10. 3D Graphics
10.1 Introduction
10 2 B
10.2
Basics
i off 3D G
Graphics
hi
10.3 Hierarchical Modeling
10.4 Projection
10 5 Visibility
10.5
10.6 Illumination
10 7 T
10.7
Texturing
t i
Thomas Ertl
Graphical-Interactive Systems WS 10/11
1
Course Structure
Introduction
Human cognition and perception (aspects of human actions and perception)
Color perception and color models
Graphical
p
output
p devices
Raster graphics
2D graphics
Interaction
GUI: foundation, basics
GUI: toolkits,, development
p
Usability engineering
3D computer
t graphics:
hi
objects, transformations, visibility, illumination, shading
Thomas Ertl
Graphical-Interactive Systems WS 10/11
2
10. 3D Graphics
10.1 Introduction
10 2 B
10.2
Basics
i off 3D G
Graphics
hi
10.3 Hierarchical Modeling
10.4 Projection
10 5 Visibility
10.5
10.6 Illumination
10 7 T
10.7
Texturing
t i
Thomas Ertl
Graphical-Interactive Systems WS 10/11
3
10.1 Introduction
3D computer graphics
Thomas Ertl
Graphical-Interactive Systems WS 10/11
4
10.1 Introduction
3D computer graphics
• 3D objects: surface modeling
modeling, freeform surfaces vs
vs.
polygons
• Curves and surfaces: Polynomials,
Polynomials Bezier
Bezier, de Casteljau
• Triangle meshes: indexed face sets, edge lists, triangle
strips
• Pipeline of geometric transformations:
p j
, vertex transformation byy homogenous
g
affine-projective,
4x4 matrices, transformation of normals
• Hierarchical modeling: matrix stacks, scene graphs
Thomas Ertl
Graphical-Interactive Systems WS 10/11
5
10.1 Introduction
3D computer graphics
• Projection
– Perspective division, homogeneous coordinates
– Mapping in direction of the projection: numerical problems and
near/far clipping planes, field-of-view
– Normalization transformation: view frustum in a unit cube
– Clipping in homogeneous coordinates
– Classification of projections:
perspective - parallel (orthogonal, oblique)
Thomas Ertl
Graphical-Interactive Systems WS 10/11
6
10.1 Introduction
3D computer graphics
• Visibility
– Culling vs. image-space methods, backface culling
– Depth sorting, painter's
painter s algorithm, BSP trees
– Z-buffer method, scanline interpolation of z values
• Illumination
– Lightsource – object, diffuse/specular reflections
– Illumination models: local – global (ray tracing, radiosity),
physics
h i b
based/software-simulation
d/ ft
i l ti – heuristic/hardwareh i ti /h d
accelerated
– Phong
g model: ambient, diffuse, specular,
p
material colors
– Shading methods: flat, Gouraud, Phong
Thomas Ertl
Graphical-Interactive Systems WS 10/11
7
10.1 Introduction
3D computer graphics
• Textures
–
–
–
–
–
General mapping methods and multi-textures
Texture sources, mapping, dimension
Texture coordinates, parameterization of standard objects
Texture filtering, mipmapping
Perspectively correct interpolation
Thomas Ertl
Graphical-Interactive Systems WS 10/11
8
10. 3D Graphics
10.1 Introduction
10 2 B
10.2
Basics
i off 3D G
Graphics
hi
11 Geometric Modeling
10.3 Hierarchical Modeling
10 4 Projection
10.4
10.5 Visibility
10 6 Ill
10.6
Illumination
i ti
10.7 Texturing
Thomas Ertl
Graphical-Interactive Systems WS 10/11
9
10.2 Basics of 3D Graphics
3D world
with objects
geometry primitive
j
subsystem object
raster
subsystem
color image
• Objects
– Points,
Points curves,
curves surfaces
surfaces, volumes
– Boundary representation (Brep): surface around an object
• Surface modeling
– Freeform surfaces
• Polynomial splines
• For example: Bézier patches, B-splines
– Polygons
• Piecewise
Pi
i linear
li
approximation
i ti
• Problems with non-flat polygons
→ split
p into triangles
g
Thomas Ertl
Graphical-Interactive Systems WS 10/11
10
10.2 Basics of 3D Graphics
Parametric polynomial curves
n + 1 points
Î interpolation polynomials of
degree n, oscillate for high order
spline = several curves with
continuous/smooth connection
piecewise cubic +
smooth connection
approximating
cubic Bézier curve
Thomas Ertl
Graphical-Interactive Systems WS 10/11
11
10.2 Basics of 3D Graphics
3D models
d l in
i computer
t graphics:
hi
triangle
ti
l meshes
h
v2
• Geometry:
coordinates of the nodes (vertices)
1
2
• Topology:
v3
connection of vertices by edges
P1
P2
v1
5
• Data structures
4
3
– Indexed face set: list of nodes + list of indices
V = (v 1,v 2 ,v 3 ,v 4 ) P1 = (1,4,2) P2 = (2,4,3 )
v4
– List of vertices + list of edges
g + indexed vertex indices
V = (v 1,v 2 ,v 3 ,v 4 ) E1 = (v 1,v 2 )
E 2 = (v 2 ,v 3 )
P1 = (E1, E 4 , E5 )
– Complex pointer structures (winged edge)
Thomas Ertl
Graphical-Interactive Systems WS 10/11
12
10.2 Basics of 3D Graphics
Ti
Triangle
l strips
ti
v0
v2
v4
The normal vector
points outward:
vertices are
counter-clockwise
.
v1
v3
v5
( v0
v1
v2
v3
v4
v0
v1
v2
v2
v1
v3
v2
v3
v4
v4
v3
v5
)
= Swap
v5
Thomas Ertl
Graphical-Interactive Systems WS 10/11
13
10.2 Basics of 3D Graphics
G
Graphics
hi pipeline:
i li
geometry
t ttransformation
f
ti
object
coordinates
di t
model
transformation affine
modelview
transform
world
coordinates
view
transformation affine
eye
coordinates
normalization
t
transformation
f
ti
(projective)
Thomas Ertl
Graphical-Interactive Systems WS 10/11
normalized
clip
li
coordinates
clipping
li i
rasterization
t i ti
14
10.2 Basics of 3D Graphics
4x4
4
4 matrices
ti
• Affine transformation in homogeneous coordinates
P ′ = M (4×4 )P
rotation
shearing
scaling
projection
⎛ ⎛.
