The document discusses various methods for representing 3D objects in computer graphics, including polygon meshes, curved surfaces defined by equations or splines, and sweep representations. It also covers 3D transformations like translation, rotation, and scaling. Key representation methods discussed are polygonal meshes, NURBS curves and surfaces, and extruded and revolved shapes. Transformation operations covered are translation using addition of a offset vector, and rotation using a rotation matrix.

1. UNIT III
THREE-DIMENSIONAL
GRAPHICS
Three dimensional concepts; Three dimensional object
representations — Polygon surfaces- Polygon tables- Plane
equations — Polygon meshes; Curved Lines and surfaces,
Quadratic surfaces; Blobby objects; Spline representations —
Bezier curves and surfaces -B-Spline curves and surfaces.
TRANSFORMATION AND VIEWING: Three dimensional
geometric and modeling transformations — Translation,
Rotation, Scaling, composite transformations; Three
dimensional viewing — viewing pipeline, viewing coordinates,
Projections, Clipping; Visible surface detection methods.
1

2. 3D Object Representation
• A surface can be analytically
generated using its function involving
the coordinates.
• An object can be represented in
terms of its vertices, edges and
polygons. (Wire Frame, Polygonal
Mesh etc.)
• Curves and surfaces can also be
designed using splines by specifying
a set of few control points.
2
y = f(x,z)
x y z . . .

3. Solid Modeling - Polyhedron
• A polyhedron is a connected mesh of simple planar polygons
that encloses a finite amount of space.
• A polyhedron is a special case of a polygon mesh that satisfies
the following properties:
• Every edge is shared by exactly two faces.
• At least three edges meet at each vertex.
• Faces do not interpenetrate. Faces at most touch along a
common edge.
• Euler’s formula : If F, E, V represent the number of faces,
vertices and edges of a polyhedron, then
V + F  E = 2.
3

4. 3D Object Representation
• The data for polygonal meshes can be represented
in two ways.
• Method 1:
• Vertex List
• Normal List
• Face List (Polygon List)
• Method 2:
• Vertex List
• Edge List
• Face List (Polygon List)
4

5. 5
0
1
2 3
4
5
6
7
Vertices and Faces - E.g. Cube
0
1
2
3
4
5
Face Index
Vertex Index

6. 6
Data representation using vertex, face and normal lists:
Vertex List Normal List Polygon List
30 30 30 0.0 0.0 1.0 0 1 2 3
-30 30 30 0.0 0.0 -1.0 4 7 6 5
-30 -30 30 0.0 1.0 0.0 0 4 5 1
30 -30 30 -1.0 0.0 0.0 1 5 6 2
30 30 -30 0.0 -1.0 0.0 2 6 7 3
-30 30 -30 1.0 0.0 0.0 3 7 4 0
-30 -30 -30
30 -30 -30

7. 7
Vertex List Edge List Polygon List
x[i] y[i] z[i] L[j] M[j] P[k] Q[k] R[k] S[k]
30 30 30 0 1 0 1 2 3
-30 30 30 1 2 4 7 6 5
-30 -30 30 2 3 0 4 5 1
30 -30 30 3 0 1 5 6 2
30 30 -30 4 5 2 6 7 3
-30 30 -30 5 6 3 7 4 0
-30 -30 -30 6 7
30 -30 -30 7 4
0 4
1 5
2 6
3 7
Data representation using vertex, face and edge lists:

8. Normal Vectors (OpenGL)
8
Assigning a normal vector to a polygon:
glBegin(GL_POLYGON);
glNormal3f(xn,yn,zn);
glVertex3f(x1,y1,z1);
glVertex3f(x2,y2,z2);
glVertex3f(x3,y3,z3);
glVertex3f(x4,y4,z4);
glEnd();
Enabling automatic conversion of normal vectors to unit vectors:
glEnable(GL_NORMALIZE);

9. Regular Polyhedra (Platonic Solids)
• If all the faces of a polyhedron are identical, and
each is a regular polygon, then the object is
called a platonic solid.
• Only five such objects exist.
9

10. Wire-Frame Models
• If the object is defined only by a set of nodes
(vertices), and a set of lines connecting the
nodes, then the resulting object representation is
called a wire-frame model.
• Very suitable for engineering applications.
• Simplest 3D Model - easy to construct.
• Easy to clip and manipulate.
• Not suitable for building realistic models.
10

11. Wire Frame Models - OpenGL
11
Some Wireframe Models in OpenGL:
Cube: glutWireCube(Gldouble size);
Sphere: glutWireSphere(…);
Torus: glutWireTorus(…);
Teapot: glutWireTeapot(Gldouble size);
Cone: glutWireCone(…);
(…) Refer text for list of arguments.

