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Graphics & Visualization

Chapter 4

Projections and Viewing Transformations

Graphics & Visualization: Principles & Algorithms Chapter 4

Graphics & Visualization: Principles & Algorithms Chapter 4 2

• In computer graphics there are:

 3D models  2D output devices (displays and printers)

• Projective mapping (projection):

 Must take place at some point in the graphics pipeline  Usually after the culling stages and before the rendering stage

• Viewing transformation:

 Defines the transition from the world coordinate

system (WCS) to canonical screen system (CSS) via the eye coordinate system (ECS)

 Specifies the clipping bounds (for frustum culling)

in ECS

Introduction

Graphics & Visualization: Principles & Algorithms Chapter 4 3

• All objects are initially defined in their own local coordinate system
• These objects are unified in WCS
• The WCS is essentially used to define the model of a 3D synthetic

world

• The transition from WCS to ECS:

 Simplifies a number of operations including culling (e.g. the specification of

the clipping bounds by the user) and projection

• The transition from ECS to CSS:

 Ensures that all objects that survived culling will be defined in a canonical

space (usually [-1,1])

• Objects defined in a canonical space:

 Can easily be scaled to the actual coordinates of any display device or monitor  Maintain high floating-point accuracy

Coordinate Systems

Graphics & Visualization: Principles & Algorithms Chapter 4 4

• Projection: techniques for the creation of the image of an object
• nto another simpler object (e.g. line, plane, surface)
• Projection lines are defined by:

 Center of projection  Points on the image being projected

• The intersection of a projector with the simpler object (e.g. the plane
• f projection) forms the image of a point of the original object.
• Projections can be defined in spaces of arbitrary dimension.
• In Computer Graphics & Visualization:

 Projections are from 3D space onto 2D space  The 2D space is referred to as the plane of projection & models the output

device

Projections

Graphics & Visualization: Principles & Algorithms Chapter 4 5

• Two projections are of interest:

 Perspective: the distance of the center of projection from the plane of

projection is finite

 Parallel: the distance of the center of projection from the plane of

projection is infinite

• Projective mappings are not affine transformations, so they cannot

be described by affine transformation matrices. Differences between affine transformations and projective mappings:

Projections(2)

Graphics & Visualization: Principles & Algorithms Chapter 4 6

• Parallel lines: are not projected onto parallel lines

their projections seem to meet at a vanishing point

• Parallel lines are projected onto parallel lines only when their plane

is parallel to the plane of projection

• A straight line will map to a straight line

Example:

• Ratios of distances are not preserved:
• Cross ratios are preserved:
• To fully describe the projective image of a line we need the image of

3 points on the line (affine transformations need only 2)

Projections(3)

     ab a b bd b d          ac a c cd c d ab a b bd b d

Graphics & Visualization: Principles & Algorithms Chapter 4 7

• Models the viewing system of our eyes
• Can be abstracted by a pinhole camera:

 Center of projection: the pinhole  Plane of projection: the image plane (where the image is formed)  Creates an inverted image  Derive an upright image by placing the image ‘in front’ of the pinhole

Perspective Projection

Graphics & Visualization: Principles & Algorithms Chapter 4 8

• Suppose that:

 Center of projection is the origin  Plane of projection is perpendicular to the

negative z-axis at distance d from the center

• A point P = [x, y, z]T is projected onto P΄ = [x΄, y΄, d]T
• P1 and P1΄ are the projections of P and P΄ onto the yz-plane
• are similar so:
• Similarly :
• The above are not linear equations (division by z)

Perspective Projection (2)

and  

1 2 1 2

OP P OP P · y y d y y d z z          

1 2 1 2 2 2

P P P P OP OP

· d x x z  

Graphics & Visualization: Principles & Algorithms Chapter 4 9

• Trick to express the perspective-projection equations in matrix form:

Use matrix which alters the homogeneous coordinate and maps the coordinates of a point [x, y, z]T as follows:

• Divide by the homogeneous coordinate:

Perspective Projection (3)

