3D Transformation in Computer Graphics Solved Examples

Tag: 3D Transformation in Computer Graphics Solved Examples

3D Shearing 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-

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

3D Shearing in Computer Graphics-

In Computer graphics,

3D Shearing is an ideal technique to change the shape of an existing object in a three dimensional plane.

In a three dimensional plane, the object size can be changed along X direction, Y direction as well as Z direction.

So, there are three versions of shearing-

Shearing in X direction
Shearing in Y direction
Shearing in Z direction

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

Let-

Initial coordinates of the object O = (X_old, Y_old, Z_old)
Shearing parameter towards X direction = Sh_x
Shearing parameter towards Y direction = Sh_y
Shearing parameter towards Z direction = Sh_z
New coordinates of the object O after shearing = (X_new, Y_new, Z_new)

Shearing in X Axis-

Shearing in X axis is achieved by using the following shearing equations-

X_new = X_old
Y_new = Y_old + Sh_y x X_old
Z_new = Z_old + Sh_z x X_old

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

Shearing in Y Axis-

Shearing in Y axis is achieved by using the following shearing equations-

X_new = X_old + Sh_x x Y_old
Y_new = Y_old
Z_new = Z_old + Sh_z x Y_old

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

Shearing in Z Axis-

Shearing in Z axis is achieved by using the following shearing equations-

X_new = X_old + Sh_x x Z_old
Y_new = Y_old + Sh_y x Z_old
Z_new = Z_old

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

PRACTICE PROBLEMS BASED ON 3D SHEARING IN COMPUTER GRAPHICS-

Problem-01:

Given a 3D triangle with points (0, 0, 0), (1, 1, 2) and (1, 1, 3). Apply shear parameter 2 on X axis, 2 on Y axis and 3 on Z axis and find out the new coordinates of the object.

Solution-

Given-

Old corner coordinates of the triangle = A (0, 0, 0), B(1, 1, 2), C(1, 1, 3)
Shearing parameter towards X direction (Sh_x) = 2
Shearing parameter towards Y direction (Sh_y) = 2
Shearing parameter towards Y direction (Sh_z) = 3

Shearing in X Axis-

For Coordinates A(0, 0, 0)

Let the new coordinates of corner A after shearing = (X_new, Y_new, Z_new).

Applying the shearing equations, we have-

X_new = X_old = 0
Y_new = Y_old + Sh_y x X_old = 0 + 2 x 0 = 0
Z_new = Z_old + Sh_z x X_old = 0 + 3 x 0 = 0

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

For Coordinates B(1, 1, 2)

Let the new coordinates of corner B after shearing = (X_new, Y_new, Z_new).

Applying the shearing equations, we have-

X_new = X_old = 1
Y_new = Y_old + Sh_y x X_old= 1 + 2 x 1 = 3
Z_new = Z_old + Sh_z x X_old = 2 + 3 x 1 = 5

Thus, New coordinates of corner B after shearing = (1, 3, 5).

For Coordinates C(1, 1, 3)

Let the new coordinates of corner C after shearing = (X_new, Y_new, Z_new).

Applying the shearing equations, we have-

X_new = X_old = 1
Y_new = Y_old + Sh_y x X_old = 1 + 2 x 1 = 3
Z_new = Z_old + Sh_z x X_old = 3 + 3 x 1 = 6

Thus, New coordinates of corner C after shearing = (1, 3, 6).

Thus, New coordinates of the triangle after shearing in X axis = A (0, 0, 0), B(1, 3, 5), C(1, 3, 6).

Shearing in Y Axis-

For Coordinates A(0, 0, 0)

Let the new coordinates of corner A after shearing = (X_new, Y_new, Z_new).

Applying the shearing equations, we have-

X_new = X_old + Sh_x x Y_old = 0 + 2 x 0 = 0
Y_new = Y_old = 0
Z_new = Z_old + Sh_z x Y_old = 0 + 3 x 0 = 0

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

For Coordinates B(1, 1, 2)

Let the new coordinates of corner B after shearing = (X_new, Y_new, Z_new).

