Category: Computer Graphics

Bezier Curve in Computer Graphics | Examples

Computer Graphics

Bezier Curve-

 

Bezier Curve may be defined as-

  • Bezier Curve is parametric curve defined by a set of control points.
  • Two points are ends of the curve.
  • Other points determine the shape of the curve.

 

The concept of bezier curves was given by Pierre Bezier.

 

Bezier Curve Example-

 

The following curve is an example of a bezier curve-

 

 

Here,

  • This bezier curve is defined by a set of control points b0, b1, b2 and b3.
  • Points b0 and b3 are ends of the curve.
  • Points b1 and b2 determine the shape of the curve.

 

Bezier Curve Properties-

 

Few important properties of a bezier curve are-

 

Property-01:

 

Bezier curve is always contained within a polygon called as convex hull of its control points.

 

 

Property-02:

 

  • Bezier curve generally follows the shape of its defining polygon.
  • The first and last points of the curve are coincident with the first and last points of the defining polygon.

 

Property-03:

 

The degree of the polynomial defining the curve segment is one less than the total number of control points.

 

Degree = Number of Control Points – 1

 

Property-04:

 

The order of the polynomial defining the curve segment is equal to the total number of control points.

 

Order = Number of Control Points

 

Property-05:

 

  • Bezier curve exhibits the variation diminishing property.
  • It means the curve do not oscillate about any straight line more often than the defining polygon.

 

Bezier Curve Equation-

 

A bezier curve is parametrically represented by-

 

 

Here,

  • t is any parameter where 0 <= t <= 1
  • P(t) = Any point lying on the bezier curve
  • Bi = ith control point of the bezier curve
  • n = degree of the curve
  • Jn,i(t) = Blending function = C(n,i)ti(1-t)n-i where C(n,i) = n! / i!(n-i)!

 

Cubic Bezier Curve-

 

  • Cubic bezier curve is a bezier curve with degree 3.
  • The total number of control points in a cubic bezier curve is 4.

 

Example-

 

The following curve is an example of a cubic bezier curve-

 

 

Here,

  • This curve is defined by 4 control points b0, b1, b2 and b3.
  • The degree of this curve is 3.
  • So, it is a cubic bezier curve.

 

Cubic Bezier Curve Equation-

 

The parametric equation of a bezier curve is-

 

 

Substituting n = 3 for a cubic bezier curve, we get-

 

 

Expanding the above equation, we get-

P (t) = B0J3,0(t) + B1J3,1(t) + B2J3,2(t) + B3J3,3(t)     ………..(1)

 

Now,

 

 

Using (2), (3), (4) and (5) in (1), we get-

 

P(t) = B0(1-t)3 + B13t(1-t)2 + B23t2(1-t) + B3t3

 

This is the required parametric equation for a cubic bezier curve.

 

Applications of Bezier Curves-

 

Bezier curves have their applications in the following fields-

 

1. Computer Graphics-

 

  • Bezier curves are widely used in computer graphics to model smooth curves.
  • The curve is completely contained in the convex hull of its control points.
  • So, the points can be graphically displayed & used to manipulate the curve intuitively.

 

2. Animation-

 

  • Bezier curves are used to outline movement in animation applications such as Adobe Flash and synfig.
  • Users outline the wanted path in bezier curves.
  • The application creates the needed frames for the object to move along the path.
  • For 3D animation, bezier curves are often used to define 3D paths as well as 2D curves.

 

3. Fonts-

 

  • True type fonts use composite bezier curves composed of quadratic bezier curves.
  • Modern imaging systems like postscript, asymptote etc use composite bezier curves composed of cubic bezier curves for drawing curved shapes.

 

PRACTICE PROBLEMS BASED ON BEZIER CURVE IN COMPUTER GRAPHICS-

 

Problem-01:

 

Given a bezier curve with 4 control points-

B0[1 0] , B1[3 3] , B2[6 3] , B3[8 1]

Determine any 5 points lying on the curve. Also, draw a rough sketch of the curve.

