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Computer graphics �subject code(22318)

BY A.I.MUSHRIF

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Syllabus

Sr.no	Name of topics
Unit 1	Basics of computer graphics
Unit 2	Raster scan Graphics
Unit 3	Overview of Transformations
Unit 4	Windowing and clipping
Unit 5	Introduction to curves

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Weightage of marks

Sr.no	Topics	Marks
1	Basics of computer Graphics	08
2	Raster Scan Graphics	18
3	Overview of Transformations	18
4	Windowing and clipping	14
5	Introductions to Curves	12
Total		70

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Computer Graphics:�Overview of�Transformations

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2D Transformations

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x

y

x

y

x

y

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2D Transformations

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x

y

x

y

x

y

OpenGl transformations:

glTranslatef (tx, ty, tz);

glRotatef (theta, vx, vy, vz)

glScalef (sx,sy,sz)

……

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2D Transformations

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x

y

x

y

x

y

Applications:

Animation
Image/object manipulation
Viewing transformation
etc.

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Applications of 2D Transformations

2D geometric transformations

Animation (demo ,demo)

Image warping

Image morphing

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2D Transformation

Given a 2D object, transformation is to change the object’s

Position (translation)
Size (scaling)
Orientation (rotation)
Shapes (shear)

Apply a sequence of matrix multiplications to the object vertices

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Point Representation

We can use a column vector (a 2x1 matrix) to represent a 2D point x

y

A general form of linear transformation can be written as:

x’ = ax + by + c

OR

y’ = dx + ey + f

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X’ a b c x

Y’ = d e f * y

1 0 0 1 1

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Translation

Re-position a point along a straight line
Given a point (x,y), and the translation distance (tx,ty)

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The new point: (x’, y’)

x’ = x + tx

y’ = y + ty

(x,y)

(x’,y’)

OR P’ = P + T where P’ = x’ p = x T = tx

y’ y ty

tx

ty

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3x3 2D Translation Matrix

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x’ = x + tx

y’ y ty

Use 3 x 1 vector

x’ 1 0 tx x

y’ = 0 1 ty * y

1 0 0 1 1

Note that now it becomes a matrix-vector multiplication

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Translation

How to translate an object with multiple vertices?

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Translate individual

vertices

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2D Rotation

Default rotation center: Origin (0,0)

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θ

> 0 : Rotate counter clockwise

< 0 : Rotate clockwise

θ

x

y

o

x

y

o

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2D Rotation

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(x,y)

(x’,y’)

θ

(x,y) -> Rotate about the origin by θ

(x’, y’)

How to compute (x’, y’) ?

φ

r

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2D Rotation

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(x,y)

(x’,y’)

θ

(x,y) -> Rotate about the origin by θ

(x’, y’)

How to compute (x’, y’) ?

φ

x = r cos (φ) y = r sin (φ)

r

x’ = r cos (φ + θ) y’ = r sin (φ + θ)

x

y

o

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2D Rotation

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(x,y)

(x’,y’)

θ

φ

r

x = r cos (φ) y = r sin (φ)

x’ = r cos (φ + θ) y = r sin (φ + θ)

x’ = r cos (φ + θ)

= r cos(φ) cos(θ) – r sin(φ) sin(θ)

= x cos(θ) – y sin(θ)

y’ = r sin (φ + θ)

= r sin(φ) cos(θ) + r cos(φ)sin(θ)

= y cos(θ) + x sin(θ)

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2D Rotation

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(x,y)

(x’,y’)

θ

φ

r

x’ = x cos(θ) – y sin(θ)

y’ = y cos(θ) + x sin(θ)

Matrix form?

x’ cos(θ) -sin(θ) x

y’ sin(θ) cos(θ) y

=

3 x 3?

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3x3 2D Rotation Matrix

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x’ cos(θ) -sin(θ) x

y’ sin(θ) cos(θ) y

=

(x,y)

(x’,y’)

θ

φ

r

x’ cos(θ) -sin(θ) 0 x

y’ sin(θ) cos(θ) 0 y

1 0 0 1 1

=

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How to rotate an object with multiple vertices?