12. Wire Frame Model - The Teapot
12

13. Polygonal Mesh
• Three-dimensional surfaces and solids can be
approximated by a set of polygonal and line elements.
Such surfaces are called polygonal meshes.
• The set of polygons or faces, together form the “skin”
of the object.
• This method can be used to represent a broad class of
solids/surfaces in graphics.
• A polygonal mesh can be rendered using hidden
surface removal algorithms.
13

14. Polygonal Mesh - Example
14

15. Solid Modeling
• Polygonal meshes can be used in solid modeling.
• An object is considered solid if the polygons fit together to
enclose a space.
• In solid models, it is necessary to incorporate directional
information on each face by using the normal vector to the
plane of the face, and it is used in the shading process.
15

16. Solid Modeling - Example
16

17. Solid Modeling - OpenGL
17
Some predefined Solid Models in OpenGL:
Cube: glutSolidCube(Gldouble size);
Sphere: glutSolidSphere(…);
Torus: glutSolidTorus(…);
Teapot: glutSolidTeapot(Gldouble size);
Cone: glutSolidCone(…);
(…) Arguments same as wire-frame models.

18. 18
Z
X
Y
y = f(x, z)
Surface Modeling
Many surfaces can be represented by an explicit function of
two independent variables, such as y = f(x, z).

19. Surface Modeling - Example
19

20. Sweep Representations
• Sweep representations are useful for both surface modeling
and solid modeling.
• A large class of shapes (both surfaces and solid models) can
be formed by sweeping or extruding a 2D shape through
space.
• Sweep representations are useful for constructing 3-D
objects that posses translational or rotational symmetries.
20

21. Extruded Shapes - Examples
21
A polyhedron obtained
by sweeping (extruding)
a polygon along a
straight line is called a
prism.

22. Surface of Revolution
• A surface of revolution is obtained by revolving a curve
(known as the base curve or profile curve) about an axis.
• In other words, a surface of revolution is generated by a
rotational sweep of a 2D curve.
• The symmetry of the surface of revolution makes it a very
useful object in presentation graphics.
22

23. 23
Z
X
Y
y = f(x)
r
y y
(x, z)
y = f(r)
x
2
2
z
x
r 

Surface of Revolution

24. 24
The three-dimensional surface obtained by
revolving the curve y = f(x) about the y-axis is
obtained by replacing x with sqrt(x*x + z*z).
The surface of revolution is thus given by
 
y f x z
 
2 2
Surface of Revolution

25. 25
Surface of Revolution
x
x
y
)
sin(

2
2
2
2
sin
z
x
z
x
y






 


26. Quad trees
Quad trees are generated by successively dividing
a 2-D region(usually a square) into quadrants.
Each node in the quadtree has 4 data elements,
one for each of the quadrants in the region. If all
the pixels within a quadrant have the same
color (a homogeneous quadrant), the
corresponding data element in the node stores
that color. For a heterogeneous region of space,
the successive divisions into quadrants
continues until all quadrants are homogeneous.
26

27. Octrees
• An octree encoding scheme divide regions of 3-D
space(usually a cube) in to octants and stores 8
data elements in each node of the tree.
• Individual elements of a 3-D space are called
volume elements or voxels.
• When all voxels in an octant are of the same type,
this type value is stored in the corresponding data
element of the node. Any heterogeneous octant is
subdivided into octants and the corresponding
data element in the node points to the next node
in the octree.
27

28. Octrees
28

29. 29
Bezier Curves
The Bezier curve only approximates the control points
and doesn’t actually pass through all of them.

30. 30
Inputs: n control points (xi, yi), i = 0, 1,2, …n-1
 
,
0
,
0
,
( ) ( )
( ) ( )
0 1
!
( ) (1 )
! !
m
b m i i
i
m
b m i i
i
i m i
m i
x u Bez u x
y u Bez u y
u
m m m
Bez u u u and
i i i m i





 
   
  
   

   


m = n1
Bezier Curves

31. 31
Inputs: n control points (xi, yi), i = 0, 1,2, …m
0
( ) ( ) , 0 1
( ) ( ( ), ( ))
( , )
m
m i i
i i
i i i
r u Bez u r u
where
r u x u y u
r x y

  



Bezier Curves

32. Properties of Bezier Curve
• Bezier curve is a polynomial of degree one less than
the number of control points
32
p0
p2
p1
p0
p2
p1
p3
Quadratic Curve Cubic Curve

33. Properties of Bezier Curve (cont.)
• Bezier curve always passes through the first and
last points; i.e.
and
33
0
(0)
x x

