3D transformation

Unit 43D Viewing Pipeline

Part - 2Projections

Madhulika (18010), Assistant Professor, LPU.

Normalized view space

Modeling Transformation

Viewing Transformation

Lighting & Shading

3D-Clipping

Projection

Scan conversion, Hiding

Primitives

Image

Object space

World space

Camera space

Image space, Device coordinates

Hidden Surface Removal

3D Viewing Pipeline

Contents

1. Introduction

2. Perspective Projections

3. Parallel Projections

Viewing and Projection

• Camera Analogy:1. Set up your tripod and point the camera at the scene (viewing transformation).2. Arrange the scene to be photographed into the desired composition (modeling transformation).3. Choose a camera lens or adjust the zoom (projection transformation).4. Determine how large you want the final photograph to be - for example, you might want it enlarged (viewport transformation).




Projections

• Our 3-D scenes are all specified in 3-D world coordinates

• To display these we need to generate a 2-D image - project objects onto a picture plane

• So how do we figure out these projections?

Picture Plane

Objects in World Space


Projections

• Projection is just one part of the process of converting from 3-D world coordinates to a 2-D image

Clip against view volume

Project onto projection

plane

Transform to 2-D device coordinates

3-D world coordinate

output primitives

2-D device coordinates

Projection Transformation




Projections

• There are two broad classes of projection:

– Parallel: Typically used for architectural and engineering drawings

– Perspective: Realistic looking and used in computer graphics

Perspective Projection Parallel Projection

Classical viewingViewing requires three basic elements

• One or more objects

• A viewer with a projection surface

• Projectors that go from the object(s) to the projection surface

Classical views are based on the relationship among these

elements

• The viewer picks up the object and orients it how she would

like to see it

Each object is assumed to constructed from flat principal

faces

• Buildings, polyhedra, manufactured objects


Classical Projections



ProjectionsProjectionsProjections

PERSPECTIVEConverging Projectors

(View Point)

PERSPECTIVEConverging Projectors

(View Point)

PARALLEL(View Direction)

PARALLEL(View Direction)

OBLIQUEProjector not to

View plane


View plane

ORTHOGRAPHICProjector to

View plane


View plane

GENERALGENERAL

MULTI VIEWView plane || to principal plane


AXONOMETRICView plane not ||

To principal plane

AXONOMETRICView plane not ||

To principal plane

1-Principal vanishing point1-Principal

vanishing point


vanishing point


vanishing pointThree viewsThree views

Auxiliary ViewAuxiliary View

Sectional ViewSectional View

ISOMETRICEqual angle with

all three axis


all three axis

DIMETRICEqual angle with

any two axis


any two axis

TRIMETRICUnequal angle with

all three axis


all three axis

CAVALIERNo foreshortening of lines

To XY-Plane


To XY-Plane

CABINETforeshortening of lines To XY-Plane by 1/2


Contents

1. Introduction




Perspective Projections

• Perspective projections are much more realistic than parallel projections and are used by artists.



• Perspective projections are described by– Centre of projection: Eye of artists or lens of camera

– View Plane: Plane containing canvas or film strip or frame buffer

• A ray called projector is drawn from COP to object point, its intersection with view plane determines the projected image point on view plane.

X-axis

Projector

COP

View Plane

Y-axis

Z-axis

Object point

Projected point

Perspective Projection


Parallel Projections




• There are a number of different kinds of perspective views

• The most common are one-point and two point perspectives


Perspective Projections• Perspective drawings are characterised by

1. Perspective foreshortening

2. Vanishing points

3. View Confusion

4. Topological Distortion

– These are also known as Perspective Anomalies.– These anomalies enhance realism in terms of depth cues, but

distorts the actual size, shape and relationship between parts of object.



1. Perspective foreshortening: an illusion that objects and lengths appear smaller as their distance form COP increases.

– We can see three balls have different dimensions, since they placed at different distances they are projected to same length

COP(0,0,-d)Z-axis

Y-axis



• Increasing the field of view angle increases the height of the view plane and so increases foreshortening



• The amount of foreshortening that is present can greatly affect the appearance of our scenes



2. Vanishing points: An illusion that certain sets of parallel lines appear to meet at a point (called vanishing point).

– These are those lines that are not parallel to view plane i.e. lines that are not to view plane normal.

