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Viewing/Projections I Week 3, Fri Jan 25

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http://www.ugrad.cs.ubc.ca/~cs314/Vjan2008

Viewing/Projections IWeek 3, Fri Jan 25

- rotate/translate/scale difficult to control
- arbitrary viewing position
- eye point, gaze/lookat direction, up vector

y

lookat

x

Pref

WCS

view

up

z

eye

Peye

y

lookat

x

Pref

WCS

view

v

VCS

up

z

eye

Peye

u

w

- translate eye to origin
- rotate view vector (lookat â€“ eye) to w axis
- rotate around w to bring up into vw-plane

- ingredients
- box, film, hole punch

- result
- picture

www.kodak.com

www.pinhole.org

www.debevec.org/Pinhole

- theoretical perfect pinhole
- light shining through tiny hole into dark space yields upside-down picture

one ray

of projection

perfect

pinhole

film plane

- non-zero sized hole
- blur: rays hit multiple points on film plane

multiple rays

of projection

actual

pinhole

film plane

- pinhole camera has small aperture (lens opening)
- minimize blur

- problem: hard to get enough light to expose the film
- solution: lens
- permits larger apertures
- permits changing distance to film plane without actually moving it
- cost: limited depth of field where image is in focus

aperture

lens

depth

of

field

http://en.wikipedia.org/wiki/Image:DOF-ShallowDepthofField.jpg

- real pinhole camera: image inverted

eye

point

image

plane

- computer graphics camera: convenient equivalent

eye

point

center of

projection

image

plane

- image plane need not be perpendicular to view plane

eye

point

image

plane

eye

point

image

plane

- our camera must model perspective

- our camera must model perspective

- planar geometric projections
- planar: onto a plane
- geometric: using straight lines
- projections: 3D -> 2D

- aka projective mappings
- counterexamples?

- properties
- lines mapped to lines and triangles to triangles
- parallel lines do NOT remain parallel
- e.g. rails vanishing at infinity

- affine combinations are NOT preserved
- e.g. center of a line does not map to center of projected line (perspective foreshortening)

-z

- project all geometry
- through common center of projection (eye point)
- onto an image plane

x

y

z

z

x

x

projectionplane

center of projection

(eye point)

how tall shouldthis bunny be?

- nonuniform foreshortening
- not affine

similar triangles

P(x,y,z)

y

P(xâ€™,yâ€™,zâ€™)

z

zâ€™=d

but

- desired result for a point [x, y, z, 1]T projected onto the view plane:
- what could a matrix look like to do this?

is homogenized version of

where w = z/d

is homogenized version of

where w = z/d

- expressible with 4x4 homogeneous matrix
- use previously untouched bottom row

- perspective projection is irreversible
- many 3D points can be mapped to same (x, y, d) on the projection plane
- no way to retrieve the unique z values

- as COP moves away, lines approach parallel
- when COP at infinity, orthographic view

- cameraâ€™s back plane parallel to lens
- infinite focal length
- no perspective convergence
- just throw away z values

- transformation of space
- center of projection moves to infinity
- view volume transformed
- from frustum (truncated pyramid) to parallelepiped (box)

x

x

Frustum

Parallelepiped

-z

-z

orthographic view volume

perspective view volume

y=top

y=top

x=left

x=left

y

y

z

x=right

VCS

z=-near

x

VCS

y=bottom

z=-far

z=-far

x

y=bottom

x=right

z=-near

- specifies field-of-view, used for clipping
- restricts domain of z stored for visibility test

z

- standardized viewing volume representation
perspective orthographic

orthogonal

parallel

x or y

x or y = +/- z

backplane

x or y

backplane

1

frontplane

frontplane

-z

-z

-1

-1

- permits standardization
- clipping
- easier to determine if an arbitrary point is enclosed in volume with canonical view volume vs. clipping to six arbitrary planes

- rendering
- projection and rasterization algorithms can be reused

- clipping

- convention
- viewing frustum mapped to specific parallelepiped
- Normalized Device Coordinates (NDC)
- same as clipping coords

- only objects inside the parallelepiped get rendered
- which parallelepiped?
- depends on rendering system

- viewing frustum mapped to specific parallelepiped

left/right x =+/- 1, top/bottom y =+/- 1, near/far z =+/- 1

NDC

Camera coordinates

x

x

x=1

right

Frustum

-z

z

left

x= -1

z=1

z= -1

z=-n

z=-f

- z axis flip changes coord system handedness
- RHS before projection (eye/view coords)
- LHS after projection (clip, norm device coords)

VCS

NDCS

y=top

y

(1,1,1)

x=left

y

z

(-1,-1,-1)

z

x

x=right

x

z=-far

y=bottom

z=-near

perspective view volume

y=top

x=left

y

z=-near

VCS

y=bottom

z=-far

x

x=right

near, far always positive in OpenGL calls

glOrtho(left,right,bot,top,near,far);

glFrustum(left,right,bot,top,near,far);

glPerspective(fovy,aspect,near,far);

orthographic view volume

y=top

x=left

y

z

x=right

VCS

x

z=-far

y=bottom

z=-near

- why near and far plane?
- near plane:
- avoid singularity (division by zero, or very small numbers)

- far plane:
- store depth in fixed-point representation (integer), thus have to have fixed range of values (0â€¦1)
- avoid/reduce numerical precision artifacts for distant objects

- near plane:

NDCS

y

(1,1,1)

z

(-1,-1,-1)

x

- scale, translate, reflect for new coord sys

VCS

y=top

x=left

y

z

x=right

x

z=-far

y=bottom

z=-near

NDCS

y

(1,1,1)

z

(-1,-1,-1)

x

- scale, translate, reflect for new coord sys

VCS

y=top

x=left

y

z

x=right

x

z=-far

y=bottom

z=-near

- scale, translate, reflect for new coord sys

- scale, translate, reflect for new coord sys

VCS

y=top

x=left

y

z

x=right

x

z=-far

y=bottom

z=-near

same idea for right/left, far/near

- scale, translate, reflect for new coord sys

- scale, translate, reflect for new coord sys

- scale, translate, reflect for new coord sys

- scale, translate, reflect for new coord sys

glMatrixMode(GL_PROJECTION);

glLoadIdentity();

glOrtho(left,right,bot,top,near,far);