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Doug James’ CG Slides, Rich Riesenfeld’s CG Slides,

Shirley, Fundamentals of Computer Graphics, Chap 7

Wen-Chieh (Steve) Lin

Institute of Multimedia Engineering

Given geometry in the world coordinate system, how do we get it to the display?

- Transform to camera coordinate system
- Transform (warp) into canonical view volume
- Clip
- Project to display coordinates
- Rasterize

ILE5014 Computer Graphics 10F

ILE5014 Computer Graphics 10F

glMatrixMode(GL_MODELVIEW)

glMatrixMode(GL_PROJECTION)

glViewport(…)

ILE5014 Computer Graphics 10F

glMatrixMode(GL_MODELVIEW)

gluLookAt(…)

glMatrixMode(GL_PROJECTION)

glFrustrum(…)

gluPerspective(…)

glOrtho(…)

glViewport(x,y,width,height)

ILE5014 Computer Graphics 10F

- Our eyes collapse 3-D world to 2-D retinal image (brain then has to reconstruct 3D)
- In CG, this process occurs by projection
- Projection has two parts:
- Viewing transformations: camera position and direction
- Perspective/orthographic transformation: reduces 3-D to 2-D

- Use homogeneous transformations

ILE5014 Computer Graphics 10F

F

P

- Stand at point P, and look through the hole—anything within the cone is visible, and nothing else is
- Reduce the hole to a point - the cone becomes a ray
- Pin hole is the focal point, eye point or center of projection

ILE5014 Computer Graphics 10F

Image

W

F

I

World

- View planeorimage plane- a plane behind the pinhole on which the image is formed
- sees anything on the line (ray) through the pinhole F
- a point W projects along the ray through F to appear at I (intersection of WF with image plane)

ILE5014 Computer Graphics 10F

Image

W

Image

F

I

F

World

World

- Projecting a shape
- project each point onto the image plane
- lines are projected by projecting end points only

Camera lens

Note: Since we don't want the image to be inverted, from now on we'll put F behind the image plane.

ILE5014 Computer Graphics 10F

World

Image

F

- When the focal point is at infinity the rays are parallel and orthogonal to the image plane
- When xy-plane is the image plane (x,y,z) -> (x,y,0) front orthographic view

ILE5014 Computer Graphics 10F

- Good model for CAD and architecture. No perspective effects.

front

side

top

ILE5014 Computer Graphics 10F

- Let’s start with a simple case where the camera is put at the origin, and looks along the negative z-axis!
- Simple “canonical” views:

orthographic projection

perspective projection

ILE5014 Computer Graphics 10F

- (l,b,n) = (left, bottom, near)
- (r,t,f) = (right, top, far)

ILE5014 Computer Graphics 10F

- Map orthographic viewing cube to the canonical view volume
- 3D window transformation
- [l, r] x [b, t] x [f, n] [-1, 1]x[-1, 1]x[-1,1]

ILE5014 Computer Graphics 10F

- 3D window transform (last class)
- [l, r] x [b, t] x [f, n] [-1, 1]x[-1, 1]x[-1,1]
- zcanonical is ignored

ILE5014 Computer Graphics 10F

y

(1,1)

x

x

(nx, ny)

(-1,-1)

y

Screen

Image Plane

ILE5014 Computer Graphics 10F

- If y-axis of screen coord. points downward
- [-1,1] x [-1, 1] [-0.5, nx-0.5] x [ny-0.5, -0.5]

translation

scale

reflection

ILE5014 Computer Graphics 10F

- Put everything together

ILE5014 Computer Graphics 10F

- What if we want the camera somewhere other than the canonical location?
- Alternative #1: derive a general projection matrix. (hard)
- Alternative #2: transform the world so that the camera is in canonical position and orientation (much simpler)

ILE5014 Computer Graphics 10F

- All we need is a single translation and angle-axis rotation (orientation), but...
- Good animation requires good camera control--we need better control knobs
- Translation knob - move to the lookfrom point
- Orientation can be specified in several ways:
- specify camera rotations
- specify a lookat point (solve for camera rotations)

ILE5014 Computer Graphics 10F

- Focal length, image size/shape and clipping planes are in the perspective transformation
- In addition:
- lookfrom: where the focal point (camera) is
- lookat:the world point to be centered in the image

- Also specify camera orientation about the lookfrom-lookfat axis

ILE5014 Computer Graphics 10F

- Translate by lookfrom, bring focal point to origin
- Rotate lookfrom - lookat to the z-axis with matrix R:
- w = (lookfrom-lookat) (normalized) and z = [0,0,1]
- rotation axis: a = (w × z)/|w × z|
- rotation angle: cosθ = w•z and sinθ = |w × z|

