CS5500 Computer GraphicsApril 23, 2007

Today’s Topic• Details of the front-end of the 3D

pipeline:– How to construct the viewing matrix?– How to construct the projection matrix?

• References:– [Ed Angel] Sections 5.3.3, 5.5.1, 5.9.– McMillan’s lecture slideshttp://www.unc.edu/courses/2003spring/comp/236/001/handouts.html

Two Tasks for Today• Deriving the viewing matrix

– e.g., For gluLookAt()• Deriving the projection matrix

– e.g., for glFrustum()

Specifying the View• Eye position• Look-at point• Up direction• Remember gluLookAt(eye, center, up)?• Note that the up vector may not be

orthogonal to the viewing direction (i.e., from eye to look-at)

Defining the Eye Space• We have two vectors: viewing direction and

up vector. Can we set up the three basis vectors for the eye space?

• v: viewing direction – The easy one = (look_at – eye)

• r: right vector– Orthogonal to both v and the up vector

• u: almost like up vector, except:– Orthogonal to both v and r

Inverse = Transpose• For an orthonormal matrix, its inverse

matrix is its transpose.

vur

vurM

vurM

133

33

100010001

Task #2 for Today• Deriving the viewing matrix

– e.g., For gluLookAt()• Deriving the projection matrix

– e.g., for glFrustum()

Simple PerspectiveConsider a simple perspective with the COP at

the origin, the near clipping plane at z = -1, and a 90 degree field of view determined by the planes

x = z, y = z

Simple Perspective

• After division by w,• Sometimes, we write is as:

10100000000100001

0 zyx

z

yx

x’ =zx y’ =

zy

10100000000100001

0 zyx

w

ywxw

Perspective in OpenGL• glFrustum( left, right,

bottom, top, near, far )

• gluPerpective( FOV_vertical, aspect_ratio,

near, far )

Scaling & Translation in X,Y

• Find Sx, Sy, Tx, Ty, so that:– (left, bottom, near, 1) (-1, -1, -1, 1)– (right, top, near, 1) (1, 1, -1, 1)

010001000000

0100010000100001

10000100

0000

yy

xx

yy

xx

TSTS

TSTS

The Z Component• So far, we have ignored the Z

coordinate.• We want to convert Z so that the range

of [near, far] becomes [-1, 1]• Note that this is NOT a “uniform”

scaling. We will see why after a few slides.

Now let’s look at the Z more carefully…

Range of Z• If Z = near, what is Z’?

-1• If Z = far, what is Z’?

1• Does Z’ change linearly with Z?

– No!– Let a= b=– Z’ = Zw / w = (a*Z+b) / Z = a + b/Z

nearfarnearfar

nearfarnearfar

2

Why Not Linear?• To make it linear, we will have to make

WZ’ = a*Z2 + bZ (so that Z’ = WZ’/W = a*Z + b)

• But that’s impossible with the 4x4 perspective matrix…

Z Resolution• Since screen Z’ is expressed in the form

of a+b/Z, most of the Z resolution is used up by the Z’s closer to the near plane.

• So, what does this mean?• You should NOT set zNear to be very

close to the eye position.

Now, some more math…

Transformation of Normals• Transformation does not necessarily

preserve the normal vectors.– If a.b=0, does T(a).T(b)=0 also?

• For example: what happen if we scale (X, Y) by (0.5, 1.0) in a 2D image?

We shouldn’t transform the two end points of a normal vector.

What we should do is to transform (three points of) the plane first, then find its normal.

What does that mean in math?

(See Appendix F of the OpenGL red book.)

Transformation of Normals• (Foley/vanDam pages 216-217)

– NT.P = 0 but is (MN)T.MP=0? Not always!!– Let (QN)T.MP=0 (i.e., transform P first, then try

to find its normal)– NTQTMP=0– So QTM=I QT= M-1 or Q=(M-1)T

• Special case when M-1=MT

– If M consists of only the composition of rotation, translation, and uniform scaling.

– Q=M