 

Solution-

 

We have-

  • The given curve is defined by 4 control points.
  • So, the given curve is a cubic bezier curve.

 

The parametric equation for a cubic bezier curve is-

 

P(t) = B0(1-t)3 + B13t(1-t)2 + B23t2(1-t) + B3t3

 

Substituting the control points B0, B1, B2 and B3, we get-

P(t) = [1 0](1-t)3 + [3 3]3t(1-t)2 + [6 3]3t2(1-t) + [8 1]t3      ……..(1)

 

Now,

To get 5 points lying on the curve, assume any 5 values of t lying in the range 0 <= t <= 1.

Let 5 values of t are 0, 0.2, 0.5, 0.7, 1

 

For t = 0:

 

Substituting t=0 in (1), we get-

P(0) = [1 0](1-0)3 + [3 3]3(0)(1-t)2 + [6 3]3(0)2(1-0) + [8 1](0)3

P(0) = [1 0] + 0 + 0 + 0

P(0) = [1 0]

 

For t = 0.2:

 

Substituting t=0.2 in (1), we get-

P(0.2) = [1 0](1-0.2)3 + [3 3]3(0.2)(1-0.2)2 + [6 3]3(0.2)2(1-0.2) + [8 1](0.2)3

P(0.2) = [1 0](0.8)3 + [3 3]3(0.2)(0.8)2 + [6 3]3(0.2)2(0.8) + [8 1](0.2)3

P(0.2) = [1 0] x 0.512 + [3 3] x 3 x 0.2 x 0.64 + [6 3] x 3 x 0.04 x 0.8 + [8 1] x 0.008

P(0.2) = [1 0] x 0.512 + [3 3] x 0.384 + [6 3] x 0.096 + [8 1] x 0.008

P(0.2) = [0.512 0] + [1.152 1.152] + [0.576 0.288] + [0.064 0.008]

P(0.2) = [2.304 1.448]

 

For t = 0.5:

 

Substituting t=0.5 in (1), we get-

P(0.5) = [1 0](1-0.5)3 + [3 3]3(0.5)(1-0.5)2 + [6 3]3(0.5)2(1-0.5) + [8 1](0.5)3

P(0.5) = [1 0](0.5)3 + [3 3]3(0.5)(0.5)2 + [6 3]3(0.5)2(0.5) + [8 1](0.5)3

P(0.5) = [1 0] x 0.125 + [3 3] x 3 x 0.5 x 0.25 + [6 3] x 3 x 0.25 x 0.5 + [8 1] x 0.125

P(0.5) = [1 0] x 0.125 + [3 3] x 0.375 + [6 3] x 0.375 + [8 1] x 0.125

P(0.5) = [0.125 0] + [1.125 1.125] + [2.25 1.125] + [1 0.125]

P(0.5) = [4.5 2.375]

 

For t = 0.7:

 

Substituting t=0.7 in (1), we get-

P(t) = [1 0](1-t)3 + [3 3]3t(1-t)2 + [6 3]3t2(1-t) + [8 1]t3

P(0.7) = [1 0](1-0.7)3 + [3 3]3(0.7)(1-0.7)2 + [6 3]3(0.7)2(1-0.7) + [8 1](0.7)3

P(0.7) = [1 0](0.3)3 + [3 3]3(0.7)(0.3)2 + [6 3]3(0.7)2(0.3) + [8 1](0.7)3

P(0.7) = [1 0] x 0.027 + [3 3] x 3 x 0.7 x 0.09 + [6 3] x 3 x 0.49 x 0.3 + [8 1] x 0.343

P(0.7) = [1 0] x 0.027 + [3 3] x 0.189 + [6 3] x 0.441 + [8 1] x 0.343

P(0.7) = [0.027 0] + [0.567 0.567] + [2.646 1.323] + [2.744 0.343]

P(0.7) = [5.984 2.233]

 

For t = 1:

 