,
0
(0)
y y

(1) m
x x
 (1) m
y y


34. Properties of Bezier Curve (cont)
• The slop at the beginning of the curve is along the line
joining the first two control points, and the slope at the end
of the curve is along the line joining the last two points.
34
p0
p2
p1
p0
p2
p1

35. Properties of Bezier Curve (cont)
• Bezier blending functions are all positive and the sum is
always 1.
• This means that the curve is the weighted sum of the
control points.
35
,
0
( ) 1
m
m i
i
Bez u




36. Design Technique using Bezier Curves:
• A closed Bezier curve can be generated by specifying the
first and last control points at the same location
36
p4
p0=p5
p1
p2
p3

37. Design Technique (Cont)
• A Bezier curve can be made to pass closer to a
given coordinate position by assigning multiple
control points to that position.
37
p0
p1 = p2 p3
p4

38. • Bezier curve can be form by piecing of several
Bezier section with lower degree.
38
p0
p1
p2= p’0
p’2
p’3
p’1

39. 39
(Not Important!)
,
0 0
( , ) ( ) ( ) ,
0 1, 0 1
( , ) ( ( , ), ( , ), ( , ))
( , , )
m l
m i l j i j
i j
ij ij ij ij
r u v Bez u Bez v r
u v
where
r u v x u v y u v z u v
r x y z
 

   



Bezier Surfaces

40. 40
A set of 16 control points The Bezier Patch
Bezier Patch

41. 41
Utah Teapot Defined Using Control Points
Bezier Patch

42. 42
Utah Teapot Generated Using Bezier Patches
Bezier Patch

43. 3D Translation 1
Translate over vector (tx, ty, tz ):
x’=x+ tx, y’=y+ ty , z’=z+ tz
or
x
y
P
P+T
T
z



































z
y
x
t
t
t
z
y
x
z
y
x
T
P
P
T
P
P'
en
,
'
'
'
'
met
,

44. 3D Translation 2
In 4D homogeneous coordinates:
x
y
P
P+T
T
z
1
1
0
0
0
1
0
0
0
1
0
0
0
1
1
'
'
'
or
,












































z
y
x
t
t
t
z
y
x
z
y
x
MP
P'

45. 3D Rotation 1
z
P
x
y
P’
a
1
0
0
0
0
1
0
0
0
0
cos
sin
0
0
sin
cos
)
(
with
,
)
(
Or
'
cos
sin
'
sin
cos
'
:
as
-
around
angle
over
Rotate













 







a
a
a
a
a
a
a
a
a
a
a
z
z
z
z
y
x
y
y
x
x
z
R
P
R
P'

46. 3D Rotation 2
z
x
y
Rotation around axis:
• Counterclockwise, viewed from rotation axis
z
x
y z
x
y

47. 3D Rotation 3
z
x
y
z
x
y z
x
y
z
y
y
x
x
z



x
z
z
y
y
x



Rotation around axes:
Cyclic permutation coordinate axes
x
z
y
x 



48. 3D Rotatie 4
z
y
y
x
x
z



1
0
0
0
0
1
0
0
0
0
cos
sin
0
0
sin
cos
)
(
with
,
)
(
Or
'
cos
sin
'
sin
cos
'
:
as
-
around
angle
over
Rotate













 







a
a
a
a
a
a
a
a
a
a
a
z
z
z
z
y
x
y
y
x
x
z
R
P
R
P'
1
0
0
0
0
cos
sin
0
0
sin
cos
0
0
0
0
1
)
(
with
,
)
(
Or
'
cos
sin
'
sin
cos
'
:
as
-
around
angle
over
Rotate






















a
a
a
a
a
a
a
a
a
a
a
x
x
x
x
z
y
z
z
y
y
x
R
P
R
P'

49. 1
3D Rotation around arbitrary axis 1
z
P
x
y
P’
a
Q
2
3
Q)P
)T(
(
T(Q)R
P'
Q
Q
Q
z 
 a
:
Or
.
over
back
Translate
3.
axis;
-
around
Rotate
2.
;
over
Translate
1.
:
rotation
2D
as
Similar
as.
-
the
example.
For
.
point
through
axis,
coordinate
to
parallel
axis,
around
Rotation
z
-
z

50. z x
y
1
P
2
P
3D Rotation around arbitrary axis 2
Rotation around axis through two points P1 and P1 .
More complex:
1. Translate such that axis goes through origin;
2. Rotate…
3. Translate back again.