– Principal vanishing points are formed by apparent intersection of lines parallel to one of the three principal axes.

– The number of principal vanishing points is determined by the number of principal axis intersected by the view plane.

X-axis

Z-axis

Y-axis COP(0,0,-d)

L1

L2L’1

L’2

O


(from Donald Hearn and Pauline Baker)


Classes of Perspective Projection

Classes of Perspective Projection

• One-Point Perspective

• Two-Point Perspective

• Three-Point Perspective

• One-Point Perspective

• Two-Point Perspective

• Three-Point Perspective

26

One-Point PerspectiveOne-Point Perspective

27

Two-point perspective projection:Two-point perspective projection:

– This is often used in architectural, engineering and industrial design drawings.

–

28

Three-point perspective projection

Three-point perspective projection

• Three-point perspective projection is used less frequently as it adds little extra realism to that offered by two-point perspective projection

• Three-point perspective projection is used less frequently as it adds little extra realism to that offered by two-point perspective projection

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3. View Confusion: An object behind the COP is projected upside down and backward onto the view plane.

X-axis

Z-axis

Y-axis

COP(0,0,-d)

L1

L2

L’1

L’2

O


Perspective Projections4. Topological Distortion: All points lying on the plane parallel to view plane and passing through the

COP are projected to ∞ by the perspective transformation. – This may make a finite line segment

to appear as two infinite rays.

X-axis

Z-axis

Y-axis

COP(0,0,-d)

O

P1

P2

P’1P’2

P3

∞ ∞





• Although a perspective projection is set up by specifying the position and size of the view plane and the position of the projection reference point called COP

• However, this can be kind of awkward


Perspective Projections• The field of view angle can be a more intuitive way to specify

perspective projections

• This is analogous to choosing a lense for a camera

Field of view



• We need one more thing to specify a perspective projections using the filed of view angle

• The aspect ratio gives the ratio between the width sand height of the view plane

Contents

1. Introduction




Parallel Projections• Parallel projections are used by drafter and engineers to create working

drawings of an object as they preserve scale and shape

• These are described by– Viewing Direction: which describe the direction of projection

– View Plane: Plane containing canvas or film strip or frame buffer

• A ray called projector is drawn || to Viewing direction and passing through object point, its intersection with view plane determines the projected image point on view plane.

X-axisView Plane

Y-axis

Z-axis

ObjectViewing Direction

Object’


Parallel Projection

• Center of projection is at infinity– Direction of projection (DOP) same for all points

DOP

ViewPlane



Parallel ProjectionsParallel Projections


View plane


View plane


View plane


View plane

GENERALGENERAL



AXONOMETRICView plane not ||To principal plane

AXONOMETRICView plane not ||To principal plane

Three viewsThree views

Auxiliary ViewAuxiliary View

Sectional ViewSectional View


all three axis


all three axis


any two axis


any two axis


all three axis


all three axis


To XY-Plane


To XY-Plane




Orthographic Projections

Top Side

Front

• DOP perpendicular to view plane


Oblique Projections

• DOP not perpendicular to view plane

Cavalier

(DOP = 45o)

Cabinet

(DOP = 63.4o)

45 4.63

• Cavalier Projection- It is obtained when the angle between the oblique projectors and the plane of projection is 45 degree and the foreshortening factors for all three principal directions are equal.

• In Cavalier projection , the resulting figure is too thick.


• Cabinet Projection- It is used to correct the deficiency that is produced by Cavalier projection.

• An oblique projection for which the foreshortening factor for the edge perpendicular to the plane of projection is one-half is called Cabinet projection.

• For a cabinet projection, the angle between the projectors and the plane of projection is 63.43.




• Identify type parallel projections

Orthographic Projection

Oblique Projection

Isometric Projection



• Isometric projections have been used in computer games from the very early days of the industry up to today

Q*Bert Sim City Virtual Magic Kingdom


Engineering

3D transformation