3D Computer Games
Peter Wonka
pwonka@gmail.com
Doom
Peter Wonka, ASU 101
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Unreal Tournament
Peter Wonka, ASU 101
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How to implement a Computer Game?
 Computer Games need to be fast
 at least 60 image per second (the monitor refresh rate)
 You need an efficient programming language
  You have to learn C++
 You cannot use Java, Visual Basic, C# for critical
routines.
 Note: A mixture of languages can be used but right now
the performance critical parts are dominated by C++ in
the industry
 Efficient Programming Language is not enough
 We need an efficient architecture
 Regular Intel / AMD CPU is too slow
Peter Wonka, ASU 101
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Computation Speed
 How can we make computation faster?
 Idea: Parallel Computing
 In the last years most speed gains come from massive
parallel computation
 Graphics hardware is a cheap parallel “supercomputer”
that can work with your PC
  You need to study computer architecture and
parallel computing to build fast hardware
  You need to study computer architecture and
parallel computing to implement fast algorithms!
 Fast implementation is not always intuitive and
requires detailed system level knowledge
Peter Wonka, ASU 101
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NVidia Architecture / DirectX 10
16 Sets of 8 “streaming processors” (SP)
128 processors are a lot faster than one
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Why is graphics hardware so fast?
 Same small program is run for a large amount of
data
 Restricted memory access for reading memory
 Very restricted memory access for writing to memory
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How to control the game?
 Keyboard, Mouse, Trackball
  You need a good library to manage input
  You need to know about operating systems
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How to store a scene?
 How to model a scene?
 Professional 3D software (e.g. Maya, 3D Max)
 Design problem, not taught in computer science!
 How to store a scene?
 Super Simple First Try (no colors)
class point
{ float x,y,z; };
class triangle { point v1, v2, v2; };
class object
{ DynamicArray< triangle > TriangleArray; };
v1
v2
Peter Wonka, ASU 101
v3
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How to draw a scene?
 Super Simple Rendering Algorithm (Wireframe)
For each object o
For each triangle t in o.TriangleArray
v1projected = project t.v1 onto image plane
v2projected = project t.v2 onto image plane
v3projected = project t.v3 onto image plane
Draw line from v1projected to v2projected
Draw line from v2projected to v1projected
Draw line from v3projected to v1projected
Peter Wonka, ASU 101
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How to project a point on an image plane?
 Points in 3D are encoded as vectors in a 4D vector
space. v1 = (x,y,z,1), e.g. v1 = (7, 5, 3, 1)
  You need to understand vectors and vector spaces
 Why do you need 4 coordinates and why is the
fourth coordinate always 1?
 These coordinates are called homogenous coordinates
 Why will not be answered in this class
y
v1
z
Peter Wonka, ASU 101
x
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How to project a point on an image plane?
 Points are projected using matrix multiplication:
 v1Projected = ProjectionMatrix * v1
 ProjectionMatrix is defined by a virtual camera in the scene
Virtual Camera 3
Virtual Camera 1
v1
Virtual Camera 2
Peter Wonka, ASU 101
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How to transform objects in the scene?
 Most important transformations: Translation and Rotation
 You need matrix vector multiplication again
 Simply multiply all vertices of an object
 Below: Rotation around the z-axis with angle theta
 x' cos 
 y '  sin 
 
 z'  0
  
1  0
y
x
Peter Wonka, ASU 101
z
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 sin 
cos 
0
0
0 0  x 
0 0  y 
.
1 0  z 
 