1 d d d             

PER

P

· · · · 1 x x d y y d z z d z                         

PER

P

· · · · / · 1 x d x d z y d y d z z z d d z                               

Graphics & Visualization: Principles & Algorithms Chapter 4 10

• Perspective shortening:

 The size of the projection of an object is inversely proportional to its distance

from the center of projection

• Known in ancient times, then forgotten
• Leonardo da Vinci studied the laws of perspective
• Older paintings had no perspective (symbolic criteria prevailed)

Perspective Projection (4)

Graphics & Visualization: Principles & Algorithms Chapter 4 11

• Center of projection is at infinite distance from the projection plane
• Projection lines are parallel to each other
• To describe parallel projection one must specify:

 The direction of projection (a vector)  The plane of projection

• Two types of parallel projections:

 Orthographic: the direction of projection is normal to the plane

• f projection

 Oblique: the direction of projection is not necessarily normal to

the plane of projection

Parallel Projection

Graphics & Visualization: Principles & Algorithms Chapter 4 12

• Usually employs one of the main planes as the plane of projection
• Suppose the xy-plane is used
• A point P = [x, y, z]T will be projected onto P΄ = [x΄,y΄,z΄]T =[x,y,0] T

as follows: where

Orthographic Projection

·  

ORTHO

P P P

1 1 1             

ORTHO

P

Graphics & Visualization: Principles & Algorithms Chapter 4 13

• Let the direction of projection be:

and the plane of projection be the xy-plane

• The projection P΄ = [x΄,y΄,z΄]T of a point P = [x, y, z]T will be

(1) for some scalar λ

• The z-coordinate of P΄ is 0, so (1) becomes:
• The other 2 coordinates of P΄ are:

Oblique Projection

[ , , ]T

x y z

DOP DOP DOP  DOP

·     P P DOP

·

• r

z z

z z DOP DOP      

· · and ·

y x x z z

DOP DOP x x DOP x z y y z DOP DOP         

Graphics & Visualization: Principles & Algorithms Chapter 4 14

• Matrix form:
• Oblique Projection (2)

1 1 ( ) 1

x z y z

DOP DOP DOP DOP                     

OBLIQUE

P DOP

( )·  

OBLIQUE

P P DOP P

Graphics & Visualization: Principles & Algorithms Chapter 3 15

EXAMPLE 1: Perspective Projection of a Cube Determine the perspective projections of a cube

• f side 1 when

(a)the plane of projection is z=-1 and (b)the plane of projection is z=-10. The cube is placed on the plane of projection. SOLUTION (a)

• Represent the vertices of the cube as a 4×8 matrix:

Projection Example 1

1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 1 1 1 1 1 1 1 1                      C

Graphics & Visualization: Principles & Algorithms Chapter 3 16

• Multiply the perspective projection matrix (d=-1) by C:
• Normalize by the homogeneous coordinates:

Projection Example 1 (2)

1 1 1 1 1 1 1 1 1 1 · · 1 1 1 1 1 2 2 2 2 1 1 1 1 1 2 2 2 2                                             

PER

P C C

1 1 1 1 2 2 1 1 1 1 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1                          

Graphics & Visualization: Principles & Algorithms Chapter 3 17

(b) The original cube is:

• Multiply the perspective projection matrix (d=-10) by C’:
• Normalize by the homogeneous coordinates:

Projection Example 1 (3)

1 1 1 1 1 1 1 1 10 10 10 10 11 11 11 11 1 1 1 1 1 1 1 1                       C

10 10 10 10 10 10 10 10 10 10 · 10 100 100 100 100 110 110 110 110 1 10 10 10 10 11 11 11 11                                              C

10 10 1 1 11 11 10 10 1 1 11 11 10 10 10 10 10 10 10 10 1 1 1 1 1 1 1 1                          

Graphics & Visualization: Principles & Algorithms Chapter 3 18

EXAMPLE 2: Perspective Projection onto an Arbitrary Plane Compute the perspective projection of a point P=[x, y, z]T onto the arbitrary plane Π which is specified by the point R0=[x0,y0,z0]T and a normal vector . The center of projection is O. SOLUTION

• P΄=[x ΄, y ΄, z ΄]T is the projection of P=[x, y, z]T
• are collinear so

for some scalar a

• In the projection equations:

(1) scalar a must be determined.

