Introduction to Computer GraphicsSoftware Rasterization Review

Colors

Red Channel

Green Channel

Blue Channel

Alpha Channel

1.00f 0.24f 0.29f 0.30f

Add and scale are applied to channels

,, ,

Rasters

0

50

50

0xFF3D4A4C

R:24%A:100%

G:29%B:30%

Linear Combination or Blend

A

.75A+.25B.75A

.25B

.75 + .25 = 1

B

Barycentric Coordinates

A

B C .8 .1 .1+ + = 1

λ1 λ2 λ3+ + = 1and

0 ≤ λx ≤ 1.8A

.1B

.1C

Barycentric Coordinates

x = λ1x1 + λ2x2 + λ3x3y = λ1y1 + λ2y2 + λ3y3

λ1 + λ2 + λ3 = 1

Barycentric Coordinates

λ1 = (y2 - y3)(x - x3) + (x3 - x2)(y - y3)(y2 - y3)(x1 - x3) + (x3 - x2)(y1 - y3)

λ2 = (y3 - y1)(x - x3) + (x1 - x3)(y - y3)(y2 - y3)(x1 - x3) + (x3 - x2)(y1 - y3)

λ3 = 1 - λ1- λ2

A

B C

Interpolationλ1 λ2 λ3+ + = 1

and0 ≤ λx ≤ 1

• λ1 + • λ2 + • λ3 + =

2D Transformations

X

Y

X

Y

XY

yx0

1 tytx1

00 0 1 1

yx0

sy 00sx

00 0 1 1

yx

00

0 0 1 1

cosθ -sinθsinθ cosθ

3D Transformations All Points

X

Y

Z

X

Y

Z

X

Y

Z

xf = xo + txyf = yo + tyzf = zo + tz

xf = xo • sxyf = yo • syzf = zo • sz

More Complex

3D Transformations

X

Y

XY

All Points

X

Y

Z

Z

Z

01 0

0100 0 1

tytx

tz0 0 0 1

yx

z1 0

sy 00sx

00 0 sz

00

00 0 0 1

yx

z1

mm m

mmmm m m

00

00 0 0 1

yx

z1

M = vector rotate matrix

X

Y

Z

00

0 0 1

cosθ -sinθsinθ cosθ

0001000

Rotation About Z-Axis

X

Y

Z

X

Y

Z

Rotation About X-Axis

Rotation About Y-Axis

-sinθ0

0 sinθ cosθ

1 00 cosθ

0001000

0sinθ

-sinθ 0 cosθ

cosθ 00 1

0001000

3D Transformations

3D Transformations

Measure α thenRotate by α thenMeasure β thenRotate by β thenRotate by θ thenRotate by -β thenRotate by -α

X

β

αY

Z

θ

X

Z

Y

a

θ

00

0 0 1

cos-α-sin-αsin-αcos-α

0001000

-sin-β0

0 sin-βcos-β

1 00 cos-β

0001000

M =

00

0 0 1

cosθ -sinθsinθ cosθ

0001000

-sinβ0

0 sinβ cosβ

1 00 cosβ

0001000

00

0 0 1

cosα -sinαsinα cosα

0001000

3D Transformations

T RSV = TRSV

R =

V = SVS

SV RSV

TRS V = TRSV

ST R TRS=10,000 Times

Notice the ordering!

10,000 Times

Either This

And This

3D TransformationsOr This1 Time

Projection - Perspectivey’ = y

z

z

?? ?

???0 0 1

??

00 0 0 1

yx

z1

x’ = xz

y/zx/z

z1

=

No possible matrix, because we don’t have z!

yx

zw

v = y/wx/w

z/ww/w

v’ =

zProjection - Perspective

Projection - Perspectivey’ = y

z

z

x’ = xz

01 0

0100 0 1

00

00 0 1 0

yx

z1

yx

zz

=

With homogeneous divide, just copy z to w

y/zx/z

z/zz/z

v’ = yx

zz

v =

Projection - Perspective

frustum(l,r,b,t,n,f)

void perspective(float fovy, float aspect, float zNear, float zFar){ float xmin, xmax, ymin, ymax;

ymax = zNear * tanf(fovy * M_PI / 360.0f); ymin = -ymax; xmin = ymin * aspect; xmax = ymax * aspect;

frustum(xmin, xmax, ymin, ymax, zNear, zFar);}

Z-Copy and Right Hand Rule

Other adjustments due to definition of frustum

Projection - Perspective

frustum(l,r,b,t,n,f)

01/tan(v/2) 0

000 0 -(f+n)/(f-n)

00

-2nf/(f-n)0 0 -1 0

scales near plane to -1 → 1 in x,y

v = fovYa = aspectn = nearf = farcopy Z

scales and biases z to -1 → 1

1/(tan(v/2)a)

Depth Buffer

for pixels in triangle calculate λ1,λ2,λ3 if λ1>0 & λ2>0 & λ3>0 interpolate z-value if z-value < z-buffer update z-buffer interpolate color store color in raster

User positioned at (eyeX,eyeY,eyeZ)

User looks at (spotX, spotY, spotZ)

Any direction can be “up”

User looks down look vector

There exists a vector describing the direction “right”

up

lookrighteye

look = spot - eyeright = look x upup = right x look

look, right, up are normalized

Camera Matrix

Camera Matrix

rightyupy upz

rightzrightxupx 0

0

00 0 0 1

Translate(-eyex, -eyey, -eyez)Translate Camera to Origin

M = BT

B =

T =

-lookx -looky -lookz

up-lookright eye

Change of Coordinate System Matrix

Back-Face Culling

Model View Matrix

Projection Matrix

Viewport Transform

Model Coordinates

EyeCoordinates

CanonicalCoordinates

RasterCoordinates

Homogeneous Divideif (n • tcenter > 0)

cull

Back-Face CullingObjects drawn with back-face culling need to be solid

Lighting

l

P

l = L P- n

vv = V P-

V

r

lr = 2(l•n)n - l

n

-l2(l•n)n

2(l•n) = 2|l||n|cosθ

r

Lighting

Ia = ambient intensity

Id = diffuse intensity

Is = specular intensity

nl •Id = ˆ ˆLdkdIa = Laka

v •Is ≃ ˆks r̂ )( αLs

ka= material ambient coefficient

ks = material specular coefficient

kd= material diffuse coefficient

= shininessα

La= light ambient intensity

Ls= light specular intensity

Ld= light diffuse intensity

Ldr Ldg Ldb, , )light diffuse color=

kdr kdg kdb, , )= material diffuse color

(

(

v

r

l

n

l = L P-v = V P-

r = 2(l•n)n - l

LightingI = IsIa Id+ +∑ ( )

lights

Only sum diffuse color if l • n is positive

Only sum specular color if l • n and v • r are positive

Normals supplied per vertexNormals interpolated across faceColor & lighting calculated per pixel

Where to apply the lighting model?

Bui Tuong Phong

Lighting

AB

C

Texture Coordinatesλ1 λ2 λ3+ + = 1

and0 ≤ λx ≤ 1

• λ1 + • λ2 + • λ3 + =

(0.6,0.2)

(0.6,0.9)

(0.15,0.1)(0.15,0.1)

(0.6,0.2)

(0.6,0.9)

(0.5,0.6)

Bump Mapping

Normals supplied per pixelNormals obtained from bump mapColor & lighting calculated per pixel

Where to apply the lighting model?