51. 3D Rotation around arbitrary axis 3
z x
y
1
P
2
P
Initial
z x
y
'
1
P
1. translate axis
'
P2
z x
y
2. rotate axis
'
1
P
'
'
P2
z x
y
3. rotate around
z-axis
'
1
P
'
'
P2
4. rotate back
z x
y
'
1
P
'
P2
z x
y
1
P
2
P
5. translate back

52. 3D Rotation around arbitrary axis 3
z x
y
1
P
2
P
Initial
z x
y
'
1
P
1. translate axis
'
P2
z x
y
2. rotate axis
'
1
P
'
'
P2
z x
y
3. rotate around
z-axis
'
1
P
'
'
P2
4. rotate back
z x
y
'
1
P
'
P2
z x
y
1
P
2
P
5. translate back
T(P1) R
Rz(a)
R1
T(P1)

53. 3D Rotation around arbitrary axis 3
z x
y
1
P
2
P
Initial
z x
y
'
1
P
1. translate axis
'
P2
z x
y
2. rotate axis
'
1
P
'
'
P2
z x
y
3. rotate around
z-axis
'
1
P
'
'
P2
4. rotate back
z x
y
'
1
P
'
P2
z x
y
1
P
2
P
5. translate back
T(P1) R
Rz(a)
R1
T(P1)
M = T(P1) R1Rz(a) RT(P1)

54. 3D Rotation around arbitrary axis 4
z x
y
'
1
P
'
P2
z x
y
2. rotate axis
'
1
P
'
'
P2
Difficult step:
Find rotation such that rotation axis aligns with z-axis.
Two options:
1. Step by step by step (see H&B, 309-312)
2. Direct derivation of matrix
R

55. 3D Rotation around arbitrary axis 5
z
x
y
u
v
w
1. Construct orthonormal axis frame u, v, w;
2. Invent rotation matrix R, such that:
u is mapped to z-axis;
v is mapped to y-axis;
w is mapped to x-axis.

56. 3D Rotation around arbitrary axis 6
z
x
y
u
v
w
Construct orthonormal axis frame u, v, w:
u = (P2P1) / |P2P1|
v = u  (1,0,0) / | u  (1,0,0) |
w = v  u
(If u = (a, 0, 0), then use (0, 1, 0))
This frame is orthonormal:
Unit length axes: u.u = v.v = w.w = 1
All axes perpendicular: u.v = v.w = w.u = 0

57. 3D Rotation around arbitrary axis 7
z
x
y
u
v
w






































































































1
0
0
1
1
and
,
1
0
1
0
1
,
1
1
0
0
1
such that
in
Fill
1
0
0
0
0
0
0
:
matrix
rotation
Generic
33
32
31
23
22
21
13
12
11
z
y
x
z
y
x
z
y
x
w
w
w
v
v
v
u
u
u
r
r
r
r
r
r
r
r
r
R
R
R
R
R

58. 3D Rotation around arbitrary axis 8
z
x
y
u
v
w
Done! But how to find R1 ?
1
0
0
0
0
0
0
:
Solution
1
0
0
1
1
and
,
1
0
1
0
1
,
1
1
0
0
1
such that
,
Fill






































































































z
y
x
z
y
x
z
y
x
z
y
x
z
y
x
z
y
x
u
u
u
v
v
v
w
w
w
w
w
w
v
v
v
u
u
u
R
R
R
R
R

59.  












z
z
z
y
y
y
x
x
x
w
v
u
w
v
u
w
v
u
w
v
u
M
Each rotation matrix is an orthonormal matrix M:
The frame u, v, w is orthonormal:
Unit length axes: u.u = v.v = w.w = 1
All axes perpendicular: u.v = v.w = w.u = 0.
Requested: M-1 such that M-1M = I
Inverse of rotation matrix 1
z
x
y
u
v
w

60. Requested: M-1 such that M-1M = I
Solution:
In words:
The inverse of a rotation matrix is the transpose.
(For a rotation around the origin).




























z
y
x
z
y
x
z
y
x
T
T
T
T
w
w
w
v
v
v
u
u
u
1
w
v
u
M
M
Inverse of rotation matrix 2

61.   I
w
w
v
w
u
w
w
v
v
v
u
v
w
u
v
u
u
u
w
v
u
w
v
u
M
M 












































1
0
0
0
1
0
0
0
1
T
T
T
T
Requested: M-1 such that M-1M = I
Solution: M-1 = M-T
Check:
Inverse of rotation matrix 3

62. 1
1
0
0
0
0
0
0
0
0
0
0
0
0
1
'
'
'
or
,












































z
y
x
s
s
s
z
y
x
z
y
x
SP
P'
3D scaling
Scale with factors sx, sy,sz :
x’= sx x, y’= sy y, z’= sz z
or