Substituting t=1 in (1), we get-

P(1) = [1 0](1-1)3 + [3 3]3(1)(1-1)2 + [6 3]3(1)2(1-1) + [8 1](1)3

P(1) = [1 0] x 0 + [3 3] x 3 x 1 x 0 + [6 3] x 3 x 1 x 0 + [8 1] x 1

P(1) = 0 + 0 + 0 + [8 1]

P(1) = [8 1]

 

Following is the required rough sketch of the curve-

 

 

To gain better understanding about Bezier Curves in Computer Graphics,

Watch this Video Lecture

 

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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-

 

 

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

 

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-

 

 

  1. Shearing in X direction
  2. Shearing in Y direction
  3. Shearing in Z direction

 

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

 

Let-

  • Initial coordinates of the object O = (Xold, Yold, Zold)
  • Shearing parameter towards X direction = Shx
  • Shearing parameter towards Y direction = Shy
  • Shearing parameter towards Z direction = Shz
  • New coordinates of the object O after shearing = (Xnew, Ynew, Znew)

 

Shearing in X Axis-

 

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

  • Xnew = Xold
  • Ynew = Yold + Shy x Xold
  • Znew = Zold + Shz x Xold

 

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-

  • Xnew = Xold + Shx x Yold
  • Ynew = Yold
  • Znew = Zold + Shz x Yold

 

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-

  • Xnew = Xold + Shx x Zold
  • Ynew = Yold + Shy x Zold
  • Znew = Zold

 

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 (Shx) = 2
  • Shearing parameter towards Y direction (Shy) = 2
  • Shearing parameter towards Y direction (Shz) = 3

 

Shearing in X Axis-

 

For Coordinates A(0, 0, 0)

 

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

 

Applying the shearing equations, we have-

  • Xnew = Xold = 0
  • Ynew = Yold + Shy x Xold = 0 + 2 x 0 = 0
  • Znew = Zold + Shz x Xold = 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 = (Xnew, Ynew, Znew).

 

Applying the shearing equations, we have-

  • Xnew = Xold = 1
  • Ynew = Yold + Shy x Xold = 1 + 2 x 1 = 3
  • Znew = Zold + Shz x Xold = 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 = (Xnew, Ynew, Znew).

 

Applying the shearing equations, we have-

  • Xnew = Xold = 1
  • Ynew = Yold + Shy x Xold = 1 + 2 x 1 = 3
  • Znew = Zold + Shz x Xold = 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 = (Xnew, Ynew, Znew).

 

Applying the shearing equations, we have-

  • Xnew = Xold + Shx x Yold = 0 + 2 x 0 = 0
  • Ynew = Yold = 0
  • Znew = Zold + Shz x Yold = 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 = (Xnew, Ynew, Znew).

 

Applying the shearing equations, we have-

  • Xnew = Xold + Shx x Yold = 1 + 2 x 1 = 3
  • Ynew = Yold = 1
  • Znew = Zold + Shz x Yold = 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 = (Xnew, Ynew, Znew).

 

Applying the shearing equations, we have-

  • Xnew = Xold + Shx x Yold = 1 + 2 x 1 = 3
  • Ynew = Yold = 1
  • Znew = Zold + Shz x Yold = 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 = (Xnew, Ynew, Znew).

 

Applying the shearing equations, we have-

  • Xnew = Xold + Shx x Zold = 0 + 2 x 0 = 0
  • Ynew = Yold + Shy x Zold = 0 + 2 x 0 = 0
  • Znew = Zold = 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 = (Xnew, Ynew, Znew).

 

Applying the shearing equations, we have-

  • Xnew = Xold + Shx x Zold = 1 + 2 x 2 = 5
  • Ynew = Yold + Shy x Zold = 1 + 2 x 2 = 5
  • Znew = Zold = 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 = (Xnew, Ynew, Znew).