⎜⎜
⎜ ⎜.
⎜ ⎜.
⎜⎜ ⎝
⎝ (.
. .⎞
⎟
. .⎟
. . ⎟⎠
. .)
⎛.⎞ ⎞
⎜ ⎟⎟
⎜.⎟ ⎟
⎜.⎟ ⎟
⎝ ⎠⎟
1 ⎟⎠
⎛x⎞
⎜ ⎟
⎜y⎟
⎜z⎟
⎜ ⎟
⎜ 1⎟
⎝ ⎠
translation
homogenization
Thomas Ertl
Graphical-Interactive Systems WS 10/11
15
10.2 Basics of 3D Graphics
Example
E
l off a 4x4
4 4 matrix
ti
• Rotation around the z axis
⎛ ⎛ cos ϕ
− sin ϕ
0 ⎞ 0⎞
⎜⎜
⎟
⎟
cos ϕ
0 ⎟ 0⎟
⎜ ⎜ sin ϕ
⎜⎜ 0
⎟ 0⎟
0
1
⎝
⎠
⎟⎟
⎜⎜
0
0
0
1
⎝
⎠
• Column vectors are images of the unit (basis) vectors
• Affine mapping in 3D needs the image of 4 points in
space (tetrahedron)
(t t h d )
• 4 (points) x 3 (coordinates)
= 3 x 3 (matrix) + 1 x 3 (translation)
Thomas Ertl
Graphical-Interactive Systems WS 10/11
16
10.2 Basics of 3D Graphics
T
Transforming
f
i th
the normall vector
t
⎛x⎞
⎜ ⎟
⎜y ⎟
point P = ⎜ ⎟ is on plane E :
z
⎜ ⎟
⎜ 1⎟
⎝ ⎠
⎛a⎞
⎜ ⎟
⎜b⎟
normal n = ⎜ ⎟
c
⎜ ⎟
⎜d ⎟
⎝ ⎠
(
ax + by + cz + d = 0
nT ⋅ P = 0
P′ = M ⋅P
)
n T M −1 ⋅ M P = 0
n T ⋅ M −1 ⋅ (M ⋅ P ) = n ′ T ⋅ P ′ = 0
(
)
P
⎛x⎞
⎜ ⎟
y
(a b c d )⎜⎜ ⎟⎟ = 0
z
⎜ ⎟
⎜ 1⎟
⎝ ⎠
nT
(
⇒ n ′ = M −1
Thomas Ertl
Graphical-Interactive Systems WS 10/11
)
T
⋅n
17
10.2 Basics of 3D Graphics
Transforming the normal vector
• Transformation rule:
(
n ′ = M −1
)
T
⋅n
• Affine
Affi ttransformation
f
ti off normall vectors:
t
– By the transposed inverse matrix
– 3x3 sub-block without translation/projection
Thomas Ertl
Graphical-Interactive Systems WS 10/11
18
10. 3D Graphics
10.1 Introduction
10 2 B
10.2
Basics
i off 3D G
Graphics
hi
10.3 Hierarchical Modeling
10.4 Projection
10 5 Visibility
10.5
10.6 Illumination
10 7 T
10.7
Texturing
t i
Thomas Ertl
Graphical-Interactive Systems WS 10/11
19
10.3 Hierarchical Modeling
y
y
x
Identity
double
chair
x
y
R(180°)
double
x
4 combo
table
y
TS1
TS2
x
double
Thomas Ertl
Graphical-Interactive Systems WS 10/11
TV1 4 combo
TV2 4 combo
TT
room
20
10.3 Hierarchical Modeling
S
Scene
graph
h (DAG = di
directed
t d acyclic
li graph)
h)
room
TV1
TV2
Transformation
of chair n in
world coordinates:
(TV1 R180 TS2)
R
TS1
n
o
I
TS2
Transformation
of chair o in
world coordinates:
TT
table
chair
(TV1 R180 TS1)
Thomas Ertl
Graphical-Interactive Systems WS 10/11
21
10.3 Hierarchical Modeling
Matrix
M
t i stacks
t k
• Efficiency requires the re-use of matrices
– Matrix
M t i stacks
t k
– Top row: current transformation matrix
Load(I)
Push
Mult(TV1)
Push Mult(R180) Push Mult(TS1) Chair
I
I
I TV1
I TV1
I TV1 R
I
I
I TV1
I TV1
I TV1 R
I TV1 R
I TV1
I
I
I TV1
I TV1
I
I
I
Thomas Ertl
Graphical-Interactive Systems WS 10/11
I TV1 R I TV1 R TS1
Pop
...