63. More 3D transformations
+/- same as in 2D:
• Matrix concatenation by multiplication
• Reflection
• Shearing
• Transformations between coordinate systems

64. z
zz
zy
zx
y
yz
yy
yx
x
xz
xy
xx
b
z
a
y
a
x
a
z
b
z
a
y
a
x
a
y
b
z
a
y
a
x
a
x












'
'
'
Affine transformations 1
Generic name for these transformations:
affine transformations

65. Affine transformations 2
Properties:
1. Transformed coordinates x’,y’ and z’ are linearly
dependent on original coordinates x, y en z.
2. Parameters aij en bk are constant and determine type
of transformation;
3. Examples: translation, rotation, scaling, reflection
4. Parallel lines remain parallel
5. Only translation, rotation reflection: angles and
lengths are maintained

66. 3D Viewing
• Viewing: virtual camera
• Projection
• Depth
• Visible lines and surfaces
• Surface rendering

67. 3D Viewing pipeline 1
• Similar to making a photo
– Position and point virtuele camera, press button;
Projection plane aka
Viewing plane
• Pipeline has +/ same structure as in 2D

68. 3D Viewing pipeline 2
MC: Modeling Coordinates
WC: World Coordinates
VC: Viewing Coordinates
PC: Projection Coordinates
NC: Normalized Coordinates
DC: Device Coordinates
Apply model transformations
To camera coordinates
Project
To standard coordinates
Clip and convert to pixels

69. 3D viewing coordinates 1
Specification of projection:
P0 : View or eye point
Pref : Center or look-at point
V: View-up vector
(projection along vertical
axis)
zvp : positie view plane
P0
zw
yw
xw
Pref
N
V
P0, Pref , V: define viewing coordinate system
Several variants possible

70. 3D viewing coordinates 2
P0
zw
yw
xw
Pref
N
V
zview
yview
xview
P0, Pref , V: define viewing coordinate system
Several variants possible

71. 3D view coordinates 3
P0
zw
yw
xw
Pref
N
V
zview
yview
xview
u
)
,
,
(
:
and
lar to
perpendicu
z
y
x u
u
u



V
n
V
u
n
V
u
v
)
,
,
(
:
and
lar to
perpendicu
z
y
x v
v
v


 u
n
v
u
n
v
)
,
,
(
:
frame
axis
Derivation
ref
0
z
y
x n
n
n




N
N
n
P
P
N
n

72. 3D viewing coördinaten 4
P0
zw
yw
xw
Pref
N
V
zview
yview
xview
n
u
v
RT
M
R
R
P
T 0


















W C,VC
z
y
x
z
y
x
z
y
x
n
n
n
v
v
v
u
u
u
:
Or
1
0
0
0
0
0
0
:
with
rotate
Next,
)
(
with
translate
First,
:
view
tion world
Transforma

73. Projection transformations
P1
P2
P’1
P’2
View plane
Parallel projection
P1
P2
P’1
P’2
View plane
Perspective projection

74. Orthogonal projections 1
P’1
P’2
Parallell projection:
Projection lines are parallel
Orthogonal projection:
Projection lines are parallel and
perpendicular to projection plane
Isometric projection:
Projection lines are parallel,
perpendicular to projection plane,
and have the same angle with axes.
P1
P2
P’1
P’2
P1
P2
x
y
z

75. Orthogonal projections 2
Orthogonal projection:
P’1
P’2
P1
P2
Trivial!
:
s)
coordinate
projection
to
s
coordinate
view
(from
)
,
,
(
of
Projection
z
z
y
y
x
x
z
y
x
p
p
p




76. Orthogonal projections 3
Clipping
window
zview
xview
yview
Near plane
Far plane
View volume

77. Orthogonal projections 4
zview
xview
yview
View volume
(xwmin, ywmin, znear)
(xwmax, ywmax, zfar)

78. Orthogonal projections 4
zview
xview
yview
View volume
(xwmin, ywmin, znear)
(xwmax, ywmax, zfar)
znorm
xnorm
ynorm
Normalized View volume
(1,1,1)
(-1,-1,-1)
Translation
Scaling
From right- to left handed

79. Perspective projection 1
P1
P2
P’1
P’2
View plane: z = zvp
Projection reference point
zview
xview
yview
View plane: orthogonal to zview axis.

80. Perspective projection 2
P = (x, y, z) R: Projection reference point
zview
xview
yview
(xr, yr, zr)
(xp, yp, zp)
To simplify,
Assume R in origin
View plane: z = zvp
Question: What is the projection of P
on the view plane?

