3D Rotation in Computer Graphics
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The 3D rotation is different from 2D rotation. In 3D Rotation we also have to define the angle of Rotation with the axis of Rotation.
For Example-Let us assume,
The initial coordinates of an object = (x0, y0, z0)
The Initial angle from origin = ?
The Rotation angle = ?
The new coordinates after Rotation = (x1, y1, z1)
In Three-dimensional plane we can define Rotation by following three ways–
- X-axis Rotation: We can rotate the object along x-axis. We can rotate an object by using following equation-
X1 = x0
Y1 = y0 cos?– z0 x sin?
Z1 = y0 x sin?+ z0 x cos?
We can represent 3D rotation in the form of matrix-
2. Y-axis Rotation: We can rotate the object along y-axis. We can rotate an object by using following equation-
x1 = z0 x sin? + x0 x cos?
y1 = y0
z1 = y0 x cos?– x0 x sin?
We can represent 3D rotation in the form of matrix–
3. Z-axis Rotation: We can rotate the object along z-axis. We can rotate an object by using following equation-
x1 = x0 x cos?– y0 x sin?
y1 = x0 x sin? + y0 x cos?
z1 = z0
We can represent 3D rotation in the form of matrix–
Example: A Point has coordinates P (2, 3, 4) in x, y, z-direction. The Rotation angle is 90 degrees. Apply the rotation in x, y, z direction, and find out the new coordinates of the point?
Solution: The initial coordinates of point = P (x0, y0, z0) = (2, 3, 4)
Rotation angle (?) = 90°
For x-axis–
Let the new coordinates = (x1, y1, z1) then,
x1= x0 = 2
y1= y0 x cos? — z0 x sin? = 3 x cos90°– 4 x sin90° = 3 x 0–4 x 1 = -4
z1= y0 x sin? + z0 x cos? = 3 x sin90°+ 4 x cos90° = 3 x 1 + 4 x 0 = 3
The new coordinates of point = (2, -4, 3)
For y-axis–
Let the new coordinates = (x1, y1, z1) then,
X1= z0 x sin? + x0 x cos? = 4 x sin90° + 2 x cos90° = 4 x 1 + 2 x 0 = 4
y1= y0 = 3
z1= y0 x cos&? — x0 x sin? = 3 x cos90°– 2 x sin90° = 3 x 0–4 x 0 = 0
The new coordinates of point = (4, 3, 0)
For z-axis–
Let the new coordinates = (x1, y1, z1) then,
x1= x0 x cos? — y0 x sin? = 2 x cos90° — 3 x sin90° = 2 x 0 + 3 x 1 = 3
y1= x0 x sin? + y0 x cos? = 2 x sin90° + 3 x cos90° = 2 x 1 + 3 x 0 = 2
z1= z0 =4
The New Coordinates of points = (3, 2, 4)