| view {elliptic} | R Documentation |
Visualization of complex functions using colourmaps and contours
view(x, y, z, scheme = 0, real.contour = TRUE, imag.contour = real.contour, default = 0, col="black", r0=1, power=1, show.scheme=FALSE, ...)
x,y |
Vectors showing real and imaginary components of complex
plane; same functionality as arguments to image() |
z |
Matrix of complex values to be visualized |
scheme |
Visualization scheme to be used. A numeric value is
interpreted as one of the (numbered) provided schemes; see source
code for details, as I add new schemes from time to time and the
code would in any case dominate anything written here.
A default of zero corresponds to Thaller (1998): see references. For no colour (ie a white background), set scheme to a
negative number.
If scheme does not correspond to a built-in function, the
switch() statement “drops through” and provides a
white background (use this to show just real or imaginary contours;
a value of -1 will always give this behaviour)
If not numeric, scheme is interpreted as a function that
produces a colour; see examples section below. See details section
for some tools that make writing such functions easier |
real.contour,imag.contour |
Boolean with default TRUE
meaning to draw contours of constant Re(z) (resp: Im(z))
and FALSE meaning not to draw them |
default |
Complex value to be assumed for colouration, if
z takes NA or infinite values; defaults to zero.
Set to NA for no substitution (ie plot z “as is”);
usually a bad idea |
col |
Colour (sent to contour()) |
r0 |
If scheme=0, radius of Riemann sphere as used by
Thaller |
power |
Defines a slight generalization of Thaller's scheme.
Use high values to emphasize areas of high modulus (white) and low
modulus (black); use low values to emphasize the argument over the
whole of the function's domain.
This argument is also applied to some of the other schemes where it makes sense |
show.scheme |
Boolean, with default FALSE meaning to
perform as advertized and visualize a complex function; and
TRUE meaning to return the function corresponding to the
value of argument scheme |
... |
Extra arguments passed to image() and contour() |
The examples given for different values of scheme are intended
as examples only: the user is encouraged to experiment by passing
homemade colour schemes (and indeed to pass such schemes to the
author).
Scheme 0 implements the ideas of Thaller: the complex plane is mapped
to the Riemann sphere, which is coded with the North pole white
(indicating a pole) and the South Pole black (indicating a zero). The
equator (that is, complex numbers of modulus r0) maps to
colours of maximal saturation.
Function view() includes several tools that simplify the
creation of suitable functions for passing to scheme.
These include:
breakup():function(x){ifelse(x>1/2,3/2-x,1/2-x)}g():function(x){0.5+atan(x)/pi}scale():function(x){(x-min(x))/(max(x)-min(x))}wrap():function(x){1/2+x/(2*pi)}
Additional ellipsis arguments are given to both image() and
contour() (typically, nlevels). The resulting
warning() from one or other function is suppressed by
suppressWarnings().
Robin K. S. Hankin
B. Thaller 1998. Visualization of complex functions, The Mathematica Journal, 7(2):163–180
n <- 55
x <- seq(from=-4,to=4,len=n)
y <- x
z <- outer(x,1i*y,"+")
view(x,y,limit(1/z),scheme=2)
view(x,y,limit(1/z),scheme=18)
view(x,y,limit(1/z+1/(z-1-1i)^2),scheme=5)
view(x,y,limit(1/z+1/(z-1-1i)^2),scheme=17)
view(x,y,z^2,scheme=15,main="try finer resolution")
view(x,y,sn(z,m=1/2+0.3i),scheme=6,nlevels=33,drawlabels=FALSE)
view(x,y,limit(P(z,c(1+2.1i,1.3-3.2i))),scheme=2,nlevels=6,drawlabels=FALSE)
view(x,y,limit(Pdash(z,c(0,1))),scheme=6,nlevels=7,drawlabels=FALSE)
view(x,x,limit(zeta(z,c(1+1i,2-3i))),nlevels=6,scheme=4,col="white")
# Now an example with a bespoke colour function:
fun <- function(z){hcl(h=360*wrap(Arg(z)),c= 100 * (Mod(z) < 1))}
view(x,x,limit(zeta(z,c(1+1i,2-3i))),nlevels=6,scheme=fun)
view(scheme=10, show.scheme=TRUE)