- Rotate about z-axis to get vup parallel to the y-axis

ILE5014 Computer Graphics 10F

y

x

vup

y

z

x

y

lookfrom

x

z

START HERE

z

w

Translate LOOKFROM

to the origin

Rotate the view vector

(lookfrom - lookat) onto

the z-axis.

y

x

Multiply by the projection matrix and everything will be in the canonical camera position

z

Rotate about z to bring vup to y-axis

ILE5014 Computer Graphics 10F

- Let e=(ex, ey, ez) be the “lookfrom” position

y

x

vup

y

z

x

lookfrom

z

START HERE

w

Translate LOOKFROM

to the origin

ILE5014 Computer Graphics 10F

- You can derive these two rotation matrices based on what you learned in the last class

y

x

y

x

z

z

Translate LOOKFROM

to the origin

Rotate the view vector

(lookat -lookfrom) onto

the z-axis.

y

x

z

Rotate about z to bring vup to y-axis

ILE5014 Computer Graphics 10F

- Alternatively, we can view these two rotations as a single rotation that aligns u-v-w-axes to x-y-z-axes!

v

y

x

z

w

w: lookat – lookfrom

v: view up direction

u = v x w

ILE5014 Computer Graphics 10F

- x, y, u, v, o, e are all vectors in global system

In global coord. (xp,yp)

In local coord. (up,vp)

ILE5014 Computer Graphics 10F

- Given two coordinate frames, how do you represent a vector specified in one frame in the other frame?

?

w

v

u

Z

P

e

Y

?

X

ILE5014 Computer Graphics 10F

- Put translation and rotation together

w

v

Z

u

P

Y

e

local coordinate

world coordinate

X

ILE5014 Computer Graphics 10F

Given 3D geometry (a set of points a)

- Compute view transformation Mv
- Compute orthographic projection Mo
- Compute M = MoMv
- For each point ai, compute p = Mai

ILE5014 Computer Graphics 10F

- Canonical case:
- camera looks along the z-axis (toward negative z-axis)
- focal point is the origin
- image plane is parallel to the xy-plane at distance d
- We call d the focal length, mainly for historical reasons

Image plane

Center of projection

ILE5014 Computer Graphics 10F

view plane

y

ys

g

e

d

z

- Diagram shows y-coordinate, x-coordinate is similar
- Point (x,y,z) projects to

e: eye position

g: gaze direction

ILE5014 Computer Graphics 10F

- Projection using homogeneous coordinates:
- transform (x,y,z) to

- 2-D image point:
- discard third coordinate
- apply viewport transformation to obtain physical pixel coordinates

Divide by 4th coordinate

(the “w” coordinate)

ILE5014 Computer Graphics 10F

- Pyramid in space defined by focal point and window in the image plane (assume window mapped to viewport)
- Defines visible region of space
- Pyramid edges are clipping planes

ILE5014 Computer Graphics 10F

- Truncated pyramid with near and far clipping planes
- Why far plane? Allows z to be scaled to a limited fixed-point value (z-buffering)
- Why near plane? Prevent points behind the camera being seen

ILE5014 Computer Graphics 10F

- Rather than derive a different projection matrix for each type of projection, we can convert all projections to orthogonal projections with the default view volume
- This strategy allows us to use standard transformations in the pipeline and makes for efficient clipping

ILE5014 Computer Graphics 10F

- After the view transformation, a simple projection and viewport transformation can generate screen coordinate.
- However, projecting all vertices are usually unnecessary.
- Clipping with 3D volume.
- Associating projection with clipping and view volume normalization.

ILE5014 Computer Graphics 10F

- Solution: transform frustum to a cube before clipping

- Converts perspective frustum to orthographic frustum
- This is yet another homogeneous transform!

ILE5014 Computer Graphics 10F

- We already know how to map orthographic view volume to canonical view volume
- Set view plane at the near plane

ILE5014 Computer Graphics 10F

- Map [x,y,n] to [x, y, n]
- Map [xf/n,yf/n,f] to [x, y, f]

ILE5014 Computer Graphics 10F

- Mp is not unique

ILE5014 Computer Graphics 10F

- Lines and planes are preserved
- Parallel lines (not parallel to the projection plan) won’t be parallel after transform
- vanishing point: parallel lines intersect at the vanishing point

vanishing point

ILE5014 Computer Graphics 10F

Given 3D geometry (a set of points a)

- Compute view transformation Mv
- Compute orthographic projection Mo
- Compute M = MoMv
- For each point ai, compute p = Mai

ILE5014 Computer Graphics 10F

Given 3D geometry (a set of points a)

- Compute view transformation Mv
- Map perspective to orthographic Mp
- Compute orthographic projection Mo
- Compute M = MoMpMv
- For each point ai, compute p = Mai

ILE5014 Computer Graphics 10F