CS 431/636
Advanced Rendering Techniques
Dr. David Breen
University Crossings 149
Tuesday 6PM  8:50PM
Presentation 2
4/7/09
Start Up
Any questions from last time?
 Go over sampling image plane?
 Or intersection algorithms?

Slide Credits
Leonard McMillan, Seth Teller,
Fredo Durand, Barb Cutler - MIT
 G. Drew Kessler, Larry Hodges - Georgia
Institute of Technology
 John Hart - University of Illinois
 Rick Parent - Ohio State University

More Geometry & Intersections
Ray/Plane Intersection
Ray is defined by R(t) = Ro + Rd*t where t  0
Ro = Origin of ray at (xo, yo, zo)
Rd = Direction of ray [xd, yd, zd] (unit vector)
Plane is defined by [A, B, C, D]
Ax + By + Cz + D = 0 for a point in the plane
Normal Vector, N = [A, B, C] (unit vector)
A2 + B2 + C2 = 1
D = - N • P0 (P0 - point in plane)
Ray/Plane (cont.)
Substitute the ray equation into the plane
equation:
A(xo + xdt) + B(yo + ydt) + C(zo +zdt) + D = 0
Solve for t:
t = -(Axo + Byo + Czo + D) / (Axd + Byd + Czd)
t = -(N • Ro - N • P0 ) / (N • Rd)
What Can Happen?
N • Rd = 0
N • Rd > 0
Ro
Ro
t<0
t>0
Ray/Plane Summary
Intersection point:
(xi, yi, zi) = (xo + xdti, yo + ydti, zo + zdti)
1.
2.
3.
4.
Calculate N • Rd and compare it to zero.
Calculate ti and compare it to zero.
Compute intersection point.
Flip normal if N • Rd is positive
Ray-Parallelepiped Intersection
Axis-aligned
 From (X1, Y1, Z1) to (X2, Y2, Z2)
 Ray P(t)=Ro+Rdt

y=Y2
y=Y1
Rd
Ro
x=X1 x=X2
Naïve ray-box Intersection
Use 6 plane equations
 Compute all 6 intersection
 Check that points are inside box
Ax+By+Cz+D ≤ 0

y=Y2
y=Y1
Rd
Ro
x=X1 x=X2
Factoring out computation
Pairs of planes have the same normal
 Normals have only one non-0 component
 Do computations one dimension at a time
 Maintain tnear and tfar (closest and farthest
so far)

y=Y2
y=Y1
Rd
Ro
x=X1 x=X2
Test if parallel

If Rdx = 0, then ray is parallel

If Rox < X1 or Rox > x2 return false
y=Y2
y=Y1
Rd
Ro
x=X1 x=X2
If not parallel

Calculate intersection distance t1 and t2
t1 = (X1-Rox)/Rdx
 t2 = (X2-Rox)/Rdx

t2
y=Y2
t1
y=Y1
Rd
Ro
x=X1 x=X2
Test 1

Maintain tnear and tfar




If t1 > t2, swap
if t1 > tnear, tnear = t1
if t2 < tfar, tfar = t2
If tnear > tfar, box is missed
t1x
tnear
t2x
tfar
y=Y2
Rd
t2y
tfar
Ro
y=Y1
t1y
x=X1 x=X2
Test 2

If tfar < 0, box is behind
Rd
y=Y2
t1x
x=X1
tfar
t2x
y=Y1
x=X2
Ro
Algorithm recap

Do for all 3 axes






Calculate intersection distance t1 and t2
Maintain tnear and tfar
If tnear > tfar, box is missed
If tfar < 0, box is behind
If box survived tests, return intersection at tnear
If tnear is negative, return tfar
y=Y2
tnear
tfar
y=Y1
t1y
Rd
Ro
x=X1 x=X2
Ray/Ellipsoid Intersection
Ray/Cylinder Intersection
Ellipsoid's surface is defined by the set of points
{(xs, ys, zs)} satisfying the equation:
(xs/ a)2 + (ys/ b)2 + (zs/ c)2 - 1 = 0
Cylinder's surface is defined by the set of points
{(xs, ys, zs)} satisfying the equation:
(xs)2 + (ys)2 - r2 = 0
Centers at origin
-z0 ≤ zs ≤ z0
Ray/Ellipsoid Intersection
Ray/Cylinder Intersection
 Substitute ray equation into surface
equations
 This is a quadratic equation in t:
 At2 + Bt + C = 0




