Lecture computer graphics and virtual reality slides lesson 7 projection transformation

Lesson 7

Projection
Transformation

Trinh Thanh Trung School of ICT, HUST

Content

1. 3D viewing and projection
2. Parallel projection
3. Perspective projection

1.
3D viewing and projection

3D viewing

■ Part of the difficulty lies in trying to display three
dimensional objects on a 2D display
■ Different views on the same object eventually
lead to different 2D representations at the
projection stage

□ The projected 2D image of a 3D object is viewer
dependent

■ Different views can be specified based on several
parameters e.g. position, orientation, field of view

3D viewing methods

■ On 2D displays:

□ perspective and depth of field
□ depth cueing
□ visible line/surface identification
□ surface rendering
□ exploded/cutaway scenes, cross-sections

3D viewing

6

3D viewing can be faked?

3D viewing methods (cont.)

■ On 3D display

□ Hologram display

Stereoscopic display

■ Faking depth in a 2D image
■ Based on natural stereoscopic eye-brain system.
■ Objects are not viewed with just one eye, but
two eyes.
■ Each eye looks at the object from a slightly
different location.

Stereoscopic display

Projections transform
points in n-space to
m-space, where m
Projections

■ In 3-D, we map points from 3-space to the
projection plane (PP) (a.k.a., image plane) along
projectors (a.k.a., viewing rays) emanating from
the center of projection (COP):

■ There are two basic types of projections:

□ Perspective – distance from COP to PP finite
□ Parallel – distance from COP to PP infinite

2.
Parallel projection

Parallel projection

■ Parallel projection is the projection which the
projection lines are parallel or start from infinity

■ Points on the object are projected to the viewing
plane along parallel lines

■ Preserves relative dimensions of the object but

does not give a realistic presentation

A’ A
B
D’
B’
centre of D
projection at
C
infinity C’

projection
plane

Parallel projection properties

■ Are actually a kind of affine transformation

□ Parallel lines remain parallel
□ Ratios are preserved
□ Angles not (in general) preserved

■ Not realistic looking
■ Good for exact measurements

Most often used in
▫ CAD,
▫ architectural drawings,
▫ etc.,
where taking exact measurement is important

Homogeneous transform

■ For projections, we specify a direction of
projection (DOP) instead of a COP.

■ We can write orthographic projection onto the
z=0 plane with a simple matrix, such that x’=x, y’=y.

 x
 x ' 1 0 0 0  
 y ' = 0 1 0 0  y
   z

1  0 0 0 1  
1 

Orthographic projection

■ Orthographic projection is a form of parallel
projection which all the projection lines are
orthogonal to the projection plane

Orthographic projection

■ For each projection plane we have a
corresponding transformation matrix

1 0 0 0 0 0 0 0 1 0 0 0

[Ty ] = 0 0 0 0 [Tx ] = 0 1 0 0 [Tz ] = 0 1 0 0

0 0 1 0 0 0 1 0 0 0 0 0
 
  0 0 0 1  
0 0 0 1 0 0 0 1

Axonometric projection

■ Axonometric projection
is a type of orthographic
projection, where the lines
of sight are perpendicular
to the plane of projection,
and the object is rotated
around one or more of its
axes to reveal multiple
sides.

Axonometric projection

Isometric

■ Three axes of space appear equally foreshortened
(~120°)

■ Corresponding rotating angles are 35.26° and 45°

2 1 − 2 sin2 
sin  =

1 − sin  2

f = cos2  = 2 / 3 = 0.8165

2 sin2 
sin  =
1 − sin 2

2sin  = sin2  1/3
2= =1/2
1 − sin  1 −1 /3

sin =  1
3