Lecture computer graphics and virtual reality slides lesson 6 viewing transformation

Trang 4 Viewing transformation■ More logical to use dimensions which are appropriate to the object.. □metres for buildings, □millimetres for assembly parts, □nanometres or microns for mo

Lesson 6

Viewing Transformation

Trinh Thanh Trung School of ICT, HUST

Content

Overview

Viewing transformation

appropriate to the object e.g

□ metres for buildings,

□ millimetres for assembly parts,

□ nanometres or microns for molecules, cells, and

atoms

physical size in the real world,

screen co-ordinates before displaying

Viewing transformation

co-ordinates to Screen Coco-ordinates

view

effect without changing the model by scaling

dimensions proportionally

screen co-ordinates?

Viewing transformation

view volume

Homogeneous Coordinates

the same point when all elements are multiplied by the same factor

□ (2,0,1,1) and (4,0,2,2) are the same points

□ To bring back to Cartesian space, need to

divide the other elements by the fourth element w

▫ (x, y, z, w) → (x/w, y/w, z/w, 1)

2D viewing transformation

Window and viewport

■ Window: the portion of the world which will be

displayed

■ Viewport: The screen where the image will be

displayed

(wx min ,wy min)

(wx max ,wy max)

(vx min ,vy min)

(vx max ,vy max)

Transformation

0 0 1

1

Yw Xw

0

0 min max

min max

0

0 0

Xwmin -

Xwmax

Xvmin -

Xvmax

] 1 [

Yw Yw

Yv Yv

0 0 1 ]

2 [

Yy Xv T

Combined transformation matrix

min

max min

min Xwmin

Xwmax

-Xvmin -

Xvmax min

min

0 min

max

min

max 0

0 0

Xwmin -

Xwmax

Xvmin -

Xvmax

]

[

] 2 [ ] 1 [ ]

Yv

Yv Yw

Yv Xw

Xv

Yw Yw

Yv

Yv T

T x S x

T

T

Example in OpenGL

//set the viewing coordinates

setWindow(xmin, xmax, ymin, ymax);

-void setWindow(GLdouble left, Gldouble right, GLdouble bottom, GLdouble top) {

-void setViewport(GLdouble left, Gldouble right, GLdouble bottom, GLdouble top)

{ glViewport(left, bottom, right – left, top - bottom);

}

3D viewing transformation

3D viewing

■ Part of the difficulty lies in trying to display three 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

□ a description of the scene geometry

□ a view definition (camera)

plane

in 2D viewing

scene geometry onto a 2D surface for display

Camera analogy

components, specified independently:

□ objects (a.k.a geometry)

□ viewer (a.k.a camera)

projection plane (usually in front of the camera).

■ Projectors emanate from the center of

projection (COP) at the center of the lens (or

pinhole)

□ The image of an object point P is at the intersection

of the projector through P and the image plane.

Camera analogy

four kinds of parameters:

□ Position: the COP.

□ Orientation: rotations about axes with origin at the COP.

□ Focal length: determines the size of the image on the film plane, or the

field of view.

□ Film plane: its width and height.

transformation

Camera parameters

■ View Reference Point (VRP): origin of our viewing system (position of the camera)

■ View Plane Normal vector (VPN): viewing direction

■ View UP vector (VUP): establishes orientation of

“camera”

),,(n1 n2 n3

=

=

N

N n

),,(u1 u2 u3

N V u

) , , ( v1 v2 v3

=



= n u v

Viewing transformation

■ (xw, yw, zw) and (u, v, n) are 2 coordinate systems with the same centre

M WC,VC = R T

24

,

,

3 2 1

3 2 1

3 2

v v v

u u

u

n v

u

100

010

001

0 0 0

z y

0

000

3 2

1

3 2

1

3 2

1

n n

n

v v

v

u u

u

R

T R

Viewing transformation

following transformation sequence:

□ Translate the view reference point to the origin of the world co-ordinate system

□ Apply Rotations to align xv, yv and zv, axes, respectively

1 0 0

0 1 0

0 0 1

0 0 0

z y x

Viewing transformation

27

1 0 0 0

0 0

0

3 2 1

3 2 1

3 2 1

v v v

u u u

u u

u

R u

Camera movement

□ The camera is positioned using a combination of translations and rotations.

□ Think if the camera being in the same location as the viewers eye.

and position

own axes

28

Camera in OpenGL

glMatrixMode(GL_MODELVIEW);

glLoadIdentity();

gluLookAt(eye.x, eye.y, eye.z,

look.x, look.y, look.z, up.x, up.y, up.z);

29

Camera Movement

1 slid in 3 directions; and

2 rotated in 3 directions

□ The camera can move along its axes.

□ This is called sliding the camera.

30

■ Besides physically moving the camera to another location, the camera can be tilted in different directions to look at different parts of the scene.

31

u v

n

The Camera - Camera Movement

■ We use a plane analogy to describe the cameras movement.

□ a rotation from the horizontal along the length is called PITCH

□ a rotation from the horizontal along the width is called ROLL

□ a rotation around the vertical is called YAW

32

u v

u n

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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