0 1  1 
More Transformations
 How to compound transformations?
 You need matrix - matrix multiplication
 How to invert a transformations?
 You need to compute inverse matrices
 How to determine which transformations can be
inverted?
 You need the concept of singular matrices
 Summary
 You need to implement fast code in C++
  You need to learn Linear Algebra!!
Peter Wonka, ASU 101
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Shooting in Games?
 Your character shoots (along a straight line)
What is hit?
 Super Simple Algorithm
construct a straight line l  CharacterPosition    ShootingDi rection
for each object o
for each triangle t in o.TriangleArray
temp_dist / temp_loc  Intersect( t, l )
if (hit_distance < smallest_distance)
hit_dist / hit_location = temp_dist / temp_loc
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Shooting in Games
 Problem: Algorithm is inefficient
 If the scene has 1M triangles we wait for a very long time
 Solution: Use hierarchical data structures
 Octree (quadtree is 2d version)
 Kd-tree
 BSP-tree
 We need knowledge about
algorithms and (spatial) data structures
Peter Wonka, ASU 101
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Quadtrees
 Quatree is a spatial data structure
 Root node encloses a quadratic (rectangular space)
 Every internal node has four children
 Space is subdivided regularly
root
0
1
0
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2
1
3
2
3
Better Intersection
 Top down
 Front to back
 Recursive
algorithm
4
3
2
1
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How to create nice Shading Effects?
 Idea:
 Create a rough 3D models with Triangles
 Pretend the surfaces are smooth and have details
  You need to use
(Vector) Calculus,
Differential Geometry,
and Geometric
Modeling
Peter Wonka, ASU 101
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How to compute nice lighting effects?
 Problem: What color has a
point x in the image?
 During the game: there is not
time
 Solution: Precomputation
 Example Formulation:
E ( x)   L( x   ) cos( N x ,  )d
x
x
Peter Wonka, ASU 101
 Translation:
The color (light energy) of
the point E(x) is the sum of
all incoming light energy in
a hemisphere of x
  You need Calculus
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Integration?
 Good news:
 no analytic computation required
 nice and simple methods exist
Peter Wonka, ASU 101
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Integration
 Low dimensional Integrals
 trapezoidal integration
 gaussian quadrature
 Higher dimensional Integrals
 f ( x)dx

f ( x1 ,..., xn )dx1...dxn
 Monte Carlo integration
 Randomized Algorithms
 Monte Carlo – use randomness but results depends on
the sequence of random numbers
 Las Vegas – use randomness but always give the same
answer in the end (differ in speed)
Peter Wonka, ASU 101
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Monte Carlo Estimator
 Algorithm Idea:
b
Question :  f ( x)
 Pick N random variables Xi
 Sample function at Xi
 Compute weighted average
a
 Question:
 How to exactly compute the
average?
a
Peter Wonka, ASU 101
b
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Monte Carlo Estimator
b
Question :  f ( x)
a
Need : supply of uniform random variables X i  [a, b]
ba N
Monte Carlo Estimator:FN 
f (Xi )

N i 1
 How can we show that the estimate is “correct”?
 What does it mean to be correct?
  You need to learn Probability and Statistics
Peter Wonka, ASU 101
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Monte Carlo Estimator
 Proof Idea: Show that the expected value is the
value of the integral
 Proof details are beyond the scope of this lecture,
just to give you an idea:
b
E ( FN )   f ( x)
a
Proof :
b  a N
 ba  N

E[ FN ]  E 
f ( X i ) 
E  f ( X i )  

N
 N i 1

 i 1

ba N
ba N b

E[ f ( X i )] 
f ( x) p ( x)dx



a
N i 1
N i 1
1

N
N
b
i 1
a

b
f ( x)dx   f ( x)dx
Peter Wonka, ASU 101
a
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Monte Carlo Estimator
 Extends easily to multiple dimensions
 Still very simple to implement
x1 y1 z1
Question :    f(x,y,z)dzdydx
x0 y0 z0
Multidimen sional Samples :
X i  ( xi , yi , zi )
1
e.g. p( X )  const 
( x1  x0 )( y1  y0 )( z1  z0 )
( x1  x0 )( y1  y0 )( z1  z0 ) N
f (Xi)
FN 

N
i 1
Peter Wonka, ASU 101
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How to cheat in games?
 Multiplayer Online Games Cheating Ideas
 Modify the rendering of the game so you can see through walls
 Send incorrect movement updates for your opponents
 Use a bot
 You connect to a bot server, the bot connects to the real game 
bot filters your interaction
 You move, but the bot automatically shoots with “perfect” aim
 Game companies need to prevent cheating
 You need to learn about
 Networking
 Security
 Artificial Intelligence
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How to animate fire, water, smoke?
 You need PDEs = partial differential equations
 E.g. Navier Stokes Equations
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Summary
 Implementation skills and algorithms are important
for computer games
 Mathematics is also important (especially if you want
to have a very good job)
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Faculty
 Peter Wonka (Computer Graphics)
 Gerald Farin (Geometric Modeling)
 Greg Nielson (Scientific Visualization)
 Ashish Amresh (Computer Games)
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