 

Applying the shearing equations, we have-

  • Xnew = Xold + Shx x Zold = 1 + 2 x 3 = 7
  • Ynew = Yold + Shy x Zold = 1 + 2 x 3 = 7
  • Znew = Zold = 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,

Watch this Video Lecture

 

Next Article- Bezier Curves

 

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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-

 

 

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

 

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 = (Xold, Yold, Zold)
  • New coordinates of the reflected object O after reflection = (Xnew, Ynew,Znew)

 

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-

  • Xnew = Xold
  • Ynew = Yold
  • Znew = -Zold

 

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-

  • Xnew = -Xold
  • Ynew = Yold
  • Znew = Zold

 

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-

  • Xnew = Xold
  • Ynew = -Yold
  • Znew = Zold

 

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 = (Xnew, Ynew, Znew).

 

Applying the reflection equations, we have-

  • Xnew = Xold = 3
  • Ynew = Yold = 4
  • Znew = -Zold = -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 = (Xnew, Ynew, Znew).

 

Applying the reflection equations, we have-

  • Xnew = Xold = 6
  • Ynew = Yold = 4
  • Znew = -Zold = -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 = (Xnew, Ynew, Znew).

 

Applying the reflection equations, we have-

  • Xnew = Xold = 5
  • Ynew = Yold = 6
  • Znew = -Zold = -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 = (Xnew, Ynew, Znew).

 

Applying the reflection equations, we have-

  • Xnew = Xold = 3
  • Ynew = -Yold = -4
  • Znew = Zold = 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 = (Xnew, Ynew, Znew).

 

Applying the reflection equations, we have-

  • Xnew = Xold = 6
  • Ynew = -Yold = -4
  • Znew = Zold = 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 = (Xnew, Ynew, Znew).

 

Applying the reflection equations, we have-

  • Xnew = Xold = 5
  • Ynew = -Yold = -6
  • Znew = Zold = 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,

Watch this Video Lecture

 

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-

 

 

  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

 

Get more notes and other study material of Computer Graphics.

Watch video lectures by visiting our YouTube channel LearnVidFun.

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-

 

 

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

 

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 = (Xold, Yold, Zold)
  • Initial angle of the object O with respect to origin = Φ
  • Rotation angle = θ
  • New coordinates of the object O after rotation = (Xnew, Ynew, Znew)

 

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-

  • Xnew = Xold
  • Ynew = Yold x cosθ – Zold x sinθ
  • Znew = Yold x sinθ + Zold 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-

  • Xnew = Zold x sinθ + Xold x cosθ
  • Ynew = Yold
  • Znew = Yold x cosθ – Xold 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-

  • Xnew = Xold x cosθ – Yold x sinθ
  • Ynew = Xold x sinθ + Yold x cosθ
  • Znew = Zold

 

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 = (Xold, Yold, Zold) = (1, 2, 3)
  • Rotation angle = θ = 90º

 

For X-Axis Rotation-

 

Let the new coordinates after rotation = (Xnew, Ynew, Znew).

 

Applying the rotation equations, we have-

  • Xnew = Xold = 1
  • Ynew = Yold x cosθ – Zold x sinθ = 2 x cos90° – 3 x sin90° = 2 x 0 – 3 x 1 = -3
  • Znew = Yold x sinθ + Zold 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 = (Xnew, Ynew, Znew).

 

Applying the rotation equations, we have-

  • Xnew = Zold x sinθ + Xold x cosθ = 3 x sin90° + 1 x cos90° = 3 x 1 + 1 x 0 = 3
  • Ynew = Yold = 2
  • Znew = Yold x cosθ – Xold 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 = (Xnew, Ynew, Znew).

 

Applying the rotation equations, we have-

  • Xnew = Xold x cosθ – Yold x sinθ = 1 x cos90° – 2 x sin90° = 1 x 0 – 2 x 1 = -2
  • Ynew = Xold x sinθ + Yold x cosθ = 1 x sin90° + 2 x cos90° = 1 x 1 + 2 x 0 = 1
  • Znew = Zold = 3

 

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

 

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

Watch this Video Lecture

 

Next Article- 3D Scaling in Computer Graphics

 

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