Computing Viewing Transforms

NOTES:

  • The GEOMETRIC MODEL is in a RIGHT-HAND world coordinate system
  • The CAMERA is in a LEFT-HAND eye coordinate system
  • Don't attempt to figure these out without making good sketches first
  • Axis shufflers are used for two purposes: 1) to transform a coordinate system into a right handed one and 2) to make rotation angles less than 90° to make angle computations easier
  • The following transformations will transform the world coordinate system into the eye coordinate system by acting on the coordinate system
  1. T1 is a translation that moves the camera location (xc, yc, zc), specified in world coordinates, to (0, 0, 0).

  2. Next, find a convenient LEFT-HANDED coordinate system whose axes are parallel to the original coordinate system and shuffle the axes into this system with a transformation, Ts. This would now be an eye coordinate system, but one whose ze axis probably points in the wrong direction. Pick the axis directions so that a couple of coordinate system rotations of less than 90° will make the ze axis point at the "look-at" point.

  3. Do one or two rotations R1 and R2 as necessary to rotate the ze axis to pass through the "look-at" point.

    Make sure that the signs of the sin terms in each rotation matrix are correct. The cannonical transformation matrices assume that

    • The current coordinate system is RIGHT-HANDED
    • Objects are being rotated, rather than the coordinate system
    • The rotation direction is CCW
    • The rotation direction is determined when standing on the positive axis and looking through the origin at the negative part of the axis

    Reversing an odd number of these 4 conditions requires the use of the matrix inverse.

  4. The final transformation is T = R2R1TsT1