New uses for complex numbers

I wish I'd known MATLAB had... complex numbers!
If you do any geometry in matlab, you'll find complex numbers really handy.

• The great advantage is, if you have 10x10 points, normally you'd need to store them either as

• X = 10x10 matrix of X-coordinates, Y = 10x10 matrix of Y-coordinates
  Or as:
• M = cat(3,X,Y),  which is a 10x10x2 matrix of coordinates; the first slice is the X's and the second slice is Y's.

   The first is irritating because if you want to copy the array, transform or pass it to functions etc... you always have 2 variables.
   The second is irritating because proper matrix operations don't work in 3 dimensions. Also you have to "squeeze" to extract coordinates e.g. M=SQUEEZE(M(:,:,1)). Also, if you are using many dimensions, you have to remember which dimension is the X/Y dimension. 

• The complex solution is easy and aesthetically pleasing!
      Z=X+1j*Y     which is a 10x10 matrix of complex numbers.
• To quickly calculate the distances of each of the points in a matrix, to another point P=x+iy,
      distance = abs(Z-P)
• and similarly for the angles, use the argument function, angle(Z-P)    (in most other languages this function is called "arg".
• The real beauty is if you wanted to rotate all the points in the matrix by 45 degrees around the point P=x+iy, simply do
      (Z-P)*exp(1j*pi/4)+P - which translates by -P, rotates by pi/4, and adds back P. Now imagine doing that with X and Y coordinates!

  If you are dealing with a trajectory of points, the complex numbers can be fed into standard SMOOTH or CONV to perform smoothing.

• If you use i and j in for loops, don't worry! For complex numbers always use 1i or 1j.
• important functions are REAL, IMAG, ABS, ANGLE

Caveat: NaN behaves in an interesting way. Although ISNAN will be true if either the X or Y coordinate is NaN, the number still retains the X and Y coordinates separately!