Applying the shearing equations, we have-

X_new = X_old + Sh_x x Y_old = 1 + 2 x 1 = 3
Y_new = Y_old = 1
Z_new = Z_old + Sh_z x Y_old = 2 + 3 x 1 = 5

Thus, New coordinates of corner B after shearing = (3, 1, 5).

For Coordinates C(1, 1, 3)

Let the new coordinates of corner C after shearing = (X_new, Y_new, Z_new).

Applying the shearing equations, we have-

X_new = X_old + Sh_x x Y_old = 1 + 2 x 1 = 3
Y_new = Y_old = 1
Z_new = Z_old + Sh_z x Y_old = 3 + 3 x 1 = 6

Thus, New coordinates of corner C after shearing = (3, 1, 6).

Thus, New coordinates of the triangle after shearing in Y axis = A (0, 0, 0), B(3, 1, 5), C(3, 1, 6).

Shearing in Z Axis-

For Coordinates A(0, 0, 0)

Let the new coordinates of corner A after shearing = (X_new, Y_new, Z_new).

Applying the shearing equations, we have-

X_new = X_old + Sh_x x Z_old = 0 + 2 x 0 = 0
Y_new = Y_old + Sh_y x Z_old = 0 + 2 x 0 = 0
Z_new = Z_old = 0

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

For Coordinates B(1, 1, 2)

Let the new coordinates of corner B after shearing = (X_new, Y_new, Z_new).

Applying the shearing equations, we have-

X_new = X_old + Sh_x x Z_old = 1 + 2 x 2 = 5
Y_new = Y_old + Sh_y x Z_old = 1 + 2 x 2 = 5
Z_new = Z_old = 2

Thus, New coordinates of corner B after shearing = (5, 5, 2).

For Coordinates C(1, 1, 3)

Let the new coordinates of corner C after shearing = (X_new, Y_new, Z_new).

Applying the shearing equations, we have-

X_new = X_old + Sh_x x Z_old = 1 + 2 x 3 = 7
Y_new = Y_old + Sh_y x Z_old = 1 + 2 x 3 = 7
Z_new = Z_old = 3

Thus, New coordinates of corner C after shearing = (7, 7, 3).

Thus, New coordinates of the triangle after shearing in Z axis = A (0, 0, 0), B(5, 5, 2), C(7, 7, 3).

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

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3D Reflection 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-

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

3D Reflection in Computer Graphics-

Reflection is a kind of rotation where the angle of rotation is 180 degree.
The reflected object is always formed on the other side of mirror.
The size of reflected object is same as the size of original object.

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

Let-

Initial coordinates of the object O = (X_old, Y_old, Z_old)
New coordinates of the reflected object O after reflection = (X_new, Y_new,Z_new)

In 3 dimensions, there are 3 possible types of reflection-

Reflection relative to XY plane
Reflection relative to YZ plane
Reflection relative to XZ plane

Reflection Relative to XY Plane:

This reflection is achieved by using the following reflection equations-

X_new = X_old
Y_new = Y_old
Z_new = -Z_old

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

Reflection Relative to YZ Plane:

This reflection is achieved by using the following reflection equations-

X_new = -X_old
Y_new = Y_old
Z_new = Z_old

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

Reflection Relative to XZ Plane:

This reflection is achieved by using the following reflection equations-

X_new = X_old
Y_new = -Y_old
Z_new = Z_old

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

PRACTICE PROBLEMS BASED ON 3D REFLECTION IN COMPUTER GRAPHICS-

Problem-01:

Given a 3D triangle with coordinate points A(3, 4, 1), B(6, 4, 2), C(5, 6, 3). Apply the reflection on the XY plane and find out the new coordinates of the object.

Solution-

Given-

Old corner coordinates of the triangle = A (3, 4, 1), B(6, 4, 2), C(5, 6, 3)
Reflection has to be taken on the XY plane

For Coordinates A(3, 4, 1)

Let the new coordinates of corner A after reflection = (X_new, Y_new, Z_new).