81. Perspective projection 3
P = (x, y, z)
zview
xview
yview
(xp, yp, zp)
(0,0, 0)
R=
.
'
and
;
'
;
'
or
,
1
0
with
,
'
:
to
(origin)
from
Line
uz
z
uy
y
ux
x
u
u





 P
X
P
R
.
hence
'
:
plane
with
crossing
At
z
z
u
z
z
vp
vp 

y
z
z
y
x
z
z
x
vp
p
vp
p 
 and
gives
on
Substituti
View plane: z = zvp

82. Perspective projection 4
P = (x, y, z) View plane
zview
yview
Viewed from the side
R
z
y
zvp
yp
y
z
z
y
z
y
z
y
vp
p
vp
p


hence
:
that
see
can
We

83. Perspective projection 5
P = (x, y, z)
Clipping window
in View plane
zview
yview
R
zvp
W
wmax
wmin
Ratio between
W=wmaxwmin and zvp
determines strenght perspective
Viewed from the side

84. Viewed from the side
Perspective projection 6
Clipping window
in View plane
zview
yview
R

Ratio between
W=wmaxwmin and zvp
determines strenght perspective.
This ratio is 2tan(/2),
with  the view angle.

85. Perspective projection 7
zview
yview
R

86. Perspective projection 7
zview
yview
R

87. Perspective projection 7
R

88. Perspective projection 8
zview
yview
R
zvp
W
How to specify the ratio between W en zvp?
How to specify ‘how much’ perspective I want to see?
Option 1: Specify view angle .


89. Perspective projection 9
zview
yview
R
zvp
W
Use camera as metaphor. User specifies focal length f .
(20 mm – wide angle, 50 mm – normal, 100 mm – tele).
Application adjusts W and/or zvp, such that W/zvp = 24/f.
For x: W/zvp = 36/f.
f (mm)
24 mm

90. View volume orthogonal…
Clipping
window
zview
xview
yview
Near
clipping plane
Far
clipping plane
View volume

91. View volume perspective
Clipping
window
zview
xview
yview
Near
clipping plane
Far
clipping plane
View volume
R

92. To Normalized Coordinates…
zview
xview
yview
R
znorm
xnorm
ynorm
Normalized View volume
(1,1,1)
(1, 1, 1)
Rectangular frustum
View Volume

93. Side view Front view
Perspective projection 10
yview
R
zview
znorm
ynorm
Perspective transformation:
Distort space, such that perpendicular projection gives
an image in perspective.

94. Perspective projection 10
yview
R
2r
zn
zf
zview
znorm
ynorm
Simplest case:
Square window,
clipping plane coincides with view plane: zn=zvp

95. Perspective projection 11
yview
R
zn
zf
zview
znorm
ynorm
(-r, -r, zn)
((zf /zn)r, (zf /zn)r, zf )
2r
(-1,-1,-1)
(1,1,1)

96. (r,  r, zn)
(rzf /zn, rzf /zn, zf )
Perspective projection 11
yview
R
zview
znorm
ynorm
(1, 1, 1)
(1,1,1)
How to put this transformation in the pipeline?
How to process division by z?
Earlier: y
z
z
y
x
z
z
x
vp
p
vp
p 
 ,

97. Homogeneous coordinates (reprise)
• Add extra coordinate:
P = (px , py , pz , ph) or
x = (x, y, z, h)
• Cartesian coordinates: divide by h
x = (x/h, y/h, z/h)
• Points: h = 1 (temporary…)perspective: h = z !

98. .
/
/
/
:
by
given
are
s
coordinate
projected
such that
1
0
1
0
0
:
by
described
be
can
ation
transform
e
Perspectiv



































































h
z
h
y
h
x
z
y
x
z
y
x
t
s
s
s
t
s
s
s
t
s
s
s
h
z
y
x
h
h
h
p
p
p
v
v
v
z
zz
zy
zx
y
yz
yy
yx
x
xz
xy
xx
h
h
h
Homogeneous coordinates (reprise)

99. Perspective projection 12
xview
R
zview
znorm
ynorm
(r, r, zn)
(rzf /zn, rzf /zn, zf )
(1, 1, 1)
(1,1,1)
(r, r, zn)
.
0
,
/
:
gives
n
Elaboratio
.
1
also
then
,
and
/
If
.
1
then
,
and
If
.
/
)
(
is
form
Generic
.
First
















x
xz
xy
n
xx
p
f
n
f
p
n
x
xz
xy
xx
p
t
s
s
r
z
s
x
z
z
z
rz
x
x
z
z
r
x
z
t
z
s
y
s
x
s
x
x

