Fundamental period parallelogram

Reduce \(z=x+iy\) to a congruent value within the fundamental period parallelogram (FPP). Function mn() gives (real, possibly noninteger) \(m\) and \(n\) such that \(z=m\cdot p_1+n\cdot p_2\).

fpp(z, p, give=FALSE)
mn(z, p)

Arguments

z

Primary complex argument

p

Vector of length two with first element the first period and second element the second period. Note that \(p\) is the period, so \(p_1=2\omega_1\), where \(\omega_1\) is the half period

give

Boolean, with TRUE meaning to return M and N, and default FALSE meaning to return just the congruent values

Details

Function fpp() is fully vectorized.

Use function mn() to determine the “coordinates” of a point.

Use floor(mn(z,p)) %*% p to give the complex value of the (unique) point in the same period parallelogram as z that is congruent to the origin.

Author

Robin K. S. Hankin

Examples

p <- c(1.01+1.123i, 1.1+1.43i)
mn(z=1:10,p) %*% p  ## should be close to 1:10
#>                   [,1]
#>  [1,]  1-1.277872e-16i
#>  [2,]  2-2.555744e-16i
#>  [3,]  3+4.057530e-15i
#>  [4,]  4-5.111488e-16i
#>  [5,]  5-6.607495e-15i
#>  [6,]  6+8.115061e-15i
#>  [7,]  7+3.546382e-15i
#>  [8,]  8-1.022298e-15i
#>  [9,]  9-5.590977e-15i
#> [10,] 10-1.321499e-14i

 #Now specify some periods:
 p2 <- c(1+1i,1-1i)

 #Define a sequence of complex numbers that zooms off to infinity:
 u <- seq(from=0,by=pi+1i*exp(1),len=2007)

 #and plot the sequence, modulo the periods:
 plot(fpp(z=u,p=p2))

 #and check that the resulting points are within the qpp:
polygon(c(-1,0,1,0),c(0,1,0,-1))