Applying the reflection equations, we have-

X_new = X_old = 3
Y_new = Y_old = 4
Z_new = -Z_old = -1

Thus, New coordinates of corner A after reflection = (3, 4, -1).

For Coordinates B(6, 4, 2)

Let the new coordinates of corner B after reflection = (X_new, Y_new, Z_new).

Applying the reflection equations, we have-

X_new = X_old = 6
Y_new = Y_old = 4
Z_new = -Z_old = -2

Thus, New coordinates of corner B after reflection = (6, 4, -2).

For Coordinates C(5, 6, 3)

Let the new coordinates of corner C after reflection = (X_new, Y_new, Z_new).

Applying the reflection equations, we have-

X_new = X_old = 5
Y_new = Y_old = 6
Z_new = -Z_old = -3

Thus, New coordinates of corner C after reflection = (5, 6, -3).

Thus, New coordinates of the triangle after reflection = A (3, 4, -1), B(6, 4, -2), C(5, 6, -3).

Problem-02:

Given a 3D triangle with coordinate points A(3, 4, 1), B(6, 4, 2), C(5, 6, 3). Apply the reflection on the XZ plane and find out the new coordinates of the object.

Solution-

Given-

Old corner coordinates of the triangle = A (3, 4, 1), B(6, 4, 2), C(5, 6, 3)
Reflection has to be taken on the XZ plane

For Coordinates A(3, 4, 1)

Let the new coordinates of corner A after reflection = (X_new, Y_new, Z_new).

Applying the reflection equations, we have-

X_new = X_old = 3
Y_new = -Y_old = -4
Z_new = Z_old = 1

Thus, New coordinates of corner A after reflection = (3, -4, 1).

For Coordinates B(6, 4, 2)

Let the new coordinates of corner B after reflection = (X_new, Y_new, Z_new).

Applying the reflection equations, we have-

X_new = X_old = 6
Y_new = -Y_old = -4
Z_new = Z_old = 2

Thus, New coordinates of corner B after reflection = (6, -4, 2).

For Coordinates C(5, 6, 3)

Let the new coordinates of corner C after reflection = (X_new, Y_new, Z_new).

Applying the reflection equations, we have-

X_new = X_old = 5
Y_new = -Y_old = -6
Z_new = Z_old = 3

Thus, New coordinates of corner C after reflection = (5, -6, 3).

Thus, New coordinates of the triangle after reflection = A (3, -4, 1), B(6, -4, 2), C(5, -6, 3).

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

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Next Article- 3D Shearing in Computer Graphics

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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-

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,

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3D Rotation 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-

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

3D Rotation in Computer Graphics-

In Computer graphics,

3D Rotation is a process of rotating an object with respect to an angle in a three dimensional plane.

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

Let-

Initial coordinates of the object O = (X_old, Y_old, Z_old)
Initial angle of the object O with respect to origin = Φ
Rotation angle = θ
New coordinates of the object O after rotation = (X_new, Y_new, Z_new)

In 3 dimensions, there are 3 possible types of rotation-

X-axis Rotation
Y-axis Rotation
Z-axis Rotation

For X-Axis Rotation-

This rotation is achieved by using the following rotation equations-

X_new = X_old
Y_new = Y_old x cosθ – Z_old x sinθ
Z_new = Y_old x sinθ + Z_old x cosθ

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

For Y-Axis Rotation-

This rotation is achieved by using the following rotation equations-

X_new = Z_old x sinθ + X_old x cosθ
Y_new = Y_old
Z_new = Y_old x cosθ – X_old x sinθ

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

For Z-Axis Rotation-

This rotation is achieved by using the following rotation equations-

X_new = X_old x cosθ – Y_old x sinθ
Y_new = X_old x sinθ + Y_old x cosθ
Z_new = Z_old

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

PRACTICE PROBLEMS BASED ON 3D ROTATION IN COMPUTER GRAPHICS-

Problem-01:

Given a homogeneous point (1, 2, 3). Apply rotation 90 degree towards X, Y and Z axis and find out the new coordinate points.