Rotate individual

Vertices

θ

2D Rotation

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2D Scaling

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Scale: Alter the size of an object by a scaling factor

(Sx, Sy), i.e.

x’ = x . Sx

y’ = y . Sy

x’ Sx 0 x

y’ 0 Sy y

=

(1,1)

(2,2)

Sx = 2, Sy = 2

(2,2)

(4,4)

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2D Scaling

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(1,1)

(2,2)

Sx = 2, Sy = 2

(2,2)

(4,4)

Not only the object size is changed, it also moved!!
Usually this is an undesirable effect
We will discuss later (soon) how to fix it

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3x3 2D Scaling Matrix

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x’ Sx 0 x

y’ 0 Sy y

=

x’ Sx 0 0 x

y’ = 0 Sy 0 * y

1 0 0 1 1

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Put it all together

Translation:

Rotation:

Scaling:

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Or, 3x3 Matrix Representations

Translation:

Rotation:

Scaling:

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x’ 1 0 tx x

y’ = 0 1 ty * y

1 0 0 1 1

x’ cos(θ) -sin(θ) 0 x

y’ sin(θ) cos(θ) 0 * y

1 0 0 1 1

=

x’ Sx 0 0 x

y’ = 0 Sy 0 * y

1 0 0 1 1

Why use 3x3 matrices?

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Why Use 3x3 Matrices?

So that we can perform all transformations using matrix/vector multiplications

This allows us to pre-multiply all the matrices together

The point (x,y) needs to be represented as

(x,y,1) -> this is called Homogeneous

coordinates!

How to represent a vector (v_x,v_y)?

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Why Use 3x3 Matrices?

So that we can perform all transformations using matrix/vector multiplications

This allows us to pre-multiply all the matrices together

The point (x,y) needs to be represented as

(x,y,1) -> this is called Homogeneous

coordinates!

How to represent a vector (v_x,v_y)? (v_x,v_y,0)

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Shearing

Y coordinates are unaffected, but x coordinates are translated linearly with y
That is:

y’ = y
x’ = x + y * h

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x 1 h 0 x

y = 0 1 0 * y

1 0 0 1 1

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x 1 0 0 x

y = g 1 0 * y

1 0 0 1 1

Shearing in Y

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Reflection

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Reflection

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Reflection

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Reflection about X-axis

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Reflection about X-axis

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x 1 0 0 x

y = 0 -1 0 * y

1 0 0 1 1

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Reflection about Y-axis

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Reflection about Y-axis

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x -1 0 0 x

y = 0 1 0 * y

1 0 0 1 1

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What’s the Transformation Matrix?

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What’s the Transformation Matrix?

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x -1 0 0 x

y = 0 -1 0 * y

1 0 0 1 1

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The standard rotation matrix is used to rotate about the origin (0,0)

cos(θ) -sin(θ) 0

sin(θ) cos(θ) 0

0 0 1

What if I want to rotate about an arbitrary center?

Rotation Revisit

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Arbitrary Rotation Center

To rotate about an arbitrary point P (px,py) by θ:

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(px,py)

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Arbitrary Rotation Center

To rotate about an arbitrary point P (px,py) by θ:

Translate the object so that P will coincide with the origin: T(-px, -py)

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(px,py)

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Arbitrary Rotation Center

To rotate about an arbitrary point P (px,py) by θ:

Translate the object so that P will coincide with the origin: T(-px, -py)
Rotate the object: R(θ)

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(px,py)

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Arbitrary Rotation Center

To rotate about an arbitrary point P (px,py) by θ:

Translate the object so that P will coincide with the origin: T(-px, -py)
Rotate the object: R(θ)
Translate the object back: T(px,py)

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(px,py)

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Translate the object so that P will coincide with the origin: T(-px, -py)
Rotate the object: R(θ)
Translate the object back: T(px,py)

