Lecture computer graphics and virtual reality slides lesson 7 projection transformation

3D viewing and projection2.. Trang 4 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

Lesson 7

Projection Transformation

Trinh Thanh Trung School of ICT, HUST

Content

3D viewing and projection

3D viewing

dimensional objects on a 2D display

lead to different 2D representations at the

projection stage

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

dependent

parameters e.g position, orientation, field of view

3D viewing

6

3D viewing can be faked?

3D viewing methods (cont.)

□ Hologram display

Stereoscopic display

two eyes

different location

Stereoscopic display

Projections transform

points in n-space to m-space, where m<n.

■ 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

Parallel projection

Parallel projection

projection lines are parallel or start from infinity

plane along parallel lines

does not give a realistic presentation

A

D

C B

projection plane

Parallel projection properties

□ Parallel lines remain parallel

□ Ratios are preserved

□ Angles not (in general) preserved

Most often used in

Homogeneous transform

projection (DOP) instead of a COP.

z=0 plane with a simple matrix, such that x’=x, y’=y.

y y

Orthographic projection

projection which all the projection lines are

orthogonal to the projection plane

Orthographic projection

corresponding transformation matrix

0 0 0 0

0 0 1 0

0 0 0 1 ]

0 1 0 0

0 0 0 0

0 0 0 1

0 1 0 0

0 0 1 0

0 0 0 0 ] [T x

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.

sin sin

3 1

2 2

/ /

/ sin

0 0 0 0

0 0 1 0

0 0 0 1

1 0

0 0

0 0

0 0

0 0

0 1

1 0

0 0

0 0

0 0

1 0

0 0

cos

sin

sin cos

cos

sin

sin cos

0

0 0 sin

cos sin

0 0 cos

0

0 0 sin

sin cos

Axonometric projection Trimetric

axes of space

appear unequally

foreshortened

Oblique projections

Axonometric

Perspective projection

Perspective projection

vanishing point

camera cause a catastrophic “passage through infinity”

produce geometrically realistic pictures

Perspective projection properties

transformations:

□ Lines map to lines

□ Parallel lines do not necessarily remain parallel

□ Ratios are not preserved

is that size varies inversely with distance – looks realistic

as exactly as we can with parallel projections

Vanishing points

different point: The vanishing point for this direction

collinear vanishing points: the horizon for that plane

□ corridor

□ higher = further away

Vanishing points

Clipping and the viewing frustum

■ The center of projection and the portion of the projection plane that map to the final image form an infinite pyramid The

sides of the pyramid are clipping planes.

■ Frequently, additional clipping planes are inserted to restrict

the range of depths These clipping planes are called the near and far or the hither and yon clipping planes.

■ All of the clipping planes bound the viewing frustum.

Field of View (FOV)

x v

-z v

Near Clip Plane

Far Clip Plane View

Volume

Left Clip Plane

Right Clip Plane

f

FOV

Projection summary

Any questions?

Lecture notes provided by School of Information and

Communication Technology, Hanoi University of Science and

Technology.

Composed by Huynh Quyet Thang, Le Tan Hung, Trinh Thanh

Trung and others

Edited by Trinh Thanh Trung

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