viewing and projection - computer science and...

Viewing and ProjectionViewing and Projection

The topics

• Interior parameters• Projection type• Field of view• Clipping• Frustum• Frustum…

• Exterior parametersp• Camera position• Camera orientation• Camera orientation

Transformation PipelineLocal coordinate

Local‐>World

World coordinate

World >Eye

ModelViewMatrixWorld‐>Eye

Eye coordinate

Projection Matrix

Cli di t

others

Clip coordinate

Screen coordinate

Projection

• The projection transforms a point from a high‐dimensional space to a low dimensional spacedimensional space to a low‐dimensional space.

• In 3D, the projection means mapping a 3DIn 3D, the projection means mapping a 3D point onto a 2D projection plane (or called image plane).

• There are two basic projection types: • Parallel: orthographic, oblique• Perspective

Orthographic ProjectionImage Plane

Direction of Projectionz-axis

z=k

xy

1 0 0 00 1 0 0

xy

k1

0 0 0 k0 0 0 1

z1

Orthographic Projection

Oblique Projection

Image Plane

Direction of Projection

Properties of Parallel Projection

• Definition: projection directions are parallel.

• Doesn’t look real.

• Can preserve parallel lines

ProjectionProjection

P ll l i 3D P ll l i 2DParallel in 3D Parallel in 2D




• Can preserve parallel lines• Can preserve ratiosp

t t 'Projection

st

s 's : t s ' : t 's : t s : t




• Can preserve parallel lines• Can preserve ratiosp• CANNOT preserve angles

Projection




• Can preserve parallel lines• Can preserve ratiosp• CANNOT preserve angles

• Often used in CAD, architecture drawings, when images can be used for measurement.when images can be used for measurement.


• No foreshortening

Image PlaneImage Plane

Perspective Projection

• Perspective projection has foreshortening:

Center of Projection

Image Plane

Perspective Projection• Images are mapped onto the image plane in different ways:different ways:

An imageAn image(640*640)


Image Plane

Projection

Image Plane

Perspective Projection• Angle of view tells us the mapping from the image to the image plane:image to the image plane:

Angle of viewField of view


Image Plane

Projection

Image Plane

Perspective Projection• Not everything will be displayed.

Frustum


Image Plane

j

Near Plane Far Plane

Perspective Projection• The frustum of perspective projection looks like:like:


Image Plane

Near PlaneNear Plane

Ortho Projection• What’s the frustum of orthographic projection?

Image Plane

Near FarNear Far

Homogenous Coordinates

• In general, the homogeneous coordinate system is define as:is define as:

wxwy

xy

I h wz

w

z1

In 3DIn homogeneousspace

• For example,

2

4

214

428

0.5 1

Homogenous Coordinates

• Both homogenous vectors and matrices are l blscalable:

wxwy

xy

y

wzw

yz1

wm00 wm01 wm02 wm03

wm10 wm11 wm12 wm13

m00 m01 m02 m03

m10 m11 m12 m13

10 11 12 13

wm20 wm21 wm22 wm23

wm30 wm31 wm32 wm33

10 11 12 13

m20 m21 m22 m23

m30 m31 m32 m33

Perspective Projection• Derivation:

x ' xyz

xy 'd1

1

1

Z Axis d

Image PlaneImage PlaneXY Plane

Perspective Projection• Derivation:

xd / z xyz

xd / zyd / zd1

1

1

Z Axis d

Image PlaneImage PlaneXY Plane

Perspective Matrix• Derivation:

x

xd / z

xyz

xd / zyd / zd

How?

1

d1

How?

xd / zyd / z

dd

xy

y

d1

dd1

yz1

The z axis now becomes useless…

OpenGL Perspective Matrix• In practice, OpenGL uses the z axis for depth test Its matrix looks like this:test. Its matrix looks like this:

d

x

x / z

dd

a b

yz

y / za b / z

1 1

1

The depth value now can be used for depth testThe depth value now can be used for depth test.We will discuss this in more details later...

Vanishing Point• Given a ray:

px

p tn x

py

pz

1

nx

ny

px tnx

py tny

pz tnz

1 nz

1

1

• Its projection: d(px tnx )

d px tnx

d

px tnx

d(py tny )

...(p tn )

dd

a b1

x x

py tny

pz tnz

1

d x x

pz tnz

dpy tny

p tn

(pz tnz ) 1 1 pz tnz

Vanishing Point• When t goes to infinity:

p tn n Parallel lines meet ath i hi i

limt

dpx tnx

pz tnz

dpy tny

dnx

nz

dny

the vanishing point.

• What if there is another ray:

dpz tnz

d

nz

y

qx tnx

q tn

qy tny

qz tnz

1

1

Properties of Perspective Projection

• Lines are mapped to lines.P ll l li t i ll l I t d• Parallel lines may not remain parallel. Instead, they may meet at the vanishing point.

