Lect25 Projections 2

7/27/2019 Lect25 Projections 2

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159.235 Graphics 1

159.235 Graphics & Graphical

Programming

Lecture - Projections - Part 2


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159.235 Graphics 2

Projections Part 2 - Outline

Mathematics of Projections

Perspective Transforms

Stereo

Orthographic

3D Clipping Homogeneous Coordinates (again)


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Mathematics of Viewing

Need to generate the transformation

matrices for perspective and parallel

projections

Should be 4x4 matrices to allow general

concatenation


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Perspective ProjectionSimplest

Case

d

x

y

z

Projection

Plane.

P(x,y,z)

Pp(xp,yp,d)

Centre of projection at the origin,

Projection plane at z=d.


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Perspective ProjectionSimplest

Case

d

x

y

z

P(x,y,z)

Pp(xp,yp,d)

z

P(x,y,z)

d

z

P(x,y,z)

d

y

x

xp

yp

dz

y

z

ydy

dz

x

z

xdx

z

y

d

y

z

x

d

x

pp

pp

/;

/

;

:trianglessimilarFrom


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Perspective Projection

01/d00

0100

0010

0001

:matrix4x4aasdrepresentebecanationtransformThe

perM

TT

T

pp dzzyxd

zyd

zxddyx

1..1


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TT

perp

T

p

dzzyxWZYX

z

y

x

PMP

WZYXP

/

101/d00

0100

0010

0001

pointprojectedgeneraltheRepresent


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ddz

y

dz

x

W

Z

W

Y

W

X

dzzyxPT

p

,/,/,,

:3Dback tocomeW toDropping

/Trouble with this formulation :

Centre of projection fixed at

the origin.


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Finding Vanishing Points

Recall : An axis vanishingpoint is the point wherethe axis intercepts the

projection plane point atinfinity.

Tvpxp 0001

:pointby themultiply

point,vanishingaxisxfindtoE.g

e.perspectivpoint1ahaveweSo

00

01/d00

0100

0010

0001

:nformulatioFor this

dP

P

P

M

zvp

yvp

xvp

per


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Alternative Formulation

z

P(x,y,z)

d

x

xp

z

P(x,y,z)

d

y

yp

Projection plane at z = 0

Centre of projection at

z = -d

1)/(;

1)/(

dbyMultiply

;

:trianglessimilarFrom

dz

y

dz

ydy

dz

x

dz

xdx

dz

y

d

y

dz

x

d

x

pp

pp


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Alternative Formulation

11/d00

00000010

0001

perM

z

P(x,y,z)

d

x

xp

z

P(x,y,z)

d

y

yp

Projection plane at z = 0,

Centre of projection at

z = -d

Now we can allow d


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159.235 Graphics 12

Stereo Projection

Stereo projection is just two perspective projections :

object

Left eye

Right eye

z

x

r

r

l

l

display screen

Max effect at 50cm

E

E


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Stereo Projection

E is the interocular separation, typically 2.5cm to 3cm.d is the distance of viewer from display, typically 50cm.

Let 2E = 5cm:

1100

0000

0010

20001

:

1100

0000

0010

20001

:

d

d

eyeRight

d

d

eyeLef t

.10

2:

71.5505

tan2tan

11

anglepreservetodEthus

dEanglestereoo


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View as an Anaglyph (Red/Green)


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Orthographic Projection

1000

0000

0010

0001

orthM

Orthographic Projection onto a plane at z = 0.

xp = x , yp = y , z = 0.


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Implementation of Viewing

Transform into world coordinates.

Perform projection into view volume oreyecoordinates.

Clip geometry outside the view volume.

Perform projection into screen coordinates. Remove hidden lines.


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3D Clipping

For orthographic projection, view volume is

a box.

For perspective projection, view volume is afrustrum.

Far c lipping plane.

Near clipping plane

left

right

Need to calculate intersection

With 6 planes.


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Speeding up Projection

Can always place camera at origin !

OpenGL , Renderman.

Much easier to clip lines to unit cube

Define canonicalview volumes.

Define normalising transformations that project

view volume into canonical view volume.

Clip, transform to screen coordinates.


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3D Clipping

Can use Cohen-Sutherland algorithm.

Now 6-bit outcode.

Trivial acceptance where both endpointoutcodes are all zero.

Perform logical AND, reject if non-zero.

Find intersect with a bounding plane and addthe two new lines to the line queue.

Line-primitive algorithm.


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3D Polygon Clipping

Sutherland-Hodgman extends easily to 3D.

Call CLIP procedure 6 times rather than 4

Polygon-primitive algorithm.


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Sutherland-Hodgman Algorithm

Four cases of polygon clipping :

Inside Outside Inside Outside Inside Outside Inside Outside

Case 3

No

output.Case 1

Output

Vertex

Case 2.

Output

Intersection

Case 4

Second

Output

First

Output


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Clipping and Homogeneous

Coordinates

Efficient to transform frustrum into perspective

canonical view volumeunit slope planes. Even better to transform to parallel canonical view

volume

Clipping must be done in homogeneous coordinates.

Points can appear withve W and cannot be

clipped properly in 3D.


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Transforming between canonical

view-volumes

The perspective canonical view-volume can be transformedto the parallel canonical view-volume with the following

matrix:

0100)1()1(

1

00

0010

0001

]1,[

d

d

d

TthendzIf ppcv

Derivation left as an exercise :-)


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Clipping in 3D.

3D parallel projection volume is defined by:

-1x1, -1 y1 , -1 z0

Replace by X/W,Y/W,Z/W:

-1X/W1, -1 Y/W1 , -1 Z/W0

Corresponding plane equations are :

X= -W, X=W, Y=-W, Y=W, Z=-W, Z=0


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Clipping in W

If W>0 , multiplication by W doesnt

change sign.

W>0: -W X W, -W Y W, -W Z 0

However if W


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Points in Homogeneous

Coordinates

X or Y

W

W=1

P1=[1 2 3 4]T

P2=[-1 2 3 4]T

Projection of P1 and P2 onto W=1 plane


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Points in Homogeneous Coordinates

X or Y

W

W=1

P1=[1 2 3 4]T

P2=[-1 2 3 4]T

Projection of P1 and P2 onto W=1 planeRegion A

Region B

Need to consider both regions when

Performing clipping.


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Clipping in Homogeneous

Coordinates

Could clip twiceonce for region B, once

for region A.Expensive.

Check for negative W values and negate

points before clipping. What about lines with endpoints in each

region ?


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Lines in Homogeneous Coordinates

X

WP1

P2

W=1

Projection of points.


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Lines in Homogeneous Coordinates

X

WP1

P2

W=1

W=-X

W=XClipped lines

Solution : Clip linesegments, negate both

endpoints, clip again.


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Final Step

Divide by W to get back to 3-D

Divide by z.

Where the perspective projection actually getsdoneperspective foreshortening.

Now we have a view volume.

Dont flatten z due to hidden line calculations.


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Projections in Java

Use Java 3D package

Or can DIY!


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Projections - Summary

Orthographic matrix - replace (z) axis withpoint.

Perspective matrixmultiply w by z.Clip in homogeneous coordinates.

Preserve z for hidden surface calculations.

Can find number of vanishing points

Acknowledgments - thanks to Eric McKenzie, Edinburgh, from whoseGraphics Course some of these slides were adapted.

Documents

Lect25 Projections 2