Projection Example 2

[ , , ]T

x y z

n n n  N

and  OP OP · a   OP OP

' , ' , ' x ax y ay z az   

Graphics & Visualization: Principles & Algorithms Chapter 3 19

• Vector is on the plane of projection, so :
• Substitute the values of x, y, z from (1), set c = nx x0 + ny y0 + nz z0 and

solve for a:

• The projection equations include a division by a combination of x,y,z
• Put in matrix form by altering the homogeneous coordinate:

Projection Example 2 (2)

 R P

· ( ) ( ) ( )

x y z x y z x y z

N

• r n x

x n y y n z z

• r n x

n y n z n x n y n z                    R P

x y z

c a n x n y n z   

, x y z

c c c n n n



              

PER

P

Graphics & Visualization: Principles & Algorithms Chapter 3 20

EXAMPLE 3: Oblique projection with Azimuth & Elevation Angles Sometimes (particularly in architectural design), oblique projections are specified in terms of the azimuth and elevation angles φ and θ that define the relation of the direction to the plane of projection. Determine the projection matrix in this case. SOLUTION

• Let xy be the plane of projection
• The direction of the projection vector is:
• So:

Projection Example 3

[cos cos ,cos sin ,sin ]T       DOP

cos 1 tan sin 1 ( , ) tan 1                           

OBLIQUE

P

Graphics & Visualization: Principles & Algorithms Chapter 3 21

EXAMPLE 4: Oblique projection onto an Arbitrary Plane Determine the oblique projection mapping onto an arbitrary plane Π that is specified by a point R0=[x0,y0,z0]T and a normal vector . The direction of projection is given by the vector SOLUTION Transform plane Π to coincide with the xy-plane, use the oblique projection matrix, undo the first transform:

• Step 1: Translate R0 to the origin,
• Step 2: Align with the positive z-axis (by using from[Ex 3.12])
• Step 3: Use the oblique projection matrix with the direction of

projection transformed according to previous steps:

Projection Example 4

[ , , ]T

x y z

DOP DOP DOP  DOP

[ , , ]T

x y z

n n n  N

( )  T R

N

( ) A N

( )· ( )·

 

 DOP A N T R DOP

Graphics & Visualization: Principles & Algorithms Chapter 3 22

• Step 4: Undo the alignment
• Step 5: undo the translation
• Finally:

Projection Example 4 (2)

( )1 A N

( ) T R

, (

) ( )· ( ) · ( )· ( )· ( )

  

 

1 OBLIQUE OBLIQUE

P DOP T R A N P DOP A N T R

Graphics & Visualization: Principles & Algorithms Chapter 4 23

• Defines:

 the process of coordinate conversion:  The clipping boundaries (for frustum culling) in ECS

(Right-handed coordinate systems)

• Viewing Transformation: a. WCS-to-ECS conversion
• b. ECS-to-CSS conversion

orthographic projections  perspective projections

• z-coordinate is maintained by the ECS-to-CSS conversion

 Stages following VT (e.g. hidden surface elimination) require 3D

information

Viewing Transformation (VT)

Graphics & Visualization: Principles & Algorithms Chapter 4 24

• ECS can be defined within the WCS by:

 The ECS origin E  The direction of view  The up direction

• The origin E represents the point of view (location of an observer)
• The vector

 defines the up direction  need not be perpendicular to

• WCS to ECS conversion steps:
• 1. Define the ECS axes xe, ye and ze
• 2. Perform the WCS-to-ECS conversion for all objects

WCS to ECS

g

up up

g

Graphics & Visualization: Principles & Algorithms Chapter 4 25

• 1. Define the ECS axes
• Sufficient information (E, , ) since the coordinate system is

right-handed

• Align xe- and ye-axes with the WCS axes with the convention:



xe is the horizontal axis & increases to the right



ye is the vertical axis & increases upwards



ze-axis points towards the observer (right-handed ECS)



