# fpp

0th

Percentile

##### 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 $z=m\cdot p_1+n\cdot p_2$.

Keywords
math
##### Usage
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.

• fpp
• mn
##### Examples
# NOT RUN {
p <- c(1.01+1.123i, 1.1+1.43i)
mn(z=1:10,p) %*% p  ## should be close to 1:10

#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))

# }

Documentation reproduced from package elliptic, version 1.4-0, License: GPL-2

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