I TV1 R
22
10. 3D Graphics
10.1 Introduction
10 2 B
10.2
Basics
i off 3D G
Graphics
hi
10.3 Hierarchical Modeling
10.4 Projection
10 5 Visibility
10.5
10.6 Illumination
10 7 T
10.7
Texturing
t i
Thomas Ertl
Graphical-Interactive Systems WS 10/11
23
10.4 Projections
Projection
P
j ti vs. Perspective
P
ti
• Projection: “One dimension is lost” (e.g., straight line Æ point)
• Perspective: “Parallel lines meet in a point”
– Projective mapping not affine,
but any affine mapping is projective
• Weaker invariants
– Not p
preserved: distances,, angles,
g , areas
– Incidence is preserved (membership of a point on a line)
– Cross ratio is preserved:
x 2 − x0 x3 − x1
x 2 − x1 x3 − x0
Thomas Ertl
Graphical-Interactive Systems WS 10/11
24
10.4 Projections
Projection
P
j ti vs. Perspective
P
ti
• Projective geometry in 2D:
– D
Duality
lit off points
i t and
d straight
t i ht lilines
• One straight line through 2 points
• Intersection of 2 straight lines:
one point
– Parallel lines intersect at infinity
– Infinite points lie on infinity line
• In 3D: duality of points and planes
• A projection is defined
f
by the image off five
f
3D points
– 5 x 3 = 4 x 4 –1 elements
– 4x4 matrix – 1 component (homogeneous)
Thomas Ertl
Graphical-Interactive Systems WS 10/11
25
10.4 Projections
2D perspective
ti projection
j ti
• Center of projection
• Projection plane
⎛x ⎞
x′
x
x
" =" ⎜⎜ h ⎟⎟
=
⇒ x′ =
z +1
D z+D
⎝ wh ⎠
D
z
Normallyy z' = 0, but we need z'
to calculate the visibility
z′ =
⎛z ⎞
z
" =" ⎜⎜ h ⎟⎟
z D +1
⎝ wh ⎠
⎛ x h ⎞ ⎛ 1 0 0 ⎞⎛ x ⎞
⎜ ⎟ ⎜
⎟⎜ ⎟
⎜ zh ⎟ = ⎜ 0 1 0 ⎟⎜ z ⎟
⎜ w ⎟ ⎜ 0 1 D 1 ⎟⎜ 1 ⎟
⎝ h⎠ ⎝
⎠⎝ ⎠
Thomas Ertl
Graphical-Interactive Systems WS 10/11
(x, z )
“screen”
(x ′, z′)
x
D
eye
26
10.4 Projections
Perspective projection
•
•
Shearing z in w-direction
Homogeneous matrices
include projections
⎛ x ⎞ ⎛ 1 0 0 ⎞⎛ x ⎞
⎜ ⎟ ⎜
⎟⎜ ⎟
⎜ z ⎟ = ⎜ 0 1 0 ⎟⎜ z ⎟
⎜ w ⎟ ⎜ 0 1 D 1 ⎟⎜ 1 ⎟
⎝ ⎠ ⎝
⎠⎝ ⎠
w
C
B' eye B
1
A'
A
a
⎛ 1 a⎞
⎟⎟
⎜⎜
⎝ 0 1⎠
1
shear
homogeneous
division
C'
–D
0
A: far,
far in front of the eye
B: just before the eye
C: behind the eye
eye
z
Î closer together
Î further to negative values
Î to positive values
Ζ∞
Thomas Ertl
Graphical-Interactive Systems WS 10/11
27
10.4 Projections
• Non-linearity
N
li
it off z transformation
t
f
ti
z‘
0
eye
–D
z
strongly nonlinear
Æ Numerical problem (overflow / underflow)
• Range restriction in z:
– near / far clipping planes
Thomas Ertl
Graphical-Interactive Systems WS 10/11
28
10.4 Projections
E
Example
l off a 2D projection
j ti
y
eye
z
near
(screen)
eye
f
far
y'
–1
1
z'
Thomas Ertl
Graphical-Interactive Systems WS 10/11
29
10.4 Projections
IIn 3D:
3D normalization
li ti ttransformation
f
ti
• View frustum Æ unit cube
y
far
clipping
plane
z
near
x
clipping
plane
eye
• View frustum often specified by
– Field of view (FOV): opening angle
– Aspect ratio (screen width/screen height)
Thomas Ertl
Graphical-Interactive Systems WS 10/11
30
10.4 Projections
Clipping
Cli
i iin homogeneous
h
coordinates
di t
• Advantage: clipping against unit cube is easier than
against view frustum in world/eye coordinates
• But: We must clip in homogeneous coordinates
– Otherwise:
Oth
i
wrong segments
t
– Issue: negative w values
– Lines
Li
th
thatt are contained
t i d iin b
both
th cones
w
e.g. − 1 <
w > 0 and − w < x < w
x
< +1
w
w < 0 and w < x < −w
Thomas Ertl
Graphical-Interactive Systems WS 10/11
x
31
10.4 Projections
Clipping
Cli
i iin homogeneous
h
coordinates
di t
• First (projective) normalization transformation
• Then clipping in (homogeneous)
(
) clip coordinates
• Then (perspective) division by w results in normalized
d i coordinates
device
di t (3D unit
it cube)
b )
• Then projection to screen coordinates
(draw at position x,
x y,
y use z for visibility)
(But: sometimes pixel-based scissoring on modern
graphics hardware)
Thomas Ertl
Graphical-Interactive Systems WS 10/11
32
10.4 Projections
Cli i iin homogeneous
Clipping
h
coordinates
di t
w
A C''
C'
C
B
1
A'
eye
B'
0
z
sshear
ea
correct segment: A´C´
wrong segment: A´C´´
homogeneous
division
view frustum
Thomas Ertl
Graphical-Interactive Systems WS 10/11
33
10.4 Projections
E
Exterior/interior
i /i
i line
li segments
Y
B
A
A'
Z
wrong interior segment
eye
B'
correct exterior segment
screen
Thomas Ertl
Graphical-Interactive Systems WS 10/11
34
10.4 Projections
H
Homogeneous
window-edge
i d
d coordinates
di t
x
> −1 x + w = WECL (P ) > 0
w
x
R : < 1 w − x = WECR (P ) > 0
w
L:
(
(assuming
g
that w > 0)
I t
Intersecti
tion off L(α ) = P1 + α (P2 − P1 )
WECL (L(α )) = x1 + α (x 2 − x1 ) + w 1 + α (w 2 − w 1 ) = 0
⇒α =
w 1 + x1
WECL (P1 )
=
(w1 + x1 ) − (w 2 + x2 ) WECL (P1 ) − WECL (P2 )
Thomas Ertl
