0 -a]
[1 0 -b]
Simplifying the multiplication, we get the transformation matrix:
[T*R*T_inv] = [0 -1 1][1 0 -a]
[1 0 -b]
This transformation matrix can be used to transform the coordinates of shape A to shape B.