• before computer graphics drawings were constructed by hand (drafting)
• applications include architecture, art, manufacturing, etc.
• there are two classes of viewing - parallel & perspective
  
  
  • perspective views
    • COP - center of projection
    • COP is the origin of the camera frame, the lens in a camera or eye
    • projectors converge at COP
      
      
      
      
  • parallel views
    • COP moved to infinity and becomes direction of projection DOP
    • projectors are parallel (never converge)
      
      

• parallel projections
  
  
  • orthographic projection - projectors are orthogonal to the projection plane
    
    
    
    
    • multiview orthographic projections
      • projection plane is parallel to one of the principle faces of the object
      • views preserve distances and angles
        
        
        
        
    • axonometric projection
      • the projection plane is not restricted to being parallel to one of the principle faces
      • circles are projected to ellipses
      • parallel lines are still parallel
      • angles are not preserved
        
        
        
        
      • three classes of axonometric projection
        
        
        
        
        • isometric
          • projection plane placed symmetrically wrt three of the principle faces
          • equal foreshortening in all three directions (we can make relative distance measurements)
        • dimetric
          • projection plane placed symmetrically with respect to two of the principle faces
          • equal foreshortening in two directions
        • trimetric
          • projection plane placed arbitrarily
          • different foreshortening in all three directions
            
            
  • oblique projection
    • projectors are not required to be orthogonal to the projection plane
    • in a plane parallel to the projection plane, distances, angles, and circles are preserved
      
      
      
  
  
• perspective projection
  • objects that are farther away appear smaller (diminuation of size)
  • parallel lines in the image converge at a vanishing point (not the COP)
    
    
    
    
  • you have multiple vanishing points depending on how many of the principle directions are parallel to the projection plane
    
    
    • suppose the principle directions of the building model are oriented on the x,y,z axes
    • three point projection
      • none of the principle directions are parallel to the projection plane
      • three vanishing points
        
        
        
        
    • two-point projection
      • one of the principle directions is parallel to the projection plane (y in this example)
      • two vanishing points
        
        
        
        
    • one-point projection
      • two of the principle directions are parallel to the projection plane (x and y)
      • one vanishing point
        
        

Viewing Transformation

• viewing is performed in two steps
  • first, position the camera relative to the world - this transformation is in the model-view matrix
  • second, perform the projection - this transformation is in the projection matrix
  • these two matrices are concatenated in the pipeline
    
    
• OpenGL supports perspective and parallel projections
• in OpenGL, the specification of the view is independent from the specification of the objects
• if you want any of the classical views, you will have to set up the matrix for it yourself
  
  
• viewing defaults
  
  • the default camera is located at the origin, looking along the negative z axis (right handed coordinate system)
    
    
    
    
  • the default viewing volume is a 2x2x2 cube centered at the origin, the default projection is an orthogonal projection
    
    
    
    
    
• we need to separate the camera and world frames
  • I find it easiest to think of moving the world frame - translate along the z axiz - T(0, 0, -d)
    
    
    
    
    
• example - positioning the camera to look along the positive x-axis towards the origin
  • you can think of positioning the camera (matches the figure below)
    • translate camera frame +d in z
    • rotate camera frame +90 around y
  • or you can think of orienting the world frame (matches the calls below that set this view)
    • rotate world frame -90 around y
    • translate world frame -d in z
  • you end up with the same arrangement, but I find the latter more intuitive
    (think of yourself as the camera, move the world where you want it)
  • when we discuss projections, we will always consider the camera to be at the origin, looking along the negative z axis, y is up
    
    
    
• calculating a viewing transformation
• the goal is to come up with a new set of coordinate axes for the camera frame
• the user specifies a view reference point (vrp), normal (n), and view plane up direction (vup)
  
  
  
  
• we calculate a transformation matrix that will position the camera at this position and orientation in the world frame
• the transformation will consist of a translation to the view reference point, and a rotation to orient the camera
  
  
• p is the view reference point (vrp in the figure)
  
  
  
  
• n is the normal to the view plane
  
  
  
  
• vup is a vector indicating the up direction for the view plane
  
  
  
  
• vup is projected onto the viewing plane to give the vector v that is orthogonal to n
  
  
  
  
• we get the third orthogonal vector by taking a cross product
  
  
  
  
• we normalize u, v, and n to obtain unit vectors
• u', v', and n' define the viewing coordinate system
• these can be used in a matrix to transform a vector in the viewing coordinate system to the world coordinate system - change of basis
  
  
  
  
• however, we want the matrix that goes the other way - we want the inverse of this matrix
  
  
  
  
• concatenating with the translation to vrp gives a matrix that performs the change of frame from world coordinates to viewing coordinates
  
  
  
  
  
• OpenGL Utility function gluLookAt( )
  • you specify an eye point e, a "look at" point a, and an up vector
  • the vector defined by (a - e) is the normal to the viewing plane, e is the vrp
    
    
    
  • gluLookAt( ) calculates the matrix described above for you and multiplies the currently selected matrix (the model-view matrix)
    
    
  • typically, we want to apply the viewing transformation last, thus it is executed first in the code
    
    
    
    
• preserving the viewing transformation
  • if you have a static view (the view is not changing)
    • it is typically set up after the projection in the reshape callback
      • set matrix mode to GL_PROJECTION
      • make call to set up projection
      • set matrix mode to GL_MODELVIEW
      • make call(s) to set up view
    • you have to be careful not to wipe out the viewing transformation in the display callback
      • one way to preserve it is to call glPushMatrix( ) on entry and glPopMatrix( ) at exit
  • if the view is changing dynamically
    • it is typically set up at the beginning of the display callback
      
      
• alternative user interfaces for specifying orientation
  
  
  • roll - pitch - yaw
    
    
    
    
    
  • elevation and azimuth
    
    
    
    

Viewing Examples

• planet.c
• Redbook Chapter 3
• play with the Nate Robbins projection tutor

Notes and figures from:

• Interactive Computer Graphics: A Top-Down Approach Using OpenGL®, Fourth Edition, by Edward Angel, Addison-Wesley, 2005