Graphical-Interactive Systems WS 10/11
35
10.4 Projections
Standard
St
d d viewing
i i conditions
diti
(e.g.,
(
OpenGL)
O
GL)
• Eye is in origin (eye coordinates)
• View in direction off negative z axis
• Specified by
– E
Eye point
i
– Line of sight
– Up vector
• Model transform + view transform = model-view matrix
Thomas Ertl
Graphical-Interactive Systems WS 10/11
36
10.4 Projections
C
Computing
ti th
the ttransformation
f
ti
matrix:
t i example
l
•
z
In 2D: based on image of four projected points
⎛0⎞
⎛0⎞
⎜ ⎟
⎜ ⎟
R ⎜ 0 ⎟ = k1⎜ − 1⎟
⎜ 1⎟
⎜0⎟
⎝ ⎠
⎝ ⎠
n (0,0 )
eye at −∞
⎛ 0 ⎞
⎛0⎞
⎜
⎟
⎜ ⎟
R ⎜ − n ⎟ = k 2 ⎜ − 1⎟
⎜ 1 ⎟
⎜1⎟
⎝
⎠
⎝ ⎠
(− n,− n )
(n,− n )
(− f ,−f )
⎛ f ⎞
⎛1⎞
⎜ ⎟
⎜ ⎟
R ⎜ − f ⎟ = k 3 ⎜1⎟
⎜ 1⎟
⎜1⎟
⎝ ⎠
⎝ ⎠
⎛ 1⎞
⎛ 1⎞
⎜ ⎟
⎜ ⎟
R⎜ 0 ⎟ = k 4 ⎜ 0 ⎟
⎜0⎟
⎜0⎟
⎝ ⎠
⎝ ⎠
o
q
x
(f ,−f )
p
z‘
(− 1,1)
p
q
x direction
remains the same
o
x‘
(1−
, 1)
n
Thomas Ertl
Graphical-Interactive Systems WS 10/11
37
10.4 Projections
• In
I 2D:
2D 12 equations
ti
ffor 12 unknown
k
values
l
• Goal: 3x3 matrix – 1 and coefficients k1, k2, k3, k4
⎛1
0
⎜
− (f + n )
R = ⎜0
f −n
⎜⎜
−1
0
⎝
0 ⎞
− 2fn ⎟
⎟
f −n ⎟
0 ⎟⎠
• In 3D with asymmetric
y
frustum (left/right,
(
g , top/bottom)
p
)
⎛ 2n
⎜
⎜r −l
⎜ 0
R =⎜
⎜
⎜ 0
⎜
⎝ 0
0
2n
t −b
0
0
Thomas Ertl
Graphical-Interactive Systems WS 10/11
r +l
r −l
t +b
t −b
− (f + n )
f −n
−1
⎞
0 ⎟
⎟
0 ⎟
⎟
− 2fn ⎟
⎟
f −n ⎟
0 ⎠
38
10.4 Projections
Cl
Classification
ifi ti off projections
j ti
Planar geometric
projections
P ll l
Parallel
Orthographic
Top
P
Perspective
ti
Oblique One-point
Cabinet
Axonometric
Cavalier
Front
elevation
Side
elevation
Two-point
Three point
Three-point
Isometric
Dimetric
Trimetric
Thomas Ertl
Graphical-Interactive Systems WS 10/11
39
10.4 Projections
Classification
Cl
ifi ti off projections
j ti
• Perspective projection
– P
Parallel
ll l straight
t i ht lines
li
only
l stay
t parallel
ll l if th
they lilie on a plane
l
parallel to the projection plane
– All the other straight
g lines ((of the same direction)) will intersect
in vanishing point(s) (1,2,3-point perspectives)
• Parallel projection
– All projection “rays” are parallel (i.e., only a single direction of
projection)
– Used for engineering and architecture because they can be
used for measurements
Thomas Ertl
Graphical-Interactive Systems WS 10/11
40
10.4 Projections
Parallel
P
ll l projection
j ti
• Orthographic
– O
Orthogonal
th
l to
t projection
j ti plane
l
– Multi-view orthographic: projection plane
orthogonal
g
or p
parallel to the p
principal
p axes
– Axonometric
multi-view orthographic
• Projection plane is not aligned with
coordinate
di t system
t
• Isometric: same contraction along each
axis,, same angle
g (120°)
(
)
• Dimetric: angles between two of the
principal axes are equal
• Trimetric:
T i t i th
three diff
differentt angles
l
axonometric
DOP: direction of projection
VPN: view plane normal
Thomas Ertl
Graphical-Interactive Systems WS 10/11
41
10.4 Projections
Parallel
P
ll l projection
j ti
• Oblique
– C
Cavalier
li projection
j ti
(45° - no contraction)
– Cabinet projection
p j
(~63° - contraction 1/2)
oblique
DOP: direction of projection
VPN: view plane normal
Thomas Ertl
Graphical-Interactive Systems WS 10/11
42
10.4 Projections
P
Parallel
ll l projections
j ti
Thomas Ertl
Graphical-Interactive Systems WS 10/11
43
10.4 Projections
P
Perspective
ti projections:
j ti
vanishing
i hi points
i t
Thomas Ertl
Graphical-Interactive Systems WS 10/11
44
10. 3D Graphics
10.1 Introduction
10 2 B
10.2
Basics
i off 3D G
Graphics
hi
10.3 Hierarchical Modeling
10.4 Projection
10 5 Visibility
10.5
10.6 Illumination
10 7 T
10.7
Texturing
t i
Thomas Ertl
Graphical-Interactive Systems WS 10/11
45
10.5 Visibility
Visible
Vi
ibl line/surface
li /
f
computation
t ti
• Determination of the visible surfaces (visibility)
• Image-space approach (z-buffer)
(
ff )
– Compare pixels’ depths
• Object-space
Obj t
approach
h (d
(depth
th sort)
t)
– Compare polygons and intersect them
• “Culling”
Culling
– View frustum culling (clipping)
– Occlusion culling
– Backface culling
• Remove invisible backfaces (approx. one half of all faces
f convex objects)
for
bj
)
• Calculate normal vector; dot product with direction of
projection
p
j
(z
( component
p
> 0))
Thomas Ertl
Graphical-Interactive Systems WS 10/11
46
10.5 Visibility
H l in
Halos
i li
line d
drawings
i
[1]
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Graphical-Interactive Systems WS 10/11
47
10.5 Visibility
Depth
D
th sortt
List priority algorithms
• Sorts
S
all polygons from
f
the front
f
to back
• Draw from back to front (painter’s algorithm),
e.g. in
i P
Postscript
t i t
• Problems
– Sorting according to which z? (center point?)
