**3D Transformations in Computer Graphics-**

We have discussed-

- Transformation is a process of modifying and re-positioning the existing graphics.
- 3D Transformations take place in a three dimensional plane.

In computer graphics, various transformation techniques are-

In this article, we will discuss about 3D Scaling in Computer Graphics.

**3D Scaling in Computer Graphics-**

In computer graphics, scaling is a process of modifying or altering the size of objects. |

- Scaling may be used to increase or reduce the size of object.
- Scaling subjects the coordinate points of the original object to change.
- Scaling factor determines whether the object size is to be increased or reduced.
- If scaling factor > 1, then the object size is increased.
- If scaling factor < 1, then the object size is reduced.

Consider a point object O has to be scaled in a 3D plane.

Let-

- Initial coordinates of the object O = (X
_{old}, Y_{old},Z_{old}) - Scaling factor for X-axis = S
_{x} - Scaling factor for Y-axis = S
_{y} - Scaling factor for Z-axis = S
_{z} - New coordinates of the object O after scaling = (X
_{new}, Y_{new}, Z_{new})

This scaling is achieved by using the following scaling equations-

- X
_{new}= X_{old}x S_{x} - Y
_{new}= Y_{old}x S_{y} - Z
_{new}= Z_{old}x S_{z}

In Matrix form, the above scaling equations may be represented as-

**PRACTICE PROBLEMS BASED ON 3D SCALING IN COMPUTER GRAPHICS-**

**Problem-01:**

Given a 3D object with coordinate points A(0, 3, 3), B(3, 3, 6), C(3, 0, 1), D(0, 0, 0). Apply the scaling parameter 2 towards X axis, 3 towards Y axis and 3 towards Z axis and obtain the new coordinates of the object.

**Solution-**

Given-

- Old coordinates of the object = A (0, 3, 3), B(3, 3, 6), C(3, 0, 1), D(0, 0, 0)
- Scaling factor along X axis = 2
- Scaling factor along Y axis = 3
- Scaling factor along Z axis = 3

**For Coordinates A(0, 3, 3)**

Let the new coordinates of A after scaling = (X_{new}, Y_{new}, Z_{new}).

Applying the scaling equations, we have-

- X
_{new}= X_{old}x S_{x}= 0 x 2 = 0 - Y
_{new}= Y_{old}x S_{y}= 3 x 3 = 9 - Z
_{new}= Z_{old}x S_{z}= 3 x 3 = 9

Thus, New coordinates of corner A after scaling = (0, 9, 9).

**For Coordinates B(3, 3, 6)**

Let the new coordinates of B after scaling = (X_{new}, Y_{new}, Z_{new}).

Applying the scaling equations, we have-

- X
_{new}= X_{old}x S_{x}= 3 x 2 = 6 - Y
_{new}= Y_{old}x S_{y}= 3 x 3 = 9 - Z
_{new}= Z_{old}x S_{z}= 6 x 3 = 18

Thus, New coordinates of corner B after scaling = (6, 9, 18).

**For Coordinates C(3, 0, 1)**

Let the new coordinates of C after scaling = (X_{new}, Y_{new}, Z_{new}).

Applying the scaling equations, we have-

- X
_{new}= X_{old}x S_{x}= 3 x 2 = 6 - Y
_{new}= Y_{old}x S_{y}= 0 x 3 = 0 - Z
_{new}= Z_{old}x S_{z}= 1 x 3 = 3

Thus, New coordinates of corner C after scaling = (6, 0, 3).

**For Coordinates D(0, 0, 0)**

Let the new coordinates of D after scaling = (X_{new}, Y_{new}, Z_{new}).

Applying the scaling equations, we have-

- X
_{new}= X_{old}x S_{x}= 0 x 2 = 0 - Y
_{new}= Y_{old}x S_{y}= 0 x 3 = 0 - Z
_{new}= Z_{old}x S_{z}= 0 x 3 = 0

Thus, New coordinates of corner D after scaling = (0, 0, 0).

To gain better understanding about 3D Scaling in Computer Graphics,

**Next Article-** **3D Reflection in Computer Graphics**

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