100. Perspective projection 13
yview
R
zview
znorm
ynorm
(r, r, zn)
(rzf /zn, rzf /zn, zf )
(1, 1, 1)
(1,1,1)
.
0
),
/
(
:
Or
.
/
)
/
(
:
gives
,
as
Same
.
Next the








y
yz
yx
n
yy
n
p
t
s
s
r
z
s
z
y
r
z
y
x
y

101. Perspective projection 14
yview
R
zview
znorm
ynorm
(r, r, zn)
(rzf /zn, rzf /zn, zf )
(1, 1, 1)
(1,1,1)
.
0
,
2
,
gives
n
Elaboratio
.
1
then
,
If
.
1
then
,
If
.
/
)
(
:
is
form
Generic
.
:
Finally


















zy
zx
f
n
f
n
z
f
n
f
n
zz
f
f
p
n
z
zz
zy
zx
p
s
s
z
z
z
z
t
z
z
z
z
s
z
z
z
z
z
z
z
t
z
s
y
s
x
s
z
z

102. Perspective projection 15
.
/
/
/
:
from
follow
s
coordinate
projected
the
where
1
0
1
0
0
2
0
0
0
0
/
0
0
0
0
/
:
by
described
be
hence
can
ation
transform
e
Perspectiv











































































h
z
h
y
h
x
z
y
x
z
y
x
z
z
z
z
z
z
z
z
r
z
r
z
h
z
y
x
v
v
v
p
p
p
v
v
v
f
n
f
n
f
n
f
n
n
n
h
h
h

103. 3D Viewport coordinates 1
znorm
xnorm
ynorm
Normalized View volume
(1,1,1)
(1, 1, 1)
zv xv
yv
3D screen
(xvmin, yvmin, 0)
(xvmax, yvvmax, 1)
scaling
translation

104. 3D Viewport coordinaten 2

















































1
1
0
0
0
2
1
2
1
0
0
0
0
2
0
0
0
0
2
:
s
coordinate
screen
3D
to
s
coordinate
view
normalized
from
tion
Transforma
min
max
min
max
p
p
p
s
s
s
z
y
x
yv
yv
xv
xv
h
z
y
x

105. OpenGL 3D Viewing 1
3D Viewing in OpenGL:
• Position camera;
• Specify projection.

106. OpenGL 3D Viewing 2
z
x
y
View volume
Camera always in origin, in direction of negative z-axis.
Convenient for 2D, but not for 3D.

107. OpenGL 3D Viewing 3
Solution for view transform: Transform your model
such that you look at it in a convenient way.
Approach 1: Do it yourself. Apply rotations,
translations, scaling, etc., before rendering the
model. Surprisingly difficult and error-prone.
Approach 2: Use gluLookAt();

108. OpenGL 3D Viewing 4
MatrixMode(GL_MODELVIEW);
gluLookAt(x0,y0,z0, xref,yref,zref, Vx,Vy,Vz);
x0,y0,z0: P0, viewpoint, location of camera;
xref,yref,zref: Pref, centerpoint;
Vx,Vy,Vz: V, view-up vector.
Default: P0 = (0, 0, 0); Pref = (0, 0, 1); V=(0, 1, 0).

109. OpenGL 3D Viewing 5
z
x
y
Orthogonal projection:
MatrixMode(GL_PROJECTION);
glOrtho(xwmin, xwmax, ywmin, ywmax, dnear, dfar);
xwmin
xwmax
ywmin
ywmax
xwmin, xwmax, ywmin,ywmax:
specification window
dnear: distance to near clipping plane
dfar : distance to far clipping plane
Select dnear and dfar right:
dnear < dfar,
model fits between clipping planes.

110. OpenGL 3D Viewing 6
Perspective projection:
MatrixMode(GL_PROJECTION);
glFrustrum(xwmin, xwmax, ywmin, ywmax, dnear, dfar);
xwmin
xwmax
ywmin
ywmax
xwmin, xwmax, ywmin,ywmax:
specification window
dnear: distance to near clipping plane
dfar : distance to far clipping plane
z
x
y
Standard projection: xwmin = -xwmax,
ywmin = -ywmax
Select dnear and dfar right:
0 < dnear < dfar,
model fits between clipping planes.