– Cycles cannot be sorted
M
P
P
MQ
Q
eye 2
eye 1
Thomas Ertl
Graphical-Interactive Systems WS 10/11
48
10.5 Visibility
Depth
D
th sort:
t b
bounding
di box
b / z extent
t t
1. Sort all polygons according to smallest (furthest) z
2. Test whether z extents overlap
3. Min/max test based on axis-aligned bounding boxes
4. Test overlap in x-y plane
5. Possibly partition the objects
Thomas Ertl
Graphical-Interactive Systems WS 10/11
49
10.5 Visibility
BSP trees
t
(binary
(bi
space partitioning)
titi i )
• Sorting the objects according to partitioning planes
(oriented half spaces)
• The drawing order is given by traversing
– Compare eye point with orientation of partitioning plane
– First, paint all objects behind
• Take object polygons as planes and split other objects
BSP tree
x
1023
1
Q dt
Quadtree
2
<
y
y
3
0
>
0
4
x
1023
Thomas Ertl
Graphical-Interactive Systems WS 10/11
y
<
> <
>
3
1
2
1 2 3 4
4
50
10.5 Visibility
Begin with level A
Sort according to normal
direction (+/-)
(+/ )
Split half spaces
Determine
D
t
i object
bj t nextt
to the eye
Sort from front to back
(or in reverse order)
Thomas Ertl
Graphical-Interactive Systems WS 10/11
51
10.5 Visibility
• Select
S l t polygon
l
planes
l
as separating
ti planes
l
• Partition intersected objects
Thomas Ertl
Graphical-Interactive Systems WS 10/11
52
10.5 Visibility
Z–buffer
Z
b ff (image-space
(i
approach)
h)
• Compute z value for each pixel and compare with the
depth from pixels already drawn
Æ needs additional z-buffer (16-32 bits)
1
2 2
3 3 3
4 4 4 4
1
2
3
4
0
2
4
4
0
5
4
4
6
5 5
4 4 4
3 3 3 3
6
5
4
4
Thomas Ertl
Graphical-Interactive Systems WS 10/11
53
10.5 Visibility
Z-buffer
Z
b ff algorithm
l
ith
Initialize
• Framebuffer
ff
Å background color
• z_buffer
Å z (of far clipping plane)
(+1 in OpenGL)
foreach polygon (in arbitrary order)
foreach pixel(x,
pixel(x y) in projected polygon
z = z(x, y) Å from interpolation
if z >=
> z_buffer(x,
z buffer(x y) then (GL_LESS
(GL LESS in OpenGL)
z_buffer(x, y) = z
framebuffer(x,
a ebu e ( , y) = p
pixel(x,
e ( , y)
Thomas Ertl
Graphical-Interactive Systems WS 10/11
54
10.5 Visibility
Z-buffer
Z
b ff algorithm
l
ith
Disadvantages:
• Storage
S
requirements
• z aliasing (especially for 16 bit buffers)
• Transparency requires sorting
• Much unnecessary work for scenes with high depth
complexity
l it ((overdraw)
d
)
Thomas Ertl
Graphical-Interactive Systems WS 10/11
55
10.5 Visibility
Scanline
S
li interpolation
i t
l ti
• In normalized vertex coordinates
y
y1
scanline
ys
r1
ra
y2
r −r
r1 − r2
= a 1
y1 − y 2 y s − y1
rp
xp
rb
x
r2
y3
r3
incremental:
rp +1 = rp + Δx
Thomas Ertl
Graphical-Interactive Systems WS 10/11
ra = r1 − (r1 − r2 )
y1 − y s
y1 − y 2
rb = r1 − (r1 − r3 )
y1 − y s
y1 − y 3
rp = rb − (rb − ra )
xb − x p
x b − xa
rb − ra
x b − xa
56
10. 3D Graphics
10.1 Introduction
10 2 B
10.2
Basics
i off 3D G
Graphics
hi
10.3 Hierarchical Modeling
10.4 Projection
10 5 Visibility
10.5
10.6 Illumination
10 7 T
10.7
Texturing
t i
Thomas Ertl
Graphical-Interactive Systems WS 10/11
57
10.6 Illumination
• Li
Lighting
hti effects
ff t essential
ti l ffor 3D iimpression
i (i
(in 2D
images)
• Interaction of
– Light sources (intensity, color, position...)
