2iv60 computer graphics set 5: viewing jack van wijk tu/e

2IV60 Computer graphicsset 5: Viewing

Jack van Wijk

TU/e

Viewing

• Transformation worldscreen

• Clipping: Removing parts outside screen

• 2D (chapter 8)

• 3D (chapter 10)

2D Viewing pipeline 1

Clipping window:

What do we want to see?

H&B 8-1:258-259

xwmin xwmax

ywmax

ywmin

Viewport:

Where do we want to see it?

Clipping window

World:

xvmin xvmax

yvmax

yvmin

Viewport

Screen:


Clipping window:

Panning…

H&B 8-2:259-261

ywmax

ywmin

xwmin xwmax

Clipping window

World:

xvmin xvmax

yvmax

yvmin

Viewport

Screen:


Clipping window:

Panning…

ywmax

ywmin

xwmin xwmax

Clipping window

World:

xvmin xvmax

yvmax

yvmin

Viewport

Screen:

H&B 8-2:259-261


Clipping window:

Zooming…

ywmax

ywmin

xwmin xwmax

World:

xvmin xvmax

yvmax

yvmin

Viewport

Screen:

H&B 8-2:259-261


MC: Modeling Coordinates

WC: World Coordinates

VC: Viewing Coordinates

NC: Normalized Coordinates

DC: Device Coordinates

Apply model transformations

Determine visible parts

To standard coordinates

Clip and determine pixels

H&B 8-2:259-261

Clipping window

xwmin xwmax

ywmax

ywmin

Clipping window

- Clipping window usually an axis-aligned rectangle- Sometimes rotation- From world to view coordinates:

possibly followed by rotation- More complex in 3D

xwmax xwmin

ywmax ywmin

minmin , ywxw T

H&B 8-2:259-261

To normalized coordinates 1

xvmax-xvmin

yvmax -yvmin

xwmax-xwmin

ywmax -ywmin

!distortion :changes ratio-aspect then the

unequal, are factors scale two theIf

,

: withScale

minmax

minmax

minmax

minmax

ywyw

yvyv

xwxw

xvxvS

H&B 8-3:261-265


xvmax-xvmin

yvmax -yvmin

xvmin xvmax

yvmax

yvmin

Viewport

1

1

minmin ,

: withTranslate

yvxvT

H&B 8-3:261-265


xwmin xwmax

ywmax

ywmin

Clipping window

xvmin xvmax

yvmax

yvmin

Viewport

1

1

),(,),(

: togetherAll

minminminmax

minmax

minmax

minmaxminmin ywxw

ywyw

yvyv

xwxw

xvxvyvxv

TST

H&B 8-3:261-265

OpenGL 2D Viewing 1

Specification of 2D Viewing in OpenGL:- Standard pattern, follows terminology.

First, this is about projection. Hence, select and the Projection Matrix (instead of the ModelView matrix) with:

glMatrixMode(GL_PROJECTION);

H&B 8-4:265-267

OpenGL 2D Viewing 2

Next, specify the 2D clipping window:gluOrtho2D(xwmin, xwmax, ywmin, ywmax);

xwmin, xwmax: horizontal range, world coordinates

ywmin, ywmax: vertical range, world coordinates

xwmin xwmax

ywmin

ywmax

H&B 8-4:265-267

OpenGL 2D Viewing 3

Finally, specify the viewport:glViewport(xvmin, yvmin, vpWidth, vpHeight);

xvmin, yvmin: coordinates lower left corner (in pixel coordinates);

vpWidth, vpHeight: width and height (in pixel coordinates);

(xvmin, yvmin)

vpWidth

vpHeight

H&B 8-4:265-267

OpenGL 2D Viewing 4

In short:

glMatrixMode(GL_PROJECTION);

gluOrtho2D(xwmin, xwmax, ywmin, ywmax);

glViewport(xvmin, yvmin, vpWidth, vpHeight);

To prevent distortion, make sure that:

(ywmax – ywmin)/(xwmax – xwmin) = vpWidth/vpHeight

H&B 8-4:265-267

Clipping

xwmin xwmax

ywmax

ywmin

Clipping window

- Clipping: removing parts outside clipping window- Many algorithms: points, lines, fill-area, curves,…

H&B 8-5:274

2D Point Clipping

xwmin xwmax

ywmax

ywmin

Clipping window

- Trivial: Save point P = (x,y) if it’s in the box:

maxmin

maxmin

ywyyw

xwxxw

H&B 8-5:274-275

2D Line Clipping 1

xwmin xwmax

ywmax

ywmin

H&B 8-6:275-281

2D Line Clipping 2

xwmin xwmax

ywmax

ywmin

P

Q Basic algorithm:Determine interval (u0, u1) inrectangle, by calculatingcrossings with edges of window.

u

Line segment:X(u) = P + u (QP), with 0 <= u <= 1

H&B 8-6:275-281

2D Line Clipping 3

xwmin xwmax

ywmax

ywmin

P

Q

u

u0 := 0; u1 = 1;(* Right side: *)If Px=Qx then If Px > xwmax then return emptyelse u := (xwmax – Px)/(Qx-Px) if Px < Qx then begin if u < u1 then u1 := u end else if u > u0 then u0 := u;If u0 > u1 then return empty

Line segment:X(u) = P + u (Q-P), with 0 <= u <= 1

u1

H&B 8-6:275-281

2D Line Clipping 4

xwmin xwmax

ywmax

ywmin

P

Q

u

(* Left side: *)If Px=Qx then If Px < xwmin then return emptyelse u := (xwmin – Px)/(Qx-Px) if Px < Qx then begin if u > u0 then u0 := u end else if u < u1 then u1 := u;If u0 > u1 then return empty

(* Lower and upper side idem *)

Line segment:X(u) = P + u (Q-P), with 0 <= u <= 1

u1

H&B 8-6:275-281

2D Line Clipping 5

xwmin xwmax

ywmax

ywmin

- Expensive: calculation crossings- Avoid this by using position end points- For instance: If end points in window, then line in window

H&B 8-6:275-281

bit 4 bit 3 bit 2 bit 1

Top Right Bottom Left

1001 1000 1010

0001 0000 0010

0101 0100 0110

2D Line Clipping 6

xwmin xwmax

ywmax

ywmin

Cohen-Sutherland algorithm:- Assign to each point a four-bit region code C;- 1 is outside, 0 is inside

H&B 8-6:275-281

bit 4 bit 3 bit 2 bit 1

Top Right Bottom Left

1001 1000 1010

0001 0000 0010

0101 0100 0110

2D Line Clipping 7

xwmin xwmax

ywmax

ywmin

Fast test on status end points line with codes C0 en C1 : C0 bitwise-or C1 = 0000: then completely in; C0 bitwise-and C1 <> 0000: then completely out;Else: fast intersection-calculation using codes.

H&B 8-6:275-281

3D Viewing

• Viewing: virtual camera

• Projection

• Depth

• Visible lines and surfaces

• Surface rendering

H&B 10-1:332-334


• Similar to making a photo– Position and point virtuele camera, press button;

Projection plane akaViewing plane

• Pipeline has +/ same structure as in 2DH&B 10-1:332-334


MC: Modeling Coordinates

WC: World Coordinates

VC: Viewing Coordinates

PC: Projection Coordinates

NC: Normalized Coordinates

DC: Device Coordinates

Apply model transformations

To camera coordinates

Project

To standard coordinates

Clip and convert to pixelsH&B 10-2:334-335

3D viewing coordinates 1

Specification of projection:

P0 : View or eye point

Pref : Center or look-at point

V: View-up vector (projection along vertical axis)

zvp : positie view plane

P0

zw

yw

xw

Pref

NV

z vp

P0, Pref , V: define viewing coordinate system

Several variants possible H&B 10-3:336-338

3D viewing coordinates 2

P0

zw

yw

xw

Pref

NV

zview

yviewxview

P0, Pref , V: define viewing coordinate system

Several variants possible H&B 10-4:338-340

3D view coordinates 3

P0

zw

yw

xw

Pref

NV

zview

yviewxview

u

),,(

: and lar toperpendicu

zyx uuu

V

nVu

nVu

v

),,(

: and lar toperpendicu

zyx vvv unv

unv

),,(

:frame axis Derivation

ref0

zyx nnn

N

Nn

PPN

n

H&B 10-4:338-340

3D viewing coördinaten 4

P0

zw

yw

xw

Pref

NV

zview

yviewxview

n

uv

RTM

R

R

PT 0

WC,VC

zyx

zyx

zyx

nnn

vvv

uuu

:Or

1000

0

0

0

: with rotate Next,

)( with translateFirst,

:viewtion worldTransforma

H&B 10-4:338-340

Projection transformations

P1

P2

P’1

P’2

View plane

Parallel projection

P1

P2

P’1

P’2

View plane

Perspective projection

H&B 10-6:340-345

Orthogonal projections 1

P’1

P’2Parallell projection:Projection lines are parallel

Orthogonal projection:Projection lines are parallel and perpendicular to projection plane

Isometric projection:Projection lines are parallel, perpendicular to projection plane,and have the same angle with axes.

P1

P2

P’1

P’2P1

P2

xy

z

H&B 10-6:340-345


Orthogonal projection:

P’1

P’2P1

P2

Trivial!

:s)coordinate projection

toscoordinate view(from

),,( of Projection

zz

yy

xx

zyx

p

p

p

H&B 10-6:340-345


Clippingwindow

zview

xview

yview

Near plane

Far plane

View volume

H&B 10-6:340-345


zview

xview

yview

View volume

(xwmin, ywmin, znear)

(xwmax, ywmax, zfar)

H&B 10-6:340-345


zview

xview

yview

View volume

(xwmin, ywmin, znear)

(xwmax, ywmax, zfar)

znorm

xnorm

ynorm

Normalized View volume

(1,1,1)

(-1,-1,-1)

TranslationScaling

From right- to left handed H&B 10-6:340-345

Perspective projection 1

H&B 10-8:351-364

P1

P2

P’1

P’2

View plane: z = zvp

Projection reference point

zview

xview

yview

View plane: orthogonal to zview axis.


P = (x, y, z) R: Projection reference point

zview

xview

yview

(xr, yr, zr)(xp, yp, zp)

To simplify, Assume R in origin

View plane: z = zvp

Question: What is the projection of Pon the view plane?

H&B 10-8:351-364


P = (x, y, z)

zview

xview

yview(xp, yp, zp)

(0,0, 0)R=

.' and;';'or

,10 with ,'

: to(origin) from Line

uzzuyyuxx

uu

PX

PR

. hence '

:plane with crossingAt

z

zuzz vp

vp

yz

zyx

z

zx vp

pvp

p and

giveson Substituti

View plane: z = zvp

H&B 10-8:351-364


P = (x, y, z) View plane

zview

yview

Viewed from the side

R

z

y

zvp

yp

yz

zy

z

y

z

y

vpp

vp

p

hence

: thatseecan We

H&B 10-8:351-364


P = (x, y, z)Clipping window

in View plane

zview

yview

R

zvpW

wmax

wmin

Ratio between W=wmaxwmin and zvp

determines strenght perspective


H&B 10-8:351-364



Clipping windowin View plane

zview

yview

R

Ratio between W=wmaxwmin and zvp

determines strenght perspective.

This ratio is 2tan(/2),with the view angle.

H&B 10-8:351-364


zview

yview

R

H&B 10-8:351-364


R

H&B 10-8:351-364


zview

yview

R

zvp

W

How to specify the ratio between W en zvp?How to specify ‘how much’ perspective I want to see?

Option 1: Specify view angle .

