Unit 2

Viewing Transformation / Window to Viewport Transformation: [VIMPT]

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2D Transformation: [VIMPT]
There are 3 basic transformations: Translation, Scaling and Rotation.
Other transformations are: Reflection and Shearing.
2D - Translation: [VIMPT]

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2D -Scaling: [VIMPT]

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2D -Rotation: [VIMPT]

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Rotation of an object around arbitrary point: [VIMPT]

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2D -Reflection: [VIMPT]

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2D – Shearing: [VIMPT]

3D Transformations: [VIMPT]
3D - Translation: [VIMPT]
In translation vector – we have tx – translation factor for x-axis.
ty- translation factor for y-axis and tz- translation factor for z-axis.

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3D -Scaling: [VIMPT]
Sx- is scaling factor wrt x-axis, Sy- is scaling factor wrt y-axis,
Sz- is scaling factor wrt z-axis.

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3D -Rotation: [VIMPT]

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Rotation of an object around arbitrary axis. [VIMPT]

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3D -Reflection:

Sums: [VIMPT] Do from Sums File.

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Projections:

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Different types of projections: [VIMPT]
After converting the description of object from world coordinates to viewing coordinates we
can project the 3D object onto the 2D view plane.
• There are 2 basic ways of projection object onto the view plane:
1. Parallel projection :
2. Perspective projection

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Differentiate between Parallel and Perspective Projections: [VIMPT]

Parallel Projections Perspective Projections

Objects are represented as if being Objects are represented in a three-dimensional
viewed through a telescope. manner.
The shape and size of objects remain Objects appear smaller the further they are from the
consistent on the plane. viewer, and larger when closer.
The object's distance from the center of The object's distance from the center of projection is
projection is infinite. finite.
Provides an accurate view of the object. Does not provide an accurate view of the object, with
shapes and sizes differing from the original object.
Projection lines are parallel to each Projection lines are not parallel to each other.
other.
There are two types of parallel There are three types of perspective projections: One
projections: Oblique and Orthographic. Point, Two Point, and Three Point.
Parallel Projection does not create a Perspective Projection creates a realistic view of
realistic view of objects. objects.

1. Different types of Parallel projections: [VIMPT]

• In parallel projection, coordinate positions are transformed to the view plane
along parallel lines.
• Good for exact measurement, Parallel lines remain parallel
• But Angles are not generally preserved, and the final look is not realistic.
• Parallel projections are basically categorized into 2 types, depending on the
relation between the direction of projection and the normal to the view plane.
• When the direction of the projection is normal to the view plane ,we have an
• orthographic parallel projection, otherwise we have an oblique parallel
projection

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1.1 Orthographic projection
- The most common type of Orthographic projections are the front
projection, top projection, side projection.
- direction of projection = normal to the projection plane
- In all these ,the projection plane is perpendicular to the principle
axis.
- Orthographic projection can display more than one face of an object.
- Types of orthographic projections are Multiview and Axonometric.
1.1.1 Multiview:
- Front, side and rear orthographic projection of an object are called
elevations and the top orthographic projection is called plan view.
- all have projection plane perpendicular to a principle axes.
- Here length and angles are accurately depicted and measured from
the drawing, so engineering and architectural drawings commonly
employee this.
- However, As only one face of an object is shown, it can be hard to
create a mental image of the object, even when several views are
available.

1.1.2. Axonometric orthographic projections

- Orthographic projections that show more than one side of an object
are called axonometric orthographic projections.
- The most common axonometric projection is an isometric projection

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 where the projection plane intersects each coordinate axis in the
model coordinate system at an equal distance.
- In this projection parallelism of lines are preserved but angles are not
preserved.
- Types of Axonometric Projection
o Isometric – has equal foreshortening along each of the three
axis directions
o Diametric – has equal foreshortening along two axis directions
and a different amount on the third axis
o Trimetric – has different foreshortening along all three axis
directions
1.2 Oblique projection: [VIMPT]
- Oblique projection is obtained by projecting points along parallel
lines that are not perpendicular to the projection plane.
- when the projection is not perpendicular to the view plane. In short,
- direction of projection ≠ normal to the projection plane.
- The view plane normal and the direction of projection are not the
same.
- The oblique projections are further classified as the cavalier and
cabinet projections.
1.2.1 Cavalier Projection:
- All lines perpendicular to the projection plane are projected with no
change in length.
- The direction of the projection makes a 45 degree angle with the
projection plane.
- Because there is no foreshortening, this causes an exaggeration of the
z axes
- Angle between projectors and projection plane is 45°. Perpendicular
faces projected at full scale
1.2.2 Cabinet projections
- Lines which are perpendicular to the projection plane (viewing
surface) are projected at 1/2 the length.
- This results in foreshortening of the z axis, and provides a more
"realistic" view.

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 - The direction of the projection makes a 63.4 degree angle with the
projection plane. This results in foreshortening of the z axis, and
provides a more "realistic" view.
- cabinet projections appear more realistic than cavalier projections
`because of the reduction in the length of perpendicular.

2. Different types of Perspective Projections: [VIMPT]

• In perspective projection, object position are transformed to the view plane
along lines that converge to a point called center of projection.
• Look is realistic as size varies inversely with the distance. [i.e. distance increases
size decreases.]
• But distance and angles are not generally preserved
• Parallel lines generally do not remain parallel.
• Center of Projection (CP) is a finite distance from object Projectors are rays (le.,
non-parallel)
• Objects appear smaller as distance from CP (eye of observer) increases •
Difficult to determine exact size and shape of object
• Most realistic, but difficult to execute
• When a 3D object is projected onto view plane using perspective transformation
equations, any set of parallel lines in the object that are not parallel to the
projection plane, converge at a vanishing point.
• There are an infinite number of vanishing points, depending on now many set of
parallel lines there are in the scene.
• If a set of lines are parallel to one of the three principle axes, the vanishing point
is called an principle vanishing point.
• There are at most 3 such points, corresponding to the number of axes cut by the
projection plane.
• Vanishing points:
• Certain set of parallel lines appear to meet at a different point
• The Vanishing point for this direction

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 • Principle vanishing points are formed by the apparent intersection of
lines parallel to one of the three principal x, y, z axes.
• The number of principal Vanishing points is determined by the number
of principal axes intersected by the view plane.
• Sets of parallel lines on the same plane lead to collinear vanishing points -
The line is called the horizon for that plane
• Classes of Perspective Projection
• One Point
• Two point
• Three point

One Point Perspective Projection – One point perspective projection occurs when any of
principal axes intersects with projection plane or we can say when projection plane is perpendicular
to principal axis. In the above figure, z axis intersects projection plane whereas x and y axis remain
parallel to projection plane.
Two Point Perspective Projection – Two point perspective projection occurs when projection
plane intersects two of principal axis. In the above figure, projection plane intersects x and y axis
whereas z axis remains parallel to projection plane.
Three Point Perspective Projection – Three point perspective projection occurs when all three
axis intersects with projection plane. There is no any principal axis which is parallel to projection
plane.
Application of Perspective Projection: The perspective projection technique is used by artists in
preparing drawings of three-dimensional objects and scenes.

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