Solution-

Given-

Old coordinates = (X_old, Y_old, Z_old) = (1, 2, 3)
Rotation angle = θ = 90º

For X-Axis Rotation-

Let the new coordinates after rotation = (X_new, Y_new, Z_new).

Applying the rotation equations, we have-

X_new = X_old = 1
Y_new = Y_old x cosθ – Z_old x sinθ = 2 x cos90° – 3 x sin90° = 2 x 0 – 3 x 1 = -3
Z_new = Y_old x sinθ + Z_old x cosθ = 2 x sin90° + 3 x cos90° = 2 x 1 + 3 x 0 = 2

Thus, New coordinates after rotation = (1, -3, 2).

For Y-Axis Rotation-

Let the new coordinates after rotation = (X_new, Y_new, Z_new).

Applying the rotation equations, we have-

X_new = Z_old x sinθ + X_old x cosθ = 3 x sin90° + 1 x cos90° = 3 x 1 + 1 x 0 = 3
Y_new = Y_old = 2
Z_new = Y_old x cosθ – X_old x sinθ = 2 x cos90° – 1 x sin90° = 2 x 0 – 1 x 1 = -1

Thus, New coordinates after rotation = (3, 2, -1).

For Z-Axis Rotation-

Let the new coordinates after rotation = (X_new, Y_new, Z_new).

Applying the rotation equations, we have-

X_new = X_old x cosθ – Y_old x sinθ = 1 x cos90° – 2 x sin90° = 1 x 0 – 2 x 1 = -2
Y_new = X_old x sinθ + Y_old x cosθ = 1 x sin90° + 2 x cos90° = 1 x 1 + 2 x 0 = 1
Z_new = Z_old = 3

Thus, New coordinates after rotation = (-2, 1, 3).

To gain better understanding about 3D Rotation 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-

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 = (X_old, Y_old, Z_old)
New coordinates of the object O after translation = (X_new, Y_new, Z_old)
Translation vector or Shift vector = (T_x, T_y, T_z)

Given a Translation vector (T_x, T_y, T_z)-

T_x defines the distance the X_old coordinate has to be moved.
T_y defines the distance the Y_old coordinate has to be moved.
T_z defines the distance the Z_old coordinate has to be moved.

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

X_new = X_old + T_x (This denotes translation towards X axis)
Y_new = Y_old + T_y (This denotes translation towards Y axis)
Z_new = Z_old + T_z (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 = (T_x, T_y, T_z) = (1, 1, 2)

For Coordinates A(0, 3, 1)

Let the new coordinates of A = (X_new, Y_new, Z_new).

Applying the translation equations, we have-

X_new = X_old + T_x = 0 + 1 = 1
Y_new = Y_old + T_y = 3 + 1 = 4
Z_new = Z_old + T_z = 1 + 2 = 3

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

For Coordinates B(3, 3, 2)

Let the new coordinates of B = (X_new, Y_new, Z_new).

Applying the translation equations, we have-

X_new = X_old + T_x = 3 + 1 = 4
Y_new = Y_old + T_y = 3 + 1 = 4
Z_new = Z_old + T_z = 2 + 2 = 4

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

For Coordinates C(3, 0, 0)

Let the new coordinates of C = (X_new, Y_new, Z_new).

Applying the translation equations, we have-

X_new = X_old + T_x = 3 + 1 = 4
Y_new = Y_old + T_y = 0 + 1 = 1
Z_new = Z_old + T_z = 0 + 2 = 2

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

For Coordinates D(0, 0, 0)

Let the new coordinates of D = (X_new, Y_new, Z_new).

Applying the translation equations, we have-

X_new = X_old + T_x = 0 + 1 = 1
Y_new = Y_old + T_y = 0 + 1 = 1
Z_new = Z_old + T_z = 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,

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Next Article- 3D Rotation in Computer Graphics

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