H&B 10-8:351-364


zview

yview

R

zvp

W

Use camera as metaphor. User specifies focal length f .(20 mm – wide angle, 50 mm – normal, 100 mm – tele). Application adjusts W and/or zvp, such that W/zvp = 24/f.For x: W/zvp = 36/f.

f (mm)

24 mm

H&B 10-8:351-364

View volume orthogonal…

Clippingwindow

zview

xview

yview

Near clipping plane

Far clipping plane

View volume

H&B 10-8:351-364

View volume perspective

Clippingwindow

zview

xview

yview

Near clipping plane

Far clipping plane

View volume

R

H&B 10-8:351-364

To Normalized Coordinates…

zview

xview

yview

R

znorm

xnorm

ynorm


(1,1,1)

(1, 1, 1)

Rectangular frustumView Volume

H&B 10-8:351-364

Side view Front view


yview

Rzview

znorm

ynorm

Perspective transformation:Distort space, such that perpendicular projection gives an image in perspective.

H&B 10-8:351-364


yview

R2r

zn

zf

zview

znorm

ynorm

Simplest case:Square window, clipping plane coincides with view plane: zn=zvp

H&B 10-8:351-364


yview

R

zn

zf

zview

znorm

ynorm

(-r, -r, zn)

((zf /zn)r, (zf /zn)r, zf )

2r

(-1,-1,-1)

(1,1,1)

H&B 10-8:351-364

(r, r, zn)

(rzf /zn, rzf /zn, zf )


yview

Rzview

znorm

ynorm

(1, 1, 1)

(1,1,1)

How to put this transformation in the pipeline?How to process division by z?

Earlier: yz

zyx

z

zx vp

pvp

p ,

H&B 10-8:351-364

Homogeneous coordinates (reprise)

• Add extra coordinate:

P = (px , py , pz , ph) or

x = (x, y, z, h)

• Cartesian coordinates: divide by h

x = (x/h, y/h, z/h)

• Points: h = 1 (temporary…) perspective: h = z !

H&B 10-8:351-364

.

/

/

/

:bygiven are scoordinate projectedsuch that

10100

:by described becan ation transformePerspectiv

hz

hy

hx

z

y

x

z

y

x

tsss

tsss

tsss

h

z

y

x

h

h

h

p

p

p

v

v

v

zzzzyzx

yyzyyyx

xxzxyxx

h

h

h

Homogeneous coordinates (reprise)

H&B 10-8:351-364


xview

Rzview

znorm

ynorm

(r, r, zn)


(1, 1, 1)

(1,1,1)(r, r, zn)

.0,/ :givesn Elaboratio

.1 also then , and / If

.1 then , and If

./)( is form Generic .First

xxzxynxx

pfnf

pn

xxzxyxxp

tssrzs

xzzzrzx

xzzrx

ztzsysxsxx

H&B 10-8:351-364


yview

Rzview

znorm

ynorm

(r, r, zn)


(1, 1, 1)

(1,1,1)

.0),/( :Or

./)/(

:gives , as Same . Next the

yyzyxnyy

np

tssrzs

zyrzy

xy

H&B 10-8:351-364


yview

Rzview

znorm

ynorm

(r, r, zn)


(1, 1, 1)

(1,1,1)

.0,2

,

givesn Elaboratio .1 then , If

.1 then , If

./)( :is form Generic . :Finally

zyzxfn

fnz

fn

fnzz

ff

pn

zzzzyzxp

sszz

zzt

zz

zzs

zzz

zzz

ztzsysxszz

H&B 10-8:351-364


.