Compute xe- and ye-axes as cross products:

WCS to ECS (2)

g

up

   

e e e e e

x up z y z x

 

e

z g

Graphics & Visualization: Principles & Algorithms Chapter 4 26

• 2. WCS-to-ECS conversion

i. Establish matrix ii. Pre-multiply all object vertices by it

• From [Ex. 3.16] this conversion requires 2 transformations:



Translation by



Change of basis (rotational transformation)

• Let the WCS coordinates of the ECS unit axis vectors be:
• Then:

WCS to ECS (3)

 WCS ECS

M

[ , , ]

T x y z

E E E   E

[ , , ] , [ , , ] and [ , , ]

T T T x y z x y z x y z

a a a b b b c c c   

e e e

x y z

1 1 · 1 1 1

x y z x x y z y x y z z

a a a E b b b E c c c E



                           

WCS ECS

M

Graphics & Visualization: Principles & Algorithms Chapter 4 27

• Orthographic projection onto the xy-plane:



Select a region of space (view volume) that will be mapped to CSS



rectangular parallelepiped



can be defined by 2 opposite vertices



xe = l, the left clip plane



xe = r, the right clip plane, (r > l)



ye = b, the bottom clip plane



ye = t, the top clip plane, (t > b)



ze = n, the near clip plane



ze = f , the far clip plane, ( f < n, since the ze axis points toward the observer)

ECS to CSS: Orthographic projection

Graphics & Visualization: Principles & Algorithms Chapter 4 28

• To maintain the z-coordinate use the ortho. proj. MORTHO = ID
• Converting the view volume into CSS:



Translation & Scaling



Map the (l, b, n) values to -1 and (r, t, f) values to 1

• Thus a WCS point Xw = [xw, yw, zw]T is mapped into CSS by:

ECS to CSS: Orthographic projection(2)

2 2 2 ( , , )· ( , , )· 2 2 2 2 2 1 2 2 2 1 · 2 2 2 1 2 1 1 1 r l t b n f r l t b f n r l r l r l r l r l t b t b t b t b t b n f n f f n f n f n



                                                                              

ORTHO ECS CSS

M S T ID              

· ·

 



ORTHO s ECS CSS WCS ECS w

X M M X

Graphics & Visualization: Principles & Algorithms Chapter 4 29

• View volume is a truncated pyramid, symmetrical about the
• ze axis
• This view volume is specified by:

 θ , the angle of the field of view in the y-direction  aspect, the width to height ratio of a cross section of the pyramid  ze = n, the near clipping plane  ze = f , the far clipping plane ( f < n)

ECS to CSS: Perspective projection

Graphics & Visualization: Principles & Algorithms Chapter 4 30

• Projection takes place onto the near clipping plane ze = n
• Compute the other clipping boundaries:
• Use a modified version of the

perspective projection matrix:

• z-coordinate :



Preserve it for hidden surface & other computations in screen space



Simply keeping ze will deform objects



Use a mapping that preserves lines & planes zs = A + B / ze , where A,B are constants

ECS to CSS: Perspective projection (2)

top: | |·tan( ) 2 bottom:

• right:

· left:

• t

n b t r t aspect l r         

1 d d d             

PER

P

Graphics & Visualization: Principles & Algorithms Chapter 4 31

• Require that (ze = n)  (zs = n) and (ze = f )  (zs = f ) so

A = ( n + f ) and B = -nf

• This mapping will not alter the boundaries ze = n , ze = f

but this will not be true for ze values between them

• Thus the perspective projection matrix is:



Makes the w-coordinate equal to ze



Must be followed by a division by ze (perspective division)



Clipping boundaries are not affected



Transforms the truncated pyramid into a rectangular parallelepiped :

ECS to CSS: Perspective projection (3)

1 n n n f nf               

VT

P

Graphics & Visualization: Principles & Algorithms Chapter 4 32

• Situation similar to the setting before the orthographic projection
• This view volume is already symmetrical about the –ze axis
• ECS-to-CSS conversion steps:
• 1. PVT
• 2. translation along ze
• 3. scaling