Analyze as before
Solve for t with quadratic formula
Plug t back into ray equation - Done
Well,… not exactly
Ray/Cylinder Intersection
 Is intersection point Pi between -Z0 and Z0?
 If not, Pi is not valid
 Also need to do intersection test with
z = -Z0 , Z0 plane
 If (Pix)2 + (Piy)2 ≤ r2, you’ve intersected a
“cap”
 Which valid intersection is closer?
Superquadrics
2
   2    2   1  2 1
 x    y     z   1
a1 
a2  
a3 


2
2
Bezier Patch

Patch of order (n, m) can be defined in terms of a set
of (n + 1)(m + 1) control points Pi+1,j+1 for integer
indices i = 0 to n, j = 0 to m.
(u,v)  [0,1]2
n
p(u,v)  
i 0
m
n
m
B
(u)B
 i
j (v)Pi1, j 1
j 0
n  i
B (u)   u (1 u) ni
i 
n
i
Tesselate Patches and
Superquadrics
Tesselate Patches and
Superquadrics
Utah Teapot


Modeled by 32 Bézier Patches
Control points available at
http://www.holmes3d.net/graphics/teapot
SMF Triangle Meshes
vertices
triangles
v
v
v
v
v
v
v
v
f
f
f
f
f
f
f
f
f
f
f
f
-1 -1
1 -1
-1 1
1 1
-1 -1
1 -1
-1 1
1 1
1 3 4
1 4 2
5 6 8
5 8 7
1 2 6
1 6 5
3 7 8
3 8 4
1 5 7
1 7 3
2 4 8
2 8 6
-1
-1
-1
-1
1
1
1
1
Draw data structure
Triangle Meshes (.iv)
Transformations &
Hierarchical Models
Homogeneous Coordinates

Add an extra dimension
• in 2D, we use 3 x 3 matrices
• In 3D, we use 4 x 4 matrices

Each point has an extra value, w
x'
a
b
c
d
x
y'
e
f
g
h
y
i
j
k
l
z
m n
o
p
w
z'
=
w'
p' =
Mp
Homogeneous Coordinates

Most of the time w = 1, and we can
ignore it
x'
a
b
c
d
x
y'
e
f
g
h
y
i
j
k
l
z
0
0
0
1
1
z'
1
=
Translate(c,0,0)
y
Translate (tx, ty, tz)

Why bother with the
extra dimension?
c
Because now translations
can be encoded in the matrix!
x'
y'
z'
1
=
1
0
0
0
0
1
0
0
0 tx
0 ty
1 tz
0 1
p'
p
x
y
z
1
x
Scale(s,s,s)
p'
y
Scale (sx, sy, sz)
p

Isotropic (uniform)
scaling: sx = sy = sz
x'
y'
z'
1

sx
0
=
0
0
0
sy
0
0
q
q'
x
0
0
sz
0
0
0
0
1
x
y
z
1
You only have to implement uniform scaling
ZRotate()
y
Rotation

p'
θ
About z axis
p
x
z
x'
y'
z'
1
=
cos 
sin 
0
0
-sin 
cos 
0
0
0
0
1
0
0
0
0
1
x
y
z
1
Rotation


0
0
0
1
0 cos  -sin  0
0 sin  cos  0
0
0
1
0
About
x axis:
x'
y'
z'
1
=
About
y axis:
x'
y'
z'
1
cos  0
1
0
=
-sin  0
0
0
sin 
0
cos 
0
0
0
0
1
x
y
z
1
x
y
z
1
y
Rotate(k, )
Rotation
θ
k
About (kx, ky, kz), an
z
arbitrary unit vector
(Rodrigues Formula)
kxkx(1-c)+c kykx(1-c)-kzs kxkz(1-c)+kys
x'
kykx(1-c)+kzs kyky(1-c)+c kykz(1-c)-kxs
y'
=
kzkx(1-c)-kys kzky(1-c)+kxs kzkz(1-c)+c
z'
0
0
0
1

