Scalars

• Numbers with the following properties:
  • addition/multiplication
  • inverses (and thus subtraction/division)
  • commutative, associative, distributive
• No geometric properties
• Used to express values such as distances
• lowercase letters from the beginning of the alphabet; lowercase Greek letters
• Forms a scalar field

Vectors / Vector Spaces

• Magnitude and direction
  • Force, velocity
• Can be thought in terms of directed line segments

  

  • length is magnitude
  • orientation is direction
• No position

  

• zero vector
  • Zero length, no direction
• Vector-Scalar multiplication
  • w = kv
  • results in a vector with same (or opposite) direction and original magnitude multipled by the scalar

    

• Vector addition
  • w = u + v
  • uses head-to-tail axiom

    

• Vector subtraction
  • w = u + v
  • uses tail-to-tail

    

• inverse of a vector
  • -v
  • When aded to vector, produces zero vector
  • same magnitude opposite direction

    

• Expression using the above operations
  • v = u + 2w - 3r

    

• lowercase letters from the end of the alphabet
• Together with scalars forms a vector space

Points / Affine Spaces

• Location in space
• Point-point subtraction
  • equivalent to point-vector addition

    

    • v P - Q
    • Q = P + v
  • Leads to the notion of a line:
    • All points of the form P(a) = P0 + αd where
      • P0 arbitrary point
      • α is a scalar that ranges over a set of values
      • d is an arbitrary vector

      form a line

      

  • The above is called a parametric form of the line
    • It's simple to generate points on the line-- simply iterate through α
      • Back to circle
        • explicit - y² = r² - x²
        • Implicit - x² + y² = r²
        • parameteric - x = cos(θ), y = sin(θ)
      • For line:
        • explicit - y = mx + b
        • Implicit - ax + by + z = 0
        • parameteric - P(a) = P0 + αd
  • P0 is the point for which α is 0.
  • For α > 0, the line extends to infinity in the direction of d
    • That directional infinite line is known as a ray
• Uppercase letters
• Added to vector space forms an affine space

Linear (In)dependence

• A set of vectors are linearly dependent if one of them can be expressed in terms of the others
  
  • Given vectors

    							 			
    v1 = [0, 0, 1] 
    v2 = [0, 2, -2]
    v3 = [1, -2, 1]
    v4 = [4, 2, 3]
    								
    							

    • The first three are linearly independant
    • The fourth can be expressed in terms of the others

      																								
      v4 = 9v1 + 5v2 + 4v3
      													
      											

    • This shouldn't be surprising as this is a 3D vector
      • No more than three vectors can be linearly independent in three dimension
        • In real life (3D), you never need to more than three directions-- e.g. how far north/south, east/west, and up/down. Any further information is superfluous.
        • Those directions are exactly vectors (magnitude and direction)
• If no such combination can be found the vectors are said to be linearly independent

Basis

• As referred to above, 2D/3d/4d vector spaces can have at most 2/3/4 linearly independent vectors
• A set of such vectors is called a basis.
• Any vector in the space may be expressed in terms of the basis vectors

   
  w = α1v1 + α2v2 + α3v3
  	
  			

• The scalars, α_i, are called the components of the vector w wth respect to the basis v₁, v₂, v₃.

  

  • Think if each component as the effect the corresponding vactor of the basis has upom the resulting vector
    • In our above example:
      • α₁ is the effect v₁ has on w
      • α₂ is the effect v₂ has on w
      • α₃ is the effect v₃ has on w
• We typically write the components as a column vector

   
     | α1 |
     | α2 |     let's call this a
     | α3 |
  	
  					 

  and this is known as the representation of the vector with respect to the basis.
• The transpose of a column vector is the corresponding row vector containing the same components in the same order. Thus:

   
  | α1 | T
  | α2 | = | α1  α2 α3 |
  | α3 |
  	
  					 

• We often also write the express the relationship between the α's and v's as:

   
               | v1 |
     |α1 α2 α3| | v2 |  = α1v1 + α2v2 + α3v3
               | v3 |
  	
  					 

  i.e., using a row vector for the α's.
• If we restrict ourselves to talking directly about columns vectors, then we write the above as

   aTv
  	
  		

  • And we can think of the above as multiplying two vectors

Coordinate Systems

v = α1e1 + α2e2 + α3e3 
		

e1 = (1, 0, 0)
e2 = (0, 1, 0)
e3 = (0, 0, 1)
	 

Frames in OpenGL

Returning to the (relative) coordinates that appear in the pipeline:

• Object or model coordinates
• >World coordinates
• Camera (eye) coordinates
• Clip coordinates
• Normalized device coordinates
• Window (screen coordinates

Model-View Transformations

Tracing this sequence in more detail:

• A vertex is specified via glVertex
  • These coordinate may be specified in a global frame (world coordinates) or in a frame local to a particular object being drawn
    • There may actually be a hierarchy of such object's (remember the discussion glPushMatrix and of drawing a nose on a face-- each nested object had its own (relative) frame/coordinate system.
• At some later point we wish to bring all the individual objets with their local frames into the world frame-- converting the local object coordinates into world coordinates
  • World and object coordinates will be different only if some transformation (for example a rotation0 is performed before drawing the vertices of the objet.
• The world frame/coordinates form the entire picture; they are then transformed into the camera frame which composes the picture (for example the angle at which the viewing is being done)

• The sequence of transformations are concatenated into a single model-view transformation. This is done by multiplying the individual transformation matrices into a single matrix known as the model-view matrix.

At this point a further set of projection transformations occur:

• The eye coordinates are trnasformed into clip coordinates which allow the system to determine which objects (or portion thereof) lie outside the clipping/viewing volume.
• A perspective division is performed (basically applying standard rules of perspective (more on that later) resulting in normalized device coordinates
• The normalized coordinates are transformed into window coordinates which still retain depth information but use the coordinate system of thedisplay device.
• Screen coordinates are then obtained by stripping out depth information

• Responsible for
  • Viewing volume (how much of the world we want to see)
  • Shape of the viewing volume (orthographic = parallelpided; perspective = frustum-- truncated pyramid)
  • Applied after ModelView in the pipeline
  • More details next chapter
• Has its own matrix
  • To select it, use glMatrixMode with a GL_PROJECTION parameter

• Specifies the size of a parallelpiped (box) viewing volume
  
  • Parallel projectors
    
  • Specify width(left/right), height(bottom/top) and depth (near/far) of box
    
  • Default is box
    • 2 wide (-1, 1)
    • 2 high (-1, 1)
    • 2 deep (-1, 1)
    • Origin at center
    
  • Projects onto z = 0 plane
    
  • Can 'see' object behgind the camera