ECS to CSS: Perspective projection (4)

2 2 2 ( , , )· (0,0, )· 2 2 1 2 1 · · 0 1 2 2 1 1 1 n f r l t b f n r l n n t b n f n f nf f n



                                                              

PERSP ECS CSS VT

M S T P 2 2 2 1 n r l n t b n f nf f n f n                          

Graphics & Visualization: Principles & Algorithms Chapter 4 33

• Thus a WCS point Xw = [xw, yw, zw]T is converted into CSS by:

followed by the perspective division by w (= ze)

• Frustum culling is performed before the perspective division

ensuring that the coordinates of every point on every object are within the clipping bounds:

• w ≤ x, y, z ≤ w
• The coordinates of every point are now in the range [-1,1]

ECS to CSS: Perspective projection (5)

· · x y z w

 

            

PERSP ECS CSS WCS ECS w

M M X

/ x y w z w             

s

X

Graphics & Visualization: Principles & Algorithms Chapter 4 34

Example:

• Take the boundary points with ECS coordinates

[l, b, n, 1]T and [0, 0, f, 1]T

• Apply the PVT matrix:
• The homogeneous coordinate is no longer 1
• Apply the scaling & translation matrices:

ECS to CSS: Perspective projection (6)

2 2

· · 1 1 l ln b bn n n f f n f                                                  

VT VT

P P

2 2

· · · · ln n bn n n n f f n n f f                                                      S T S T

Graphics & Visualization: Principles & Algorithms Chapter 4 35

Example(continued):

• Due to the symmetry of the truncated pyramid about –ze :

r = -l, t = -b so r - l = -2l, t - b = -2b

• Perspective division:

which are the CSS values of the points

ECS to CSS: Perspective projection (7)

1 1 / / 1 1 1 1 n n n f n f n f                                                        

Graphics & Visualization: Principles & Algorithms Chapter 4 36

• A. Truncated Pyramid Not Symmetrical about ze-Axis

e.g. stereo viewing where two viewpoints are slightly offset on the xe –axis

Extended Viewing Transformation

Graphics & Visualization: Principles & Algorithms Chapter 4 37

• A. Truncated Pyramid Not Symmetrical about ze-Axis
• Parameters of the clipping planes:

 ze = n0, the near clipping plane (as before)  ze = f0, the far clipping plane, f0 < n0 (as before)  ye = b0, the ye-coordinate of the bottom clipping plane at its

intersection with the near clipping plane

 ye = t0, the ye-coordinate of the top clipping plane at its intersection

with the near clipping plane

 xe = l0, the xe-coordinate of the left clipping plane at its intersection

with the near clipping plane

 xe = r0, the xe-coordinate of the right clipping plane at its intersection

with the near clipping plane

Extended Viewing Transformation(2)

Graphics & Visualization: Principles & Algorithms Chapter 4 38

• A. Truncated Pyramid Not Symmetrical about ze-Axis (steps)
• Convert the non-symmetrical pyramid to symmetrical about ze with a shear

transformation on the xy-plane

(1)

• Determine the A, B parameters



Take the shear on the y-coordinate: map the midpoint of t0b0 to 0



Similarly for the xe shear factor:

• So (1) becomes:

Extended Viewing Transformation(3)

1 1 1 1 A B             

xy

SH

· 2 2 b t b t B n B n       

2 l r A n   

1 2 1 2 1 1 l r n b t n



                      

NON SYM

SH

Graphics & Visualization: Principles & Algorithms Chapter 4 39

• A. Truncated Pyramid Not Symmetrical about ze-Axis (steps)
• Change the clipping boundaries, to reflect the symmetrical shape of

the new pyramid:

• Substitute the above into

which is equivalent to the original with the clipping bounds replaced by the initial clipping bounds.

Extended Viewing Transformation(4)

2 2 2 2 n n f f l r l r l l r r b t b t b b t t              

 PERSP ECS CSS

M

2 2 2 1 n r l n t b n f n f f n f n



                            

PERSP ECS CSS

M

 PERSP ECS CSS

M

Graphics & Visualization: Principles & Algorithms Chapter 4 40

• A. Truncated Pyramid Not Symmetrical about ze-Axis
• Thus it is not necessary to have initial clipping bounds; can name

them n, f, l, r, b, t from the start.