– Objects (material properties
properties, orientation wrt the light source)
• Types of reflection
– Specular
p
reflection
• Angle of incidence =
angle of reflection
– Diffuse
Diff
reflection
fl ti
• Same reflection
into all directions
Thomas Ertl
Graphical-Interactive Systems WS 10/11
eye
ϕϕ
58
10.6 Illumination
Lambert
L
b t reflection
fl ti
• Perfectly diffuse reflection
N
ϑ
L
dA´
ϑ
dA
effectively
effecti
el illuminated
ill minated ssurface
rface dA
dA´ = dA cos ϑ
calculating intensity
I ~ cos ϑ
Thomas Ertl
Graphical-Interactive Systems WS 10/11
59
10.6 Illumination
Reflection
R
fl ti coefficient
ffi i t
• Real objects often reflect diffuse and specular light
simultaneously: “glossy”
glossy surfaces
• Material described by diffuse and specular reflection
coefficients
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Graphical-Interactive Systems WS 10/11
60
10.6 Illumination
eye
Reflection
R
fl ti
models
d l
• Local reflection:
– C
Consider
id only
l directly
di tl visible
i ibl
(point- ) light sources (as seen
from the object )
– No further reflections
diff se/spec lar
diffuse/specular
eye
• Global reflection:
– Consider multiply reflected light
reflective
Thomas Ertl
Graphical-Interactive Systems WS 10/11
61
10.6 Illumination
Illumination
Ill
i ti calculation
l l ti
• Global models:
– R
Ray ttracing
i
(according to Whitted):
• Some p
point light
g sources
• Several perfectly reflecting
reflections
• Sharp
Sh
shadow
h d
edges
d
– Radiosity:
• All surfaces emit and/or
reflect light diffusely
• Soft shadow edges
• Indirect lighting
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Graphical-Interactive Systems WS 10/11
62
10.6 Illumination
Ray tracing
Radiosity
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63
10.6 Illumination
Illumination
Ill
i ti calculation
l l ti
• Physics-based illumination models
– C
Compute
t physical
h i l quantities
titi ((radiant
di t flflux, radiance)
di
)
– Physical material laws: interaction between light and matter
– Complex numerical simulation
• Hardware-accelerated graphics (e.g. OpenGL)
– “Arbitrary”
y units
– Heuristic, local illumination models (Phong)
– “Faking” realism: textures
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Graphical-Interactive Systems WS 10/11
64
10.6 Illumination
Phong ill
Ph
illumination
i ti
• 3 components of intensity: I = Ia + Id + Is
– A
Ambient:
bi t
• Environment light from all directions
• “Simulates”
Simulates multiple scattering / multiple reflection
– Diffuse: Lambert law
– Specular: Not perfectly reflecting
N
ϑ
L
• Diffuse component
Id = k d ⋅ IL ⋅ cos ϑ = k d ⋅ IL ⋅ (N • L )
intensity of light source
diffuse reflection coefficient
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Graphical-Interactive Systems WS 10/11
65
10.6 Illumination
Phong ill
Ph
illumination
i ti
• Specular component
– Extended
E t d d hi
highlights
hli ht
– Perfect reflection only towards direction R:
α
R + L = N ⋅ (N • L ) ⋅ 2
– Strongly decreasing values for other directions
• Model according
g to cosnα (p
(perfectlyy reflecting:
g n → ∞))
Is = k s ⋅ IL ⋅ cos n α = k s ⋅ IL ⋅ (R • V )
n
eye
N
R
V
Thomas Ertl
Graphical-Interactive Systems WS 10/11
α
L
ϑ ϑ
66
10.6 Illumination
Blinn-Phong
Bli
Ph
ill
illumination:
i ti
halfway
h lf
vector
t
• Alternative for the specular component
n
Is = k s ⋅ IL ⋅ cos n α = k s ⋅ IL ⋅ (R • V )
by using the halfway vector H =
(α ≈ β )
N
H
eye
L +V
L +V
Is = k s ⋅ IL ⋅ (N • H )
n
β
R
L
V
α
ϑ
ϑ
Thomas Ertl
Graphical-Interactive Systems WS 10/11
67
10.6 Illumination
Blinn-Phong
Bli
Ph
illumination
ill i ti
• In OpenGL additional aspects
vertex color = material emission + global ambient light
+ ∑ attenuate (ambient + diffus + specular ) ⋅ spotlight
LQ
I λ = OEλ + Iaλ ⋅ Oaλ
(
+ ∑ faiλ ⋅ Iai λ ⋅ Oaλ + max (N • L i ,0 ) ⋅ Idi λ ⋅ Od λ
i
+ (N • H i < 0 || N • L i < 0 ? 0 : N • H i ) ⋅ Isi λ ⋅ Osλ
n
)
• Instead of scalar reflection factors ka, kd, ks
– Frequency-dependent material colors Oaλ, Odλ, Osλ
– Diffuse
Diff
reflection
fl i (usually)
(
ll ) has
h the
h color
l off the
h surface
f
– Specular reflection (usually) has the color of the source of
light (except for metals)
Thomas Ertl
Graphical-Interactive Systems WS 10/11
68
10.6 Illumination
Shading
Sh
di
• Illumination calculation in world or in eye coordinates
• Iff illumination is only available at polygon vertices (not
(
for each pixel)
– How can we compute the pixel color from the vertex color?
Æ Shading!
• Flat shading
g
– Constant color in the triangle (Æ Mach bands)
e.g., one normal vector for each triangle
• Gouraud shading (smooth shading)
.
