3D Computer Games

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3D Computer Games

• Peter Wonka
• pwonka_at_gmail.com

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Doom
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Unreal Tournament
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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

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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

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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 DynamicArraylt triangle gt
  TriangleArray

v1
v2
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

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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
x
z
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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
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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

y
x
z
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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!!

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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
• for each object o for each triangle t in
  o.TriangleArray
• temp_dist / temp_loc ? Intersect( t, l )
• if (hit_distance lt 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

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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
3
2
0
1
2
3
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Better Intersection

• Top down
• Front to back
• Recursivealgorithm

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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 GeometricModeling

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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
• TranslationThe color (light energy) of the
  point E(x) is the sum of all incoming light
  energy in a hemisphere of x
• ? You need Calculus

x
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Integration?

• Good news
• no analytic computation required
• nice and simple methods exist

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Integration

• Low dimensional Integrals
• trapezoidal integration
• gaussian quadrature
• Higher dimensional Integrals
• 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)

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Monte Carlo Estimator

• Algorithm Idea
• Pick N random variables Xi
• Sample function at Xi
• Compute weighted average
• Question
• How to exactly compute the average?

a
b
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Monte Carlo Estimator

• How can we show that the estimate is correct?
• What does it mean to be correct?
• ? You need to learn Probability and Statistics

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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

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Monte Carlo Estimator

• Extends easily to multiple dimensions
• Still very simple to implement

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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)