where c = cos  & s = sin 
x
0 x
0 y
0 z
1 1
How are transforms combined?
Scale then Translate
(1,1)
(2,2)
Scale(2,2)
(5,3)
Translate(3,1)
(3,1)
(0,0)
(0,0)
Use matrix multiplication: p' = T ( S p ) = ((TS) p)
TS =
1 0 3
0 1 1
0 0 1
2 0 0
0 2 0
0 0 1
=
2 0 3
0 2 1
0 0 1
Caution: matrix multiplication is NOT commutative!
Non-commutative Composition
Scale then Translate: p' = T ( S p ) = TS p
(1,1)
(2,2)
Scale(2,2)
(5,3)
Translate(3,1)
(3,1)
(0,0)
(0,0)
Translate then Scale: p' = S ( T p ) = ST p
(8,4)
(1,1)
(0,0)
(4,2)
Translate(3,1)
(3,1)
Scale(2,2)
(6,2)
Non-commutative Composition
Scale then Translate: p' = T ( S p ) = TS p
TS =
1 0 3
0 1 1
0 0 1
2 0 0
0 2 0
0 0 1
=
2 0 3
0 2 1
0 0 1
Translate then Scale: p' = S ( T p ) = ST p
ST =
2 0 0
0 2 0
0 0 1
1 0 3
0 1 1
0 0 1
=
2 0 6
0 2 2
0 0 1
Transformations in Ray Tracing
Transformations in Modeling
Position objects in a scene
 Change the shape of objects
 Create multiple copies of objects
 Projection for virtual cameras
 Animations

Scene Description
Scene
Camera
Lights
Background
Materials
Objects
Simple Scene Description File
Camera {
center 0 0 10
direction 0 0 -1
up 0 1 0 }
Lights {
numLights 1
DirectionalLight {
direction -0.5 -0.5 -1
color 1 1 1 } }
Background { color 0.2 0 0.6 }
Materials {
numMaterials <n>
<MATERIALS> }
Group {
numObjects <n>
<OBJECTS> }
Hierarchical Models

Logical organization of scene
Simple Example with Groups
Group {
numObjects 3
Group {
numObjects 3
Box { <BOX PARAMS> }
Box { <BOX PARAMS> }
Box { <BOX PARAMS> } }
Group {
numObjects 2
Group {
Box { <BOX PARAMS> }
Box { <BOX PARAMS> }
Box { <BOX PARAMS> } }
Group {
Box { <BOX PARAMS> }
Sphere { <SPHERE PARAMS> }
Sphere { <SPHERE PARAMS> } } }
Plane { <PLANE PARAMS> } }
Adding Materials
Group {
numObjects 3
Group {
numObjects 3
Box { <BOX PARAMS> }
Box { <BOX PARAMS> }
Box { <BOX PARAMS> } }
Group {
numObjects 2
Group {
Box { <BOX PARAMS> }
Box { <BOX PARAMS> }
Box { <BOX PARAMS> } }
Group {
Box { <BOX PARAMS> }
Sphere { <SPHERE PARAMS> }
Sphere { <SPHERE PARAMS> } } }
Plane { <PLANE PARAMS> } }
Adding Transformations
Using Transformations

Position the logical
groupings of objects
within the scene

Transformation in group
Directed Acyclic Graph
is more efficient and useful
Processing Model
Transformattions

Goal

Get everything into world coordinates
Traverse graph/tree in depth-first order
 Concatenate transformations
 Can store intermediate transformations
 Apply/associate final transformation to
primitive at leaf node
 What about cylinders, superquadrics, etc.?