Put in matrix form: T(px,py) R(θ) T(-px, -py) * P

x’ 1 0 px cos(θ) -sin(θ) 0 1 0 -px x

y’ = 0 1 py sin(θ) cos(θ) 0 0 1 -py y

1 0 0 1 0 0 1 0 0 1 1

Arbitrary Rotation Center

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Arbitrary Rotation Center

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Example of arbitrary rotation

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Example of arbitrary rotation

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Example of arbitrary rotation

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Projections

PROJECTION METHODS

1. Parallel Projection

2.Perspective Projection

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Types of projections

1.Parallel projection

2.Perspective projection

Parallel projection

A parallel projection is a projection of an object in three-dimensional space onto a fixed plane, known as the projection plane or image plane, where the rays, known as lines of sight or projection lines, are parallel to each other.

Projections

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Parallel Projection

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Parallel Projections

Types of parallel projection

Orthographic projection
Oblique projection

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Parallel Projections

Types of parallel projection

Orthographic projection

The projection of a single view of an object (such as a view of the front) onto a projection plane in which the lines of projection are perpendicular to the projection plane.

when the projection is perpendicular to the view plane. In short,

– direction of projection = normal to the projection plane.

– the projection is perpendicular to the view plane

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Parallel Projections

Types of parallel projection

Orthographic projection
Isometric Projection: All projectors make equal angles generally angle is of 30°.
Diametric: In these two projectors have equal angles. With respect to two principle axis.
Trimetric: The direction of projection makes unequal angle with their principle axis.

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Orthographic Projections

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Parallel Projections

Oblique projection

when the projection is not perpendicular to the view plane. In short,

Direction of projection is not normal to the projection plane.
Not perpendicular.

.

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Parallel Projections

Oblique projection

Cavalier: All lines perpendicular to the projection plane are projected with no change in length.

Cabinet: All lines perpendicular to the projection plane are projected to one half of their length. These give a realistic appearance of object.

.

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Parallel Projections

Oblique projection

Cavalier: All lines perpendicular to the projection plane are projected with no change in length.

.

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Parallel Projections

Oblique projection

Cabinet: All lines perpendicular to the projection plane are projected to one half of their length. These give a realistic appearance of object.

.

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Parallel Projections

.

Fig orthographic projection

Fig Oblique projection

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Perspective projection

It is a type of drawing that graphically approximates on a planar (two-dimensional) surface (e.g. paper) the images of three-dimensional objects so as to approximate actual visual perception. It is sometimes also called perspective view or perspective drawing or simply perspective

Perspective projection

The object position are transformed to the view plane along lines that converge to a point called projection reference point or (center of projection)

Projections

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Perspective Projections

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Characteristics of perspective projection:

Center of Projection (CP) is a finite distance from object
Projectors are rays (i.e., non-parallel)
Vanishing points: The projection lines appear to converge on a vanishing point.
Objects appear smaller as distance from CP (eye of observer) increases
Difficult to determine exact size and shape of object
Most realistic, difficult to execute

Perspective Projections

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Perspective Projections

Types of perspective projection

One-Point Perspective:
Two-Point Perspective
Three-Point Perspective

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Perspective Projections

Types of perspective projection

One-Point Perspective:
The projection is one-point perspective when it contains only one vanishing point on the horizon line.
This type of perspective is typically used for images of roads, railway tracks, hallways, or buildings viewed so that the front is directly facing the viewer.
These parallel lines converge at the vanishing point.

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Perspective Projections

Fig : One point Perspective

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Perspective Projections

Two-Point Perspective:
Two point of perspective is seen when there are two vanishing point from observers point of view .
Two-point perspective drawings are often used in architectural drawings and interior designs; they can be used for drawings of both interiors and exteriors.

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Perspective Projections

Fig : Two point Perspective

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Perspective Projections

Three-Point Perspective:
Three point of perspective is seen when there are three vanishing point from observers point of view .
Three-point perspective drawings are often used in building drawings

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Perspective Projections

Fig : Three point Perspective