Tag: 3D Translation Rotation Scaling

3D Scaling in Computer Graphics | Definition | Examples

Computer Graphics

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-

 

 

  1. Translation
  2. Rotation
  3. Scaling
  4. Reflection
  5. Shear

 

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 = (Xold, Yold,Zold)
  • Scaling factor for X-axis = Sx
  • Scaling factor for Y-axis = Sy
  • Scaling factor for Z-axis = Sz
  • New coordinates of the object O after scaling = (Xnew, Ynew, Znew)

 

This scaling is achieved by using the following scaling equations-

  • Xnew = Xold x Sx
  • Ynew = Yold x Sy
  • Znew = Zold x Sz

 

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 = (Xnew, Ynew, Znew).

 

Applying the scaling equations, we have-

  • Xnew = Xold x Sx = 0  x 2 = 0
  • Ynew = Yold x Sy = 3 x 3 = 9
  • Znew = Zold x Sz = 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 = (Xnew, Ynew, Znew).

 

Applying the scaling equations, we have-

  • Xnew = Xold x Sx = 3  x 2 = 6
  • Ynew = Yold x Sy = 3 x 3 = 9
  • Znew = Zold x Sz = 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 = (Xnew, Ynew, Znew).

 

Applying the scaling equations, we have-

  • Xnew = Xold x Sx = 3  x 2 = 6
  • Ynew = Yold x Sy = 0 x 3 = 0
  • Znew = Zold x Sz = 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 = (Xnew, Ynew, Znew).

 

Applying the scaling equations, we have-

  • Xnew = Xold x Sx = 0  x 2 = 0
  • Ynew = Yold x Sy = 0 x 3 = 0
  • Znew = Zold x Sz = 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,

Watch this Video Lecture

 

Next Article- 3D Reflection in Computer Graphics

 

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3D Translation in Computer Graphics | Definition | Examples

Computer Graphics

3D Transformation in Computer Graphics-

 

In Computer graphics,

Transformation is a process of modifying and re-positioning the existing graphics.

 

  • 3D Transformations take place in a three dimensional plane.
  • 3D Transformations are important and a bit more complex than 2D Transformations.
  • Transformations are helpful in changing the position, size, orientation, shape etc of the object.

 

Transformation Techniques-

 

In computer graphics, various transformation techniques are-

 

 

  1. Translation
  2. Rotation
  3. Scaling
  4. Reflection
  5. Shear

 

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

 

3D Translation in Computer Graphics-

 

In Computer graphics,

3D Translation is a process of moving an object from one position to another in a three dimensional plane.

 

Consider a point object O has to be moved from one position to another in a 3D plane.

 

Let-

  • Initial coordinates of the object O = (Xold, Yold, Zold)
  • New coordinates of the object O after translation = (Xnew, Ynew, Zold)
  • Translation vector or Shift vector = (Tx, Ty, Tz)

 

Given a Translation vector (Tx, Ty, Tz)-

  • Tx defines the distance the Xold coordinate has to be moved.
  • Ty defines the distance the Yold coordinate has to be moved.
  • Tz defines the distance the Zold coordinate has to be moved.

 

 

This translation is achieved by adding the translation coordinates to the old coordinates of the object as-

  • Xnew = Xold + Tx     (This denotes translation towards X axis)
  • Ynew = Yold + Ty     (This denotes translation towards Y axis)
  • Znew = Zold + Tz     (This denotes translation towards Z axis)

 

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

 

 

Also Read- 2D Translation in Computer Graphics

 

PRACTICE PROBLEM BASED ON 3D TRANSLATION IN COMPUTER GRAPHICS-

 

Problem-

 

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

 

Solution-

 

Given-

  • Old coordinates of the object = A (0, 3, 1), B(3, 3, 2), C(3, 0, 0), D(0, 0, 0)
  • Translation vector = (Tx, Ty, Tz) = (1, 1, 2)

 

For Coordinates A(0, 3, 1)

 

Let the new coordinates of A = (Xnew, Ynew, Znew).

 

Applying the translation equations, we have-

  • Xnew = Xold + Tx = 0 + 1 = 1
  • Ynew = Yold + Ty = 3 + 1 = 4
  • Znew = Zold + Tz = 1 + 2 = 3

 

Thus, New coordinates of A = (1, 4, 3).

 

For Coordinates B(3, 3, 2)

 

Let the new coordinates of B = (Xnew, Ynew, Znew).

 

Applying the translation equations, we have-

  • Xnew = Xold + Tx = 3 + 1 = 4
  • Ynew = Yold + Ty = 3 + 1 = 4
  • Znew = Zold + Tz = 2 + 2 = 4

 

Thus, New coordinates of B = (4, 4, 4).

 

For Coordinates C(3, 0, 0)

 

Let the new coordinates of C = (Xnew, Ynew, Znew).

 

Applying the translation equations, we have-

  • Xnew = Xold + Tx = 3 + 1 = 4
  • Ynew = Yold + Ty = 0 + 1 = 1
  • Znew = Zold + Tz = 0 + 2 = 2

 

Thus, New coordinates of C = (4, 1, 2).

 

For Coordinates D(0, 0, 0)

 

Let the new coordinates of D = (Xnew, Ynew, Znew).

 

Applying the translation equations, we have-

  • Xnew = Xold + Tx = 0 + 1 = 1
  • Ynew = Yold + Ty = 0 + 1 = 1
  • Znew = Zold + Tz = 0 + 2 = 2

 

Thus, New coordinates of D = (1, 1, 2).

Thus, New coordinates of the object = A (1, 4, 3), B(4, 4, 4), C(4, 1, 2), D(1, 1, 2).

 

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

Watch this Video Lecture

 

Next Article- 3D Rotation in Computer Graphics

 

Get more notes and other study material of Computer Graphics.

Watch video lectures by visiting our YouTube channel LearnVidFun.