Transform ray!
Transform the Ray

Map the ray from World Space
to Object Space
r minor
r major
(x,y)
r=1
(0,0)
World Space
pWS = M
pOS
pOS = M-1 pWS
Object Space
Transform Ray

New origin:
originOS = M-1 originWS

New direction:
directionOS = M-1 (originWS + 1 * directionWS) - M-1 originWS
directionOS = M-1 directionWS
originWS
directionWS
qWS = originWS + tWS * directionWS
originOS
directionOS
qOS = originOS + tOS * directionOS
World Space
Object Space
Transforming Points &
Directions

Transform point

Transform direction
Homogeneous Coordinates: (x,y,z,w)
W = 0 is a point at infinity (direction)

Map intersection point and normal back to
world coordinates
Transform Normals

Why? They’re used for shading
object color only
Diffuse Shading
Transforming Normals

A surface normal is a property, not a
geometric entity

Correct normal transformation matrix:
See http://www.cgafaq.info/wiki/Transforming_Normals
Constructive Solid Geometry
(CSG)
Given overlapping shapes A and B:
Union
Intersection
Subtraction
How can we implement CSG?
Points on B,
Outside of A
Points on A,
Outside of B
Points on B,
Inside of A
Union
Points on A,
Inside of B
Intersection
Subtraction
Collect all the intersections
Union
Intersection
Subtraction
Implementing CSG
Test "inside" intersections:
1.
•
•
Find intersections with A,
test if they are inside/outside B
Find intersections with B,
test if they are inside/outside A
Overlapping intervals:
2.
•
•
Find the intervals of "inside"
along the ray for A and B
Compute
union/intersection/subtraction
of the intervals
Constructive Solid
Geometry
Ray Tracing CSG
Models
Rick Parent
Ohio State U.
CSG

Form object as booleans of primitive
objects
Primitives: sphere, cube, cylinder, cone
 Boolean operators: union, intersection,
difference


Tree structure used to manage operations
Leaf nodes are primitive objects
 Intermediate nodes specify combination
operator

B
Union
C
+
C
Ray intersects union: at first intersection
CSE 681
Min (tCmin, tBmin )
B
Possible ways for 2
spans to overlap
B
Intersection
C
++
C
B
First time in B and in C
If ((tCmin< tBmin ) and (tCmax> tBmin ) ): tBmin
Else If ((tBmin< tCmin ) and (tBmax> tCmin ) ): tCmin
Else: none
B
Difference
C
-+
C
First time in B not in C
If ((tBmin< tCmin ): tBmin
Else if (tCmax< tBmax ): tCmax
Else: none
B
B
Difference
+-
C
C
First time in C not in B
If ((tCmin< tBmin): tCmin
Else if (tBmax< tCmax ): tBmax
Else: none
B
Primitives
Anything that can be intersected (easily) with a ray
Conics: solve analytically using R(t)
Convex polyhedra
A plane (a cutting plane is useful)
can be used as a modeling tool (boolean operations)
surface model (e.g., polyhedron) computed from CGS
or
Can be used as a model representation
keep tree structure and ray trace directly
Controlling the Combinations
+
-
+
?
T2
T1
- +
Tree Structure
+
T3
T5
+ +
T4
+
-
T2
rectangle
T1
circle
T3
rectangle
Tree Structure #1
T2
T1
- +
Tree Structure
+
T3
T5
+ +
T4
-
+
T2
rectangle
T1
circle
T3
rectangle
Tree Structure #2
Tree Structure
• Intersect ray with leaf nodes (primitive objects)
• Combine intersection spans according to
intermediate nodes
• union
• intersection
• difference
• Might create multiple spans
Union of Spans
Intersection of Spans
Difference of Spans
Normals of CSG intersections
Normal of some surface (or its negation)
Union or intersection:
positive normal of intersected surface
Difference normals
• Intersection is one of:
• tmin of positive object – normal of surface
• tmax of negative object – negated normal
Add transformations to tree
http://www.cs.mtu.edu/~shene/COURSES/cs3621/NOTES/model/csg.html
Construction
•Use bounding
volumes at leaf nodes
• Union bounding
volumes at interior
nodes
Bounding Volumes
T5
+ +
Traversal
•Top-down
•Test bounding
volume at interior
T4
+
-
T2
rectangle
T1
circle
T3
rectangle
CSE 681
Example
Example
Example (Simon Chorley)
Example (Simon Chorley)
For example:
Wrap Up

Discuss status/problems/issues with
this week’s programming assignment