– Interpolation of the vertex colors
• Phong
Ph
shading
h di
– Calculation of the illumination for every pixel
Thomas Ertl
Graphical-Interactive Systems WS 10/11
69
10.6 Illumination
Gouraud
G
d (smooth)
(
th) shading
h di
• Scanline interpolation of the vertex color (RGB)
• Lighting computation in the geometry stage
• Normal at vertices by weighted average from the
surrounding
di ttriangles
i
l
N
• Supported in
(RGB)
standard OpenGL
(RGB)
N
(RGB)
(RGB)
N
Thomas Ertl
Graphical-Interactive Systems WS 10/11
70
10.6 Illumination
Gouraud
G
d (smooth)
(
th) shading
h di
• Problems:
– B
Bumps are reduced
d
d / removed
d
– No highlights inside the polygons
– Creases / sharp edges have to be modeled by using several
normal vectors per vertex
– If all normal vectors are equal: flat shading
Thomas Ertl
Graphical-Interactive Systems WS 10/11
71
10.6 Illumination
Phong shading
Ph
h di
• Lighting for each pixel
Æ Needs
N d normall vector
t att each
h pixel
i l
• Scanline interpolation of the three components of the
vertex normal,, then normalization to unit length
g
• High quality but high computational costs
– 3 components, normalization, lighting per pixel in eye
coordinates in rasterization stage
– Not in standard OpenGL
N
– Fragment/pixel shader
• Acceleration:
– Interpolate scalar products
– No normalization
– Highlight textures
N
N
Thomas Ertl
Graphical-Interactive Systems WS 10/11
72
10.6 Illumination
Shading
Sh
di problems
bl
• Smooth surfaces but non-smooth edges
– At silhouette
ilh
tt lilines
• Vertex inconsistencies
– T-vertex
T vertex (hanging node)
• Linear interpolation only correct for triangles
– Color depends on orientation Æ split into triangles
1
4
2
4
3
1
2
3
Thomas Ertl
Graphical-Interactive Systems WS 10/11
73
10. 3D Graphics
10.1 Introduction
10 2 B
10.2
Basics
i off 3D G
Graphics
hi
10.3 Hierarchical Modeling
10.4 Projection
10 5 Visibility
10.5
10.6 Illumination
10 7 T
10.7
Texturing
t i
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74
10.7 Texturing
Simple
Si
l illumination
ill i ti
• No lighting
• Constant
C
colors
(flat shading)
• Polygons
P l
•
•
•
•
Parallel light
Diff
Diffuse
reflection
fl ti
Smooth shading
Freeform surfaces
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Graphical-Interactive Systems WS 10/11
75
10.7 Texturing
IImproved
d ill
illumination
i ti
• Parallel light
• Specular
S
reflection
f
• Phong shading
• Several local light
sources
• Different materials
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Graphical-Interactive Systems WS 10/11
76
10.7 Texturing
Texturing
T
t i
• 2D image texture
• Bump mapping
• Reflection textures
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77
10.7 Texturing
Motivation
M
ti ti
• Realistic look: fine structures on surfaces
• Combining
C bi i pictures
i
and
d geometry
y
z
x
geometry
screen
image
t
s
vertices
geometry pipeline
rasterizer
image
g
Thomas Ertl
Graphical-Interactive Systems WS 10/11
pixel pipeline
78
10.7 Texturing
Texturing:
T
t i
modulation
d l ti off iimage synthesis
th i process
• Reflection properties
– Color,
C l reflection
fl ti coefficients,
ffi i t ttransparency
– Classical texture mapping
• Illumination
– Environment mapping / reflection mapping
– Shadow mapping,
pp g, illumination mapping
pp g
• Geometry
– Displacement mapping: move vertices
• Normal vectors
– Bump mapping / normal mapping
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Graphical-Interactive Systems WS 10/11
79
10.7 Texturing
E
Examples
l
• Solid texturing
• Bump mapping
Reflection mapping
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Graphical-Interactive Systems WS 10/11
80
10.7 Texturing
Multi-textures
M
lti t t
• Complex optical effects
• Combining
C
a number off textures
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Graphical-Interactive Systems WS 10/11
81
10.7 Texturing
Texture
T
t
sources
• Images (e.g., photos, simulations, videos, ...)
–
–
–
–
–
Simple
Si
l acquisition
i iti
Limited resolution
High memory requirements
Attention: “frozen illumination”
1D images (color tables) and 3D images (volumes)
• Algorithms (shaders)
– Non-trivial programming
– Unlimited resolution, anti-aliasing possible
– Any algorithm
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Graphical-Interactive Systems WS 10/11
82
10.7 Texturing
Dimensionality
Di
i
lit off textures
t t
• Texture as a discretely sampled mapping
• nD texture defines
f
the domain:
–
–
–
–
1D: parameter of a line, height, temperature,...
2D: parameter of a surface (u,v
u v),
) direction (θ
(θ,ϕ),
ϕ) ...
3D: point in space (solid texture), direction vector,...
4D: lines in space
p
((reflection))
• Image space: dimensionality of data values:
–
–
–
–
1D: parameter for reflection (kd): intensity/luminance
2D: intensity and transparency
3D: color, normal vector, reflection vector
4D color/vector
4D:
l /
t and
d ttransparency
Thomas Ertl
Graphical-Interactive Systems WS 10/11
83
10.7 Texturing
Texture
T
t
mapping
i
• Point coordinates (x,y,z)
• Surface
S f
coordinates (u,v)
( ) (e.g.,
(
Bézier patch))
• Texture coordinates (s,t,r), sometimes (u,v)
• Direction vector (R, N, H,...)
• Function of the above parameters (in a shader)
– Projection onto other surfaces (sphere, cylinder, box,...)
