conformalFit: Conformal mapping

In conformal: Conformal mapping from a polygon to a disk

Description

Solve the Schwartz-Christoffel equations for transforming a polygon plus point to the unit circle plus origin.

Usage

conformalFit(xypoly, wc = c(mean(xypoly$x), mean(xypoly$y)), nptsq = 12)

Arguments

xypoly	a list with components named ‘x’ and ‘y’ representing the polygon to be transformed to the unit circle. It should be in anticlockwise order.
wc	a vector of length 2 representing the complex value to be transformed to 0.
nptsq	the number of points to be used per subinterval in Gauss-Jacobi quadrature. Recommended value is equal to one more than the number of digits of accuracy desired in the answer. Default 12.

Details

The solution of the Schwartz-Christoffel equations involves finding the prevertices z, which are the points on the unit circle mapping to the vertices of the polygon, normalized so that z[length(z)] == 1. The formula for the mapping is as follows, where betam_k is the external angle in the polygon at vertex k divided by -pi.

w=w_c+c\int_0^z ∏_{k=1}^n (1-z/z_k)^{β_k}dz

Value

A list containing the prevertices and other information required to perform the transformation.

Author(s)

Nick Ellis, [e-mail]

References

Trefethen, LN (1980) Numerical computation of the Schwarz-Christoffel transformation, Siam J. Sci. Stat. Comp. 1, 82-102.

See Also

conformalPredict

Examples

CircleGrid <- function(nradial) { # create a random pattern to fill a disk
  r <- seq(0,1,len=nradial)
  nth <- pmax(1,ceiling(2*pi*r*(nradial-1)))
  th0 <- runif(nth,-pi/nth,pi/nth)
  list(
    x=unlist(lapply(1:nradial, function(i,r,nth,th0)
      r[i]*cos(th0[i]+seq(0,2*pi,len=nth[i]+1)[-nth[i]-1]),r=r,nth=nth,th0=th0)),
    y=unlist(lapply(1:nradial, function(i,r,nth,th0)
      r[i]*sin(th0[i]+seq(0,2*pi,len=nth[i]+1)[-nth[i]-1]),r=r,nth=nth,th0=th0))
  )
}
disk <- listxy2R(CircleGrid(30))
poly <- list(
  x=c(0.81,0.00,-0.73,-0.72,0.07,0.59,0.90),
  y=c(0.26,0.52,0.18,-0.44,-0.75,-0.39,0.01)
)
#poly <- locator(type='l')
fit <- conformalFit(poly)
pred <- conformalPredict(fit, disk, polygon2disk=FALSE)
cDisk <- R2C(disk)
cPred <- R2C(pred)
cols <- hsv(h=Arg(cDisk)/2/pi+0.5,s=pmin(1,Mod(cDisk)))
par(mfrow=c(1,2))
plot(cDisk, col=cols, asp=1, pch=16, main="Colour wheel")
lines(poly)
plot(cDisk, type='n', asp=1, main="Colour wheel\nmapped to polygon")
points(cPred, col=cols,pch=16)
circ <- abs(Mod(cDisk)-1) < 1e-6
segments(Re(cDisk[circ]),Im(cDisk[circ]),Re(cPred[circ]),Im(cPred[circ]), col=cols[circ])
lines(poly)

conformal documentation built on May 2, 2019, 5:47 p.m.