• Ratios are not preserved• Ratios are not preserved.

• It has foreshortening effects. So it looks real.

• Distances cannot be directly measured, as in parallel projectionparallel projection.

Basic OpenGL Projection• Everything will be considered in the eye space:

• Geometry objects have been transformed into the eye y j ycoordinate system using the GL_MODELVIEW matrix.

• You define the projection matrix in GL_PROJECTION, also in the eye space.in the eye space.

• OpenGL always assume that the viewing direction is the –z direction.

• OpenGL automatically processes each vertex using GL_PROJECTION:

Aft j ti th f t i t d i t• After projection, the frustum is converted into a canonical view volume ( [‐1, 1] in all coordinates)

OpenGL Orthographic Projection

glOrtho(left,�right,�bottom,�top,�near,�far)X range Y range Z range

(right, top, far)(right, top, far)

(1, 1, 1)( , , )

(left, bottom, near) (‐1, ‐1, ‐1)

OpenGL Orthographic Projection

glOrtho(l,�r,�b,�t,�n,�f)

• Translation so that the center is the origin.• Scaling so that the size becomes (2 2 2)Scaling so that the size becomes (2, 2, 2).

l 2 2 r l 1

r l2

1 t b

2

2r l

2t b

2r l

r lr l

2t b

t bt b

2

1 f n

21

t b2

f n1

t b t b2

f n

f nf n1

1 1 1

OpenGL Perspective ProjectionglFrustum(left,�right,�bottom,�top,�near,�far)

Always positive

(left, right, bottom, top)may not be centered along

Always positive,although it’s facingthe –z direction

Frustum

y gthe axis

Center of ProjectionProjection

Near Plane(Image Plane) Far Plane

OpenGL Perspective ProjectiongluPerspective(fov,�aspect_ratio,�near,�far)

Always positive

ratio=image_width/image_height

Always positive,although it’s facingthe –z direction

Frustumf

top: near*ctan(fov/2)right: near*ratio*ctan(fov/2)


fov

Projection

Near Plane(Image Plane) Far Plane

OpenGL Perspective Projection

glFrustum(…) is less useful than gluPerspective(...).glFrustum(…) is less useful than gluPerspective(...).

But we can still use it for ut we can still use it fordemonstration purpose next.


ProjectionProjection

xd / z

n

x

yd / z

n

nn

n

yz

1

1 1


Projection

xd / zyd / z

nn

xy

in (‐1, 1)yd / z

a b / z1

na b1

yz1

1 1 in (‐n, ‐f), why???


a b / (n) 1 (z n)

xd / z

a b / ( n) 1 (z n)a b / ( f ) 1 (z f )

yd / za b / z

a

f n1

a

f n

b 2 fn

b f n


Projection

(r/n, t/n, 1)

Projection

(l/n, b/n, ‐1)

11

x

1

f nf n

2 fnf n

yz

f n f n

1

1


(r/n, t/n, 1) (1, 1, 1)

Transform

(l/n, b/n, ‐1) (‐1, ‐1, ‐1)

2nr l

2n

1r l2n

t b

2nr l

r lr l

2n t b

2n

t b1

1

1t b2n

11

2nt b

t bt b

11

1 1 1

TranslationScaling


(1, 1, 1)

Projection

(‐1, ‐1, ‐1)

2nr l

r lr l

2 t b

11

2nr l

r lr l

2n t b

2n

t bt bt b

11

f nf n

2 fnf n

1

t b t b

f nf n

2 fnf n

1

1

Original ProjectionTransformation

What happens after projection?• Clipping

• Viewport transformation

clipping

• Viewport transformation(1, 1)

viewport

(800, 800)

• Rasterization(‐1, ‐1)

viewport

(0, 0)

rasterizerasterize

Depth Test• Before rasterization, all processes are done based on vertices.on vertices.

• The z coordinate at each vertex is transformed into a new z value (or called the depth value)a new z value (or called the depth value).

• During rasterization, the z value of each pixel is i t l t d f tiinterpolated from vertices.

• The z value then stored in the depth buffer for• The z value then stored in the depth buffer, for occlusion tests. (smaller z means closer).

Depth Test

• The depth buffer is part of the frame buffer:

bl d bl h d h b ff

glutInitDisplayMode(GLUT_DOUBLE|GLUT_RGB|GLUT_DEPTH);

• To enable or disable the depth buffer:glEnable(GL_DEPTH_TEST);

• Without the depth test the occlusion is

glDisable(GL_DEPTH_TEST);

Without the depth test, the occlusion is determined by the drawing order.

Common Issues

• When you set up the perspective matrix:• Near cannot be zero!• far/near cannot be too large! (far cannot be / g (too large, or near cannot be too small.) Why?

2n r l 2nr l

r lr l

2n t b

t b t b

f nf

2 fnf

f n f n

1

viewing and projection - computer science and...

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