111. OpenGL 3D Viewing 7
Finally, specify the viewport (just like in 2D):
glViewport(xvmin, yvmin, vpWidth, vpHeight);
xvmin, yvmin: coordinates lower left corner (in pixel coordinates);
vpWidth, vpHeight: width and height (in pixel coordinates);
(xvmin, yvmin)
vpWidth
vpHeight

112. Visible surface detection
methods

113. Visibility
 Assumption: All polygons are opaque
 What polygons are visible with respect to your view frustum?
 Outside: View Frustum Clipping
 Remove polygons outside of the view volume
 For example, Liang-Barsky 3D Clipping
 Inside: Hidden Surface Removal
 Backface culling
 Polygons facing away from the viewer
 Occlusion
 Polygons farther away are obscured by closer polygons
 Full or partially occluded portions
 Why should we remove these polygons?
 Avoid unnecessary expensive operations on these polygons later

114. No Lines Removed

115. Hidden Lines Removed

116. Hidden Surfaces Removed

117. Occlusion: Full, Partial, None
Full Partial None
l The rectangle is closer than the triangle
l Should appear in front of the triangle

118. Backface Culling
 Avoid drawing polygons facing away from the viewer
 Front-facing polygons occlude these polygons in a closed
polyhedron
 Test if a polygon is front- or back-facing?
front-facing
back-facing
Ideas?

119. Detecting Back-face Polygons
 The polygon normal of a …
 front-facing polygon points towards the viewer
 back-facing polygon points away from the viewer
If (n  v) > 0  “back-face”
If (n  v) ≤ 0  “front-face”
v = view vector
 Eye-space test … EASY!
 “back-face” if nz < 0
 glCullFace(GL_BACK)
back
front

120. Polygon Normals
 Let polygon vertices v0, v1, v2,..., vn - 1 be in
counterclockwise order and co-planar
 Calculate normal with cross product:
n = (v1 - v0) X (vn - 1 - v0)
 Normalize to unit vector with n/║n║
v0
v1
v2
v3
v4
n

121. Normal Direction
 Vertices counterclockwise  Front-facing
 Vertices clockwise  Back-facing
0
1
2
0
2
1
Front facing Back facing

122. Painter’s Algorithm (1)
 Assumption: Later projected polygons overwrite earlier
projected polygons
Graphics Pipeline
1 1
2 2
3 3
Oops! The red polygon
Should be obscured by
the blue polygon

123. Painter’s Algorithm (2)
 Main Idea
 A painter creates a picture
by drawing background
scene elemens before
foreground ones
 Requirements
 Draw polygons in back-to-
front order
 Need to sort the polygons
by depth order to get a
correct image
from Shirley

124. Painter’s Algorithm (3)
 Sort by the depth of each polygon
Graphics Pipeline
1 1
2 2
3 3
depth

125. Painter’s Algorithm (4)
 Compute zmin ranges for each polygon
 Project polygons with furthest zmin first
(z) depth
zmin
zmin
zmin
zmin

126. Painter’s Algorithm (5)
 Problem: Can you get a total sorting?
zmin
zmin
zmin
zmin
Correct?

127. Painter’s Algorithm (6)
 Cyclic Overlap
 How do we sort these three polygons?
 Sorting is nontrivial
 Split polygons in order to get a total ordering
 Not easy to do in general

128. Visibility
 How do we ensure that closer polygons
overwrite further ones in general?

129. Z-Buffer
 Depth buffer (Z-Buffer)
 A secondary image buffer that holds depth values
 Same pixel resolution as the color buffer
 Why is it called a Z-Buffer?
 After eye space, depth is simply the z-coordinate
 Sorting is done at the pixel level
 Rule: Only draw a polygon at a pixel if it is closer
than a polygon that has already been drawn to
this pixel

130. Z-Buffer Algorithm
 Visibility testing is done during rasterization

131. Z-buffer: A Secondary Buffer
DAM Entertainment
Color buffer Depth buffer

132. Z-Buffer
 How do we calculate the depth values on the polygon interior?
P1
P2
P3
P4
ys za zp zb
Scanline order
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
b
a
p
a
a
b
a
p
s
b
s
a
x
x
x
x
z
z
z
z
y
y
y
y
z
z
z
z
y
y
y
y
z
z
z
z















2
1
1
1
2
1
4
1
1
1
4
1
Bilinear Interpolation

133. Z-buffer - Example
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
Z-buffer
Screen

134. 
 
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
  
 

  
 

Parallel with
the image plane

135. 


   
   
   
  
 

 

   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
Not Parallel

136. Z-Buffer Algorithm
 Algorithm easily handles this case

137. Z-buffering in OpenGL
 Create depth buffer by setting GLUT_DEPTH flag in
glutInitDisplayMode()or the appropriate
flag in the PIXELFORMATDESCRIPTOR.
 Enable per-pixel depth testing with
glEnable(GL_DEPTH_TEST)
 Clear depth buffer by setting GL_DEPTH_BUFFER_BIT
in glClear()