/

/

/

:from follow scoordinate projected thewhere

10100

200

00/0

000/

:by described be hencecan ation transformePerspectiv

hz

hy

hx

z

y

x

z

y

x

zz

zz

zz

zzrz

rz

h

z

y

x

v

v

v

p

p

p

v

v

v

fn

fn

fn

fn

n

n

h

h

h

H&B 10-8:351-364

3D Viewport coordinates 1

H&B 10-9:365-365

znorm

xnorm

ynorm


(1,1,1)

(1, 1, 1)

zv xv

yv

3D screen

(xvmin, yvmin, 0)

(xvmax, yvvmax, 1)

scalingtranslation

3D Viewport coordinaten 2

110002

1

2

100

002

0

0002

:scoordinatescreen 3D to

scoordinate viewnormalized fromtion Transforma

minmax

minmax

p

p

p

s

s

s

z

y

xyvyv

xvxv

h

z

y

x

H&B 10-9:365-365

OpenGL 3D Viewing 1

H&B 10-10:365-371

3D Viewing in OpenGL:

- Position camera;- Specify projection.

OpenGL 3D Viewing 2

z

x

y

View volume

H&B 10-10:365-371

Camera always in origin, in direction of negative z-axis.

Convenient for 2D, but not for 3D.

OpenGL 3D Viewing 3

H&B 10-10:365-371

Solution for view transform: Transform your model such that you look at it in a convenient way.

Approach 1: Do it yourself. Apply rotations, translations, scaling, etc., before rendering the model. Surprisingly difficult and error-prone.

Approach 2: Use gluLookAt();

OpenGL 3D Viewing 4

H&B 10-10:365-371

MatrixMode(GL_MODELVIEW); gluLookAt(x0,y0,z0, xref,yref,zref, Vx,Vy,Vz);

x0,y0,z0: P0, viewpoint, location of camera;

xref,yref,zref: Pref, centerpoint;

Vx,Vy,Vz: V, view-up vector.

Default: P0 = (0, 0, 0); Pref = (0, 0, 1); V=(0, 1, 0).

OpenGL 3D Viewing 5

H&B 10-10:365-371

z

xy

Orthogonal projection:

MatrixMode(GL_PROJECTION); glOrtho(xwmin, xwmax, ywmin, ywmax, dnear, dfar);

xwmin

xwmaxywmin

ywmax

dnear

dfar

xwmin, xwmax, ywmin,ywmax:

specification window

dnear: distance to near clipping plane

dfar : distance to far clipping plane

Select dnear and dfar right: dnear < dfar, model fits between clipping planes.

OpenGL 3D Viewing 6

H&B 10-10:365-371

Perspective projection:

MatrixMode(GL_PROJECTION); glFrustrum(xwmin, xwmax, ywmin, ywmax, dnear, dfar);

xwminxwmaxywmin

ywmax

xwmin, xwmax, ywmin,ywmax:

specification window

dnear: distance to near clipping plane

dfar : distance to far clipping plane

z

xy

dnear

dfar

Standard projection: xwmin = -xwmax, ywmin = -ywmax

Select dnear and dfar right: 0 < dnear < dfar, model fits between clipping planes.

OpenGL 3D Viewing 7

Finally, specify the viewport (just like in 2D):glViewport(xvmin, yvmin, vpWidth, vpHeight);

xvmin, yvmin: coordinates lower left corner (in pixel coordinates);

vpWidth, vpHeight: width and height (in pixel coordinates);

(xvmin, yvmin)

vpWidth

vpHeight

H&B 8-4:265-267

OpenGL 2D Viewing 8

In short:

glMatrixMode(GL_PROJECTION);glFrustrum(xwmin, xwmax, ywmin, ywmax, dnear, dfar);glViewport(xvmin, yvmin, vpWidth, vpHeight);glMatrixMode(GL_MODELVIEW);

gluLookAt(x0,y0,z0, xref,yref,zref, Vx,Vy,Vz);

To prevent distortion, make sure that: (ywmax – ywmin)/(xwmax – xwmin) = vpWidth/vpHeight

Make sure that you can deal with resize/reshape of the (OS) window.

H&B 8-4:265-267

Next…

• We now know how to project objects.

• But how to model objects?

2iv60 computer graphics set 5: viewing jack van wijk tu/e

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