Thomas Ertl
Graphical-Interactive Systems WS 10/11
84
10.7 Texturing
F
Forward
d / backward
b k
d mapping
i
Backward mapping is standard in graphics hardware
Thomas Ertl
Graphical-Interactive Systems WS 10/11
85
10.7 Texturing
C
Compound
d fforward
d mapping
i (2
(2-level
l
l mapping)
i )
Thomas Ertl
Graphical-Interactive Systems WS 10/11
86
10.7 Texturing
Texture
T
t
coordinates
di t
• Texture parameter generally in [0,1]
– T
Texture
t
coordinates
di t att the
th vertices
ti
off a surface
f
• Calculate of the texture coordinates of a pixel
– Need a parameterization inside the surface
• Triangle (linear scanline interpolation)
• Rectangle (bilinear interpolation)
s = a*b*s4 + (1-a)*b*s3 + a*(1-b)*s2 + (1-a) *(1-b)*s1
(s3, t3)
1
b
0
( s1, t1) 0
Thomas Ertl
Graphical-Interactive Systems WS 10/11
(s4, t4)
p
(s2, t2)
a
1
87
10.7 Texturing
Standard
St
d d parameterizations
t i ti
• Cylinder coordinates (φ,z)
• Polar coordinates (φ,θ)
(
• Analog for cone, torus, …
Thomas Ertl
Graphical-Interactive Systems WS 10/11
88
10.7 Texturing
St d d parameterizations
Standard
t i ti
Thomas Ertl
Graphical-Interactive Systems WS 10/11
89
10.7 Texturing
T t
Texture
coordinates
di t
Thomas Ertl
Graphical-Interactive Systems WS 10/11
90
10.7 Texturing
Texture
T
t
wrapping
i
• Continuation of a texture beyond [0,1]
clamp/
clamp
clamp/
repeat
repeat/
p
clamp
repeat/
repeat
Thomas Ertl
Graphical-Interactive Systems WS 10/11
91
10.7 Texturing
Texture
T
t
filtering
filt i
• Magnification:
– Few texels are mapped to
pixel
many pixels
– Nearest
N
t neighbor
i hb sampling
li
• Take the nearest texel
– Bilinear
Bili
iinterpolation
t
l ti
texture
• Interpolate between
the four next texels
• Use fractional coordinates (floating-point
computations)
• Results in smoother images
Thomas Ertl
Graphical-Interactive Systems WS 10/11
92
10.7 Texturing
T t
Texture
filtering:
filt i
magnification
ifi ti
nearest neighbor
bilinear
Thomas Ertl
Graphical-Interactive Systems WS 10/11
93
10.7 Texturing
Texture
T
t
filtering
filt i
• Minification:
– Many texels are mapped
to few pixels
pixel
– Footprint
F t i t
– Aliasing:
• A high-frequency
hi h f
signal
i
l
texture
is sampled at low frequency
– Filtering:
• Average over (many) corresponding pixels
• Too expensive to be done during runtime
Thomas Ertl
Graphical-Interactive Systems WS 10/11
94
10.7 Texturing
T t
Texture
filtering:
filt i
minification
i ifi ti / magnification
ifi ti
pixel center
pixel center
nearest texel
4 texels for
f bilinear interpolation
Thomas Ertl
Graphical-Interactive Systems WS 10/11
95
10.7 Texturing
MipMapping
Mi
M
i
• Simple prefiltering of textures
• Multum In Parvo (MIP)
(
)
• Store recursively textures of half sizes
(1/3 memory increase)
i
)
• Filtering/averaging for each block of 2x2 texels
Thomas Ertl
Graphical-Interactive Systems WS 10/11
96
10.7 Texturing
MipMapping
Mi
M
i
• Use texture resolution so that
– Texelsize(n)
T
l i ( ) < pixel's
i l' extent
t t on ttexture
t
< ttexturesize(n+1)
t
i ( 1)
• Reconstruction
– Trillinear interpolation of the 8 nearest texels
v
d
v
u
d
u
Thomas Ertl
Graphical-Interactive Systems WS 10/11
97
10.7 Texturing
Mi M
MipMapping
i
No MipMapping
Thomas Ertl
Graphical-Interactive Systems WS 10/11
With MipMapping
98
10.7 Texturing
T t
Texture
iinterpolation
t
l ti
Wrong interpolation
in screen space
p0
viewpoint
Line with parameters
p0,p1 at vertices
0.5 • (p0 + p1)
view plane
p1
1
[3]
Thomas Ertl
Graphical-Interactive Systems WS 10/11
99
10.7 Texturing
Perspectively
P
ti l correctt interpolation
i t
l ti
• Nonlinear z transformation can generate texture
distortions
• Linear interpolation preserves
division ratios
• Projection does not
• Correct: hyperbolic (rational) interpolation
Thomas Ertl
Graphical-Interactive Systems WS 10/11
100
10.7 Texturing
Di id iin 1,
Divide
1 4
4, 64
64... parts
t
Thomas Ertl
Graphical-Interactive Systems WS 10/11
101
10.7 Texturing
P
Perspective
ti texture
t t
interpolation
i t
l ti
Object-affine
j
spaces
p
Objectt
Obj
space
World space
p
Screen-affine spaces
div by 1/w
T t
Texture
space
Homogeneous texture space
(s,t,1)
(s/w,t/w,1/w)
div by w
Homogeneous
Screen space
screen space
(x,y,z
x y z,1)
1)
(xw,yw,zw,w)
• Linear interpolation off s/w,
/ t/w
/ and 1/w
/
• Divide per pixel: s/w by 1/w
Thomas Ertl
Graphical-Interactive Systems WS 10/11
102
10.7 Texturing
Texture
T
t
ffunctions
ti
How is the texture color applied to the pixel?
• Replace:
replace the pixel's color with the texture's color
• Modulate:
M d l t
scale the pixel's color (illumination)
• Blend:
interpolate between the pixel's and the texture's color
• Decal:
replace the pixel's color with alpha blending
Thomas Ertl
Graphical-Interactive Systems WS 10/11
103
10.7 Texturing
O
OpenGL
GL Pipeline
Pi li
Thomas Ertl
Graphical-Interactive Systems WS 10/11
104
References
[1] Foley, James; van Dam, Andries; Feiner, Steven K.; Hughes, John:
Computer Graphics, Addison Wesley, 1997
[2] Watt, Alan: 3D-Computergraphik, Addison Wesley, 3. Auflage, 2002
[3] Hill, F.S. JR, Computer Graphics – Using OpenGL, 2nd Edition, Prentice
Hall,, 1990
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