Video Game Rendering Mathematics

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Video Game Rendering Mathematics

• Zack Booth Simpson

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How to we model a body?

• Break each object up into lots of triangles.
  Each face is connected to a mesh of vertices
• Make each part of a body rigid
• Connect the rigid parts together with joints
• Let the joints rotate

From unrealtechnology.com
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Rendering Meshes

• Each vertex is positioned byan artist. The
  programmers getthis vertex list and tell
  thecomputer to connect the dots.
• But the vertices are in 3D?!
• How do we get from a 3D model to a 2D view on the
  screen?
• Create a virtual camera

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Pretend Cameras

• What does the camera see of this box?
• Two problems to solve
• Relative position and orientation of the box
  versus the camera
• The perspective distortion

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PerspectiveThings further away appear smaller
Create a pretend projection plane that
represents the computer screen
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Exercise The Perspectometer
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Answers The Perspectometer

• The length of the line on the board is 3 feet.
• The units are feet, because the distance from
  your eye to the board was measured in feet.
• Note that the length of your arm and the apparent
  height of the line were both measured in inches.

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Position and Orientation

• Meet vector
• From Latin vehere to carry same root as
  vehicle
• Each object in the world is placed and oriented.
• Vectors let us get from place to place and orient
  ourselves.

Where are the vertices of the box from the point
of view of the camera? (Never mind perspective
for now)
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Translate Move without rotate

• Relative position is easy
• Vector addition and subtraction!
• a b - c d
• d is relative to the camera we sayd is the
  position of the vertex in camera space

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Rotation

• Now things get harder
• From the cameras point of view the box is
  rotated
• From the boxs point of view the camera is rotated

• Coping with differing points of view is always a
  challenge!

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Whos point of view is right?
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Both / Neither side is right

• We need a way to measure each vector using the
  other one as the unit ruler.
• In other words we need tools to convert from one
  point of view to the other.

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Projection

• a/b from bs perspective b/a from as
  perspective
• Remember alwaysa/b means measure a against
  b orusing b as the ruler, how long is a?
• It is easy to forget/get confused about this I
  wish a teacher had beaten this into my head with
  a stick!

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Oh boy, here we go again
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Projection from different directions

• Each side can be projected onto the other.

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• As a is rotated away from b, the projection
  diminishes until at 90 degrees it is zero
• Perpendicular vectors can not measure each other
  they have zero relative projections!

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Projection with Cosine

• We dont want no stinking cosines theyre a
  pain!
• Heres a great trick

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Ever seen this lovely proof of the Pythagorean
theorem?
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Rectangular Version of the Pythagorean Theorem
Dot product graphic removed pending publication

• Cool! We can compute the projection with just two
  multiplies and add! Games have millions of
  vertices so this is very important! Considering
  speed is the difference between mathematics and
  computation!

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Projection with the dot productfrom bs point of
view

• TADA! p can be computed without angles or
  cosines!

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More Dot Product Goodness

• Dot product sign tells you about the relationship
  of the vectors.
• Same general direction
• Orthogonal
• Opposite directions

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Exercise Dot Products
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Answers Dot Products
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Back to Rotations

• Ok, so now we can project a vector onto another
• Whats this got to do with rotations again?
• Represent the cameras POV as aligned vectors.
  Call these basis vectors as in base
• The projection onto thesevectors gives us the
  rotationwe want! i.e. the boxfrom the POV of
  camera

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Basis Vector Demo

• Run demo app
• Skew, rotation, scale, reflection, projection
• Orthogonal basis vectors create a rotation or
  reflection

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The Matrix

• A cool way to organize basis vectors The Matrix
• The default basis vectors are one unit in each
  direction.
• Write each basis vector as a column in the matrix
• Write a block of numbers but visualize vectors!
• The matrix encodes any linear transformation

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The Power of The Matrix
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The Matrix as a Teleporter

• The matrix is a teleportation machine which
  moves points from one space to another

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A Teleporter inside a Teleporter?!

• What happens if you put a teleporter inside a
  teleporter?
• Compounding teleportations!

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Matrix Concatenation

• Map the basis vectors of one space into the
  another

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Matrix ConcatenationUsing Dot product
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Matrix ConcatenationUsing Dot product
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Matrix ConcatenationUsing Dot product
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Matrix ConcatenationUsing Dot product
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Exercise Matrix Arithmetic
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Answers Matrix Arithmetic
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Rotate, Scale, and Translate?

• So now we can rotate, scale, etc. but what ever
  happened to translations?
• The ugly way
• Would be nice to matrixify it!

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Stuffin the Matrix

• Make up a pretend dimension for the translation
• Hardwire 1 unit of this extra dimension in all
  vertices to be transformed.
• Where is this extra dimension exactly?
• Who cares! It works! Math is so cool

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Moving to 3D

• Just add a dimension again.
• Now matrices are 4x4
• Right-hand rule

From math.montana.edu
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Exercise Matrix Transforms
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Answers Matrix Transformations
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Game Matrix Mine Sweeper

• Theres a bomb in a house
• Theres two rounds
• You must move the bomb with one of the three
  given matrices in each round
• You can not go off the map
• Plan ahead!!

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Game Matrix Mine Sweeper

• Theres a bomb in a house
• Theres two rounds
• You must move the bomb with one of the three
  given matrices in each round
• You can not go off the map
• Plan ahead!!
• GO!

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Answers Matrix Mine SweeperRound 1
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Game Matrix Mine Sweeper
Round 2. GO!!
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Answers Matrix Mine SweeperRound 2
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Answers Matrix Mine SweeperBonus Round!!

• If you are outside a building, youre a hero!

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Answers Matrix Mine SweeperBonus Round!!

• If you are outside a building, youre a hero!
• If you are inside, this is your last chance! You
  are almost dead! Quick! Make up your
  own matrix!
• You have 20 seconds! GO!

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BOOM!
From www.prostunts.net
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A final desperate solution!The zero matrix!

• Collapse all paths to a singularity!