Computer Graphics World, View and Projection Matrices

1
Computer GraphicsWorld, View and Projection
Matrices

• CO2409 Computer Graphics
• Week 8

2
Lecture Contents

• Viewing a 3D Scene
• Camera Positioning
• The View Matrix
• Camera Internals
• Projecting 3D Vertices to 2D
• Projection Matrices

3
Model Space

• Scene models are defined in their own local
  coordinate system called Model Space
• E.g. this square is defined as four vertices in
  model space
• A(5,5,0), B(-5,5,0), C(-5,-5,0), D(5,-5,0)

• The origin in model space is set to the model
  centre
• Allows us to apply rotation and scaling with a
  sensible centre

4
Model Space to World Space

• Each model has a positioning matrix

• To transform it from model space into world space
• Built from basic transforms
• Model 1 is simply translated away from the origin
• Model 2 is rotated (Z) around the origin (first),
  then translated
• This matrix is called the World Matrix (or
  Transform)
• Transforms the vertices of the model into world
  space

5
Model Positioning

• Recall the position and rotation of a model can
  be specified by defining its local space
• Position P and axes X,Y Z
• Defined in world space
• Can write this in a matrix
• but this time add a fourth column from the
  identity (like last week)
• It turns out that this is the world matrix of the
  model
• Same as combining the basic transformations to
  position it

• So two ways to think about a world matrix

6
Building World Matrices

• Two ways to build a world matrix for a model
• Make a matrix from the desired position and axes
• Combine basic transformations (considering order)
• E.g. WorldMatrix Rot(X) Rot(Y) Trans(P)
• Can rebuild the matrix each time we render
• From stored variables
• Or can store it persistently
• To move or reorient a model we
• Apply another transform to its world matrix
• Or entirely rebuild the world matrix (either
  method)

7
Viewing a 3D Scene

• Have only considered the objects in a scene
• Not considered the viewer
• Treat the viewer as an object in the 3D
  environment, looking around
• Different from 2D where the viewer is outside of
  the scene, looking in
• So the viewer has a position and an orientation
• The viewer (or viewpoint) is usually considered
  to be a person (shown as an eye) or a camera
• We will use the camera analogy

8
Camera Positioning

• The camera is positioned like any other object in
  world space
• So it has a local coordinate system called Camera
  Space
• Define a matrix for the cameras position in the
  usual way
• This matrix can be translated and rotated like
  any other matrix
• To position and orient the camera
• Scaling the camera is unusual
• Effectively shrinks the world

9
World to Camera Space

• To render pixels on the screen, need to know how
  the models are positioned relative to the camera

• What is in front of the camera, what is behind
  etc
• Need to convert the models from world space into
  camera space
• We might think that the camera matrix can be used
  to do this
• but it converts wrong way
• From camera space to world space

10
Inverting the Camera Matrix

• Need another matrix that performs the inverse of
  the camera matrix
• Finding the inverse of a matrix is a well-known
  mathematical technique
• But it is expensive and best avoided where
  possible
• Can find the inverse easily if the camera matrix
  is built from basic transforms
• If Cam Rot(Z) Rot(Y) Rot(X) Translate(P)
• then Cam-1 Translate(-P) Rot(-X) Rot(-Y)
  Rot(-Z)
• Negate the translations and rotations and reverse
  the order of the multiplication

11
The View Matrix

• This inverted camera matrix is called the View
  Matrix (Transform)
• In fact we can calculate its general form
• This is the same result as the multiplication
  method given on the last slide
• Again two ways to think about this matrix
• And two possibilities when creating it

(Dot products on the bottom line)
12
Using World View Matrices

• Transform each model vertex by the world matrix
• To convert the models from model space into world
  space
• Then transform these world space models by view
  matrix
• Into camera space
• Now have a the entire 3D world as viewed from the
  perspective of the camera
• NB We view down the Z axis

13
Camera Internals 1

• Finally we project the 3D camera space models
  into 2D geometry
• The viewport is assumed to be a fixed distance in
  front of the camera
• Rays are projected from the 3D camera space
  vertices to the position of the camera
• Passing through the viewport

14
Camera Internals 2

• The viewport has a fixed size, and covers an
  angle from the camera called the field of view
  (FOV)
• FOV can be different in X Y
• Rays from the 3D models that hit the viewport
  define visible 2D vertices
• The key camera settings to perform this
  projection are
• The viewport distance
• The field of view

15
Projection Matrices

• The details of the camera calculations are in an
  additional set of notes for this week
• The process is performed in two steps
• Projection
• Perspective divide and scaling to pixel
  coordinates
• The first step uses the projection matrix
• Complex and unlike previous matrices you have
  seen
• It includes the FOV and viewport distance
• We will look how they can be created in the lab
• The second step is typically performed by the
  graphics API (based on the viewport size)

16
Process Overview

• For each vertex in our original model
• Multiply by the world, view, then projection
  matrix
• These matrices can themselves be combined
• Resulting vertices are in viewport space
• A final perspective divide and scale is needed to
  create a pixel position for the vertex
• In DirectX
• We only need to create the three matrices
• Create view / projection matrix once per render
  for camera
• Create a different world matrix for each model
• Vertex transformations done by our shader
  (programmable pipeline) or by DirectX (fixed
  pipeline)
• Final perspective divide is done by API in both
  cases