The function riemannMask can be used for laying a circular mask over
an existing phasePortrait (as generated with the function
phasePortrait). This mask shades the plot region outside the
unit circle. The unshaded area is a projection on the southern or northern
Riemann hemisphere. The standard projection used by
phasePortrait, i.e. invertFlip = FALSE hereby
corresponds to the southern Riemann hemisphere with the origin being the
south pole. If phasePortrait was called with invertFlip =
TRUE, then the unit circle contains the northern Riemann hemisphere with the
point at infinity in the center (see the vignette for more details). Options
for adding annotation, landmark points are available
(see Wegert (2012)
, p. 41).
Several parameters are on hand for adjusting the mask's transparency, color,
and similar features. some details, this function behaves less nicely under
Windows than under Linux (see Details).
riemannMask( colMask = "white", alphaMask = 0.5, circOutline = TRUE, circLwd = 1, circleSteps = 360, circleCol = par("fg"), gridCross = FALSE, annotSouth = FALSE, annotNorth = FALSE, xlim = NULL, ylim = NULL )
| colMask | Color for the shaded area outside the unit circle. Defaults to
"white". Can be any kind of color definition R accepts. I recommend,
however, to use a color definition without a transparency value, because
this would be overridden by the parameter |
|---|---|
| alphaMask | Transparency value for the color defined with
|
| circOutline | Boolean - if |
| circLwd | Line width of the unit circle outline. Obviously relevant
only when |
| circleSteps | Number of vertices to draw the circle. Defaults to 360 (one degree between two vertices). |
| circleCol | Color of the unit circle, default is the default foreground
color ( |
| gridCross | Boolean - if |
| annotSouth | Boolean - add landmark points and annotation for a
southern Riemann hemisphere, defaults to |
| annotNorth | Boolean - add landmark points and annotation for a
northern Riemann hemisphere, defaults to |
| xlim, ylim | optional, if provided must by numeric vectors of length 2
defining plot limits as usual. They define the outer rectangle of the
Riemann mask. If |
There is, unfortunately, a somewhat different behavior of this function under
Linux and Windows systems. Under Windows, the region outside the unit circle
is only shaded if the whole unit circle fits into the plot region. If only a
part of the unit circle is to be displayed, the shading is completely omitted
under Windows (annotation etc. works correctly, however), while it works
properly on Linux systems. Obviously, the function polypath,
which we are using for creating the unit circle template, is interpreted
differently on both systems.
Wegert E (2012). Visual Complex Functions. An Introduction with Phase Portraits. Springer, Basel Heidelberg New York Dordrecht London. ISBN 978-3-0348-0179-9.
# Tangent with fully annotated Riemann masks. # The axis tick marks on the second diagram (Northern hemisphere) # have to be interpreted as the real and imaginary parts of 1/z # (see vignette). The axis labels in this example have been adapted # accordingly. # \donttest{ # x11(width = 16, height = 8) # Screen device commented out # due to CRAN test requirements. # Use it when trying this example op <- par(mfrow = c(1, 2), mar = c(4.7, 4.7, 3.5, 3.5)) phasePortrait("tan(z)", pType = "pma", main = "Southern Riemann Hemisphere", xlim = c(-1.2, 1.2), ylim = c(-1.2, 1.2), xlab = "real", ylab = "imaginary", xaxs = "i", yaxs = "i", nCores = 2) # Max. two cores on CRAN, not a limit for your use riemannMask(annotSouth = TRUE, gridCross = TRUE) phasePortrait("tan(z)", pType = "pma", main = "Northern Riemann Hemisphere", invertFlip = TRUE, xlim = c(-1.2, 1.2), ylim = c(-1.2, 1.2), xlab = "real (1/z)", ylab = "imaginary (1/z)", xaxs = "i", yaxs = "i", nCores = 2) # Max. two cores on CRAN, not a limit for your use riemannMask(annotNorth = TRUE, gridCross = TRUE) par(op) # } # Rational function with Riemann masks without annotation. # The axis tick marks on the second diagram (Northern hemisphere) # have to be interpreted as the real and imaginary parts of 1/z # (see vignette). The axis labels in this example have been adapted # accordingly. # \donttest{ # x11(width = 16, height = 8) # Screen device commented out # due to CRAN test requirements. # Use it when trying this example op <- par(mfrow = c(1, 2), mar = c(4.7, 4.7, 3.5, 3.5)) phasePortrait("(-z^17 - z^15 - z^9 - z^7 - z^2 - z + 1)/(1i*z - 1)", pType = "pma", main = "Southern Riemann Hemisphere", xlim = c(-1.2, 1.2), ylim = c(-1.2, 1.2), xlab = "real", ylab = "imaginary", xaxs = "i", yaxs = "i", nCores = 2) # Max. two cores on CRAN, not a limit for your use riemannMask(annotSouth = FALSE, gridCross = FALSE, circOutline = FALSE, alphaMask = 0.7) phasePortrait("(-z^17 - z^15 - z^9 - z^7 - z^2 - z + 1)/(1i*z - 1)", pType = "pma", main = "Northern Riemann Hemisphere", invertFlip = TRUE, xlim = c(-1.2, 1.2), ylim = c(-1.2, 1.2), xlab = "real (1/z)", ylab = "imaginary (1/z)", xaxs = "i", yaxs = "i", nCores = 2) # Max. two cores on CRAN, not a limit for your use riemannMask(annotNorth = FALSE, gridCross = FALSE, circOutline = FALSE, alphaMask = 0.7) par(op) # }