• The ECSCSS mapping in the case of non-symmetrical

perspective projection is thus:

Extended Viewing Transformation(5)

2 1 2 2 1 · · 2 2 1 1 1 n l r r l n n b t t b n n f nf f n f n

    

                                                 

PERSP NON SYM PERSP ECS CSS ECS CSS NON SYM

M M SH 2 2 2 1 n l r r l r l n b t t b t b n f nf f n f n                                 

Graphics & Visualization: Principles & Algorithms Chapter 4 41

• B. Oblique Projection
• Useful, e.g., in the computation of oblique views for 3D displays:



is not sufficient



Direction of projection must be taken into account

• View volume: a six-sided parallelepiped specified by:



The 6 parameters used for the non-symmetrical pyramid (n0, f0, l0, r0, b0, t0)



The direction of projection vector

• Steps:

1. Translate the view volume so that (l0, b0, n0)-point moves to the ECS origin 2. Perform a shear in the xy-plane

Extended Viewing Transformation(6)

 ORTHO ECS CSS

M

DOP

Graphics & Visualization: Principles & Algorithms Chapter 4 42

• B. Oblique Projection (steps)
• Take the point defined by the origin and



Shear the (DOPy) coordinate to 0:



Similarly for the x-shear factor:

• The required transformation is:
• Alter the clipping bounds to reflect the new rectangular parallelepiped

n = 0, f = f0 - n0, l = 0, r = r0 - l0, b = 0, t = t0 - b0

• ECSCSS mapping for a general parallel projection is thus:

Extended Viewing Transformation(7)

[ , , ]T

x y z

DOP DOP DOP  DOP

·

y y z z

DOP DOP B DOP B DOP     

x z

DOP A DOP  

1 1 1 1 · · 0 1 1 1 1

x z y z

DOP DOP l DOP b DOP n                                    

PARALLEL PARALLEL

SH T

· ·

 



PARALLEL ORTHO ECS CSS ECS CSS PARALLEL PARALLEL

M M SH T

Graphics & Visualization: Principles & Algorithms Chapter 4 43

• Frustum culling:



Eliminates primitives (parts or whole) that lie outside the view volume(frustum)



Is implemented by 3D clipping algorithms

• The viewing transformation defines the 3D clipping boundaries.
• Clipping takes place in CSS, after the application of or

but before the division by w.

• Clipping boundaries for: perspective projection: -w ≤ x, y, z ≤ w
• rthographic or parallel projection: -1 ≤ x, y, z ≤ 1
• Why clip in 3D and not in 2D, after throwing away z?



In perspective projection, objects behind the center of projection E would appear upside-down (no sufficient information in clipping)



Avoids possible perspective division by 0



The near and far clipping planes limit the depth range & enable the optimal allocation of the bits of the depth buffer

Frustum Culling & the Viewing Transformation

 PERSP ECS CSS

M

 ORTHO ECS CSS

M

Graphics & Visualization: Principles & Algorithms Chapter 4 44

• Viewport: rectangular part of the screen where the contents
• f the view volume are displayed.
• Defined by its bottom-left and top-right corners



[xmin, ymin]T and [xmax, ymax]T in pixel coordinates



[xmin, ymin, zmin]T and [xmax, ymax, zmax]T to maintain the z-coordinate

• Viewport transformation converts objects from CSS into the

viewport coordinate system (VCS) with a scaling & a translation:

• This is a generalization of the 2D window-to-viewport transformation [Ex. 3.8].
• The size of the viewport defines the final size of the objects on screen

The Viewport Transformation

1 2 2 2 2 1 · 2 2 2 2 1 2 2 1 1

min max max min max min min max min max max min max min min max min max max min max min

x x x x x x x x y y y y y y y y z z z z z z



                                                    

VIEWPORT CSS VCS

M 2 2 1

min max

z z                     