Efficiency and Optimization

We have already seen that we can reduce the final application of the transformation matrix to 4 * and 4 +. We can make approximations to further reduce this.

For rapid animation - rotation

x' = x cos q - y sin q -> 4* & 2 +
y' = y cos q + x sin q

If q small then cos q is close to 1 then can do ( 2* & 2+):

(1)
x' = x - y sin q
y' = x sin q + y

A better approximation is:

(2)
x' = x - y sin q
y' = x' sin q + y

This is better because the determinant of the corresponding matrix = 1, which is true for all rotations but not true for (1). Look at computed results for above for q = 1°, xo = 1.0, and yo = 1.0:

  Nrot    x(1)     y (1)      x(2)        y(2)
 
  360    1.057     1.055    1.00824      .999756  

  3600   1.75      1.72     1.0024       .99756  
 

Note: the coordinates will eventually will "blow up" so we must save old coordinates and periodically restore them.

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