# 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)**