cusp (version 2.3.8)

cusp3d.surface: Generate 3D plot of the Cusp surface

Description

This function generates a 3D display of the cusp equilibrium surface.

Usage

cusp3d.surface(alpha = c(-5, 5), beta = c(-3, 3), y = 41, 
xlim = range(alpha), ylim = range(beta), zlim = c(-5, 4), 
xlab = expression(alpha), ylab = expression(beta), zlab = "equilibrium states", 
main = NULL, sub = NULL, phi = 20, theta = 160, 
r = sqrt(3), d = 1, scale = TRUE, expand = 1, hue = 240, 
chroma = 35, surf.alpha = 0.75, gamma = 1.5, bcol = NA, 
lcol = "gray", ltheta = 90, lphi = 70, box = TRUE, 
axes = FALSE, nticks = 5, ticktype = "simple", floor.lines = TRUE, ...)

Value

cusp3d.surface returns the viewing transformation matrix, say VT, a 4 x 4 matrix suitable for projecting 3D coordinates (x,y,z) into the 2D plane using homogeneous 4D coordinates (x,y,z,t). It can be used to superimpose additional graphical elements on the 3D plot, by lines() or points(), using the simple function trans3d().

Arguments

alpha

numeric 2-vector specifying the normal/symmetry factor axis range

beta

numeric 2-vector specifying the bifurcation/splitting factor axis range

y

numeric specifying the iso contours used to render the surface (see details below)

xlim,ylim,zlim

numeric 2-vectors (see persp)

xlab,ylab,zlab,main,sub

strings (see persp)

phi,theta

numeric, determine viewing direction (see persp)

r

numeric, distance to center of the plotting box (see persp)

d

numeric, strength of perspective transformation (see persp)

scale,expand

logical, see persp

hue,chroma,surf.alpha

hue, chroma and alpha (transparency) of the surface segments (see hcl)

gamma

gamma for shading of surface (see cusp3d)

bcol

color, NA, or string "surface". Color of the border of each surface element; NA gives transparent borders; "surface" tries to hide the border as much as possible by giving it the same color as the surface segment.

lcol

color of the lines on the floor of the plotting cube

ltheta,lphi

numeric, direction of illumination of the surface (similar to persp)

box,axes,nticks,ticktype

(see persp)

floor.lines

logical, if TRUE (default) iso-contours are projected on the floor of the plotting cube (revealing the bifurcation set)

...

extra arguments that are passed to lines and polygon

Author

Raoul Grasman

Details

If y has length 1, it is interpreted as the number of contours. Otherwise it is interpreted as a vector of contour levels from which the surface must be determined. If y is a number, the exact range of y is determined by the ranges of alpha and beta through the cusp equilibrium equation below.

The surface is constructed from the iso-contours of the cusp equilibrium surface that makes up the solutions to $$\alpha + \beta*y - y^3 = 0$$ as a (multi-)function of the asymmetry variable \(\alpha\) and bifurcation variable \(\beta\). For each possible solution \(y\) the iso-contours are given by the equation $$\alpha = (\beta*y - y^3)/y,$$ which are linear in \(\beta\). For each value of \(y\) the values of \(alpha\) are determined for the end points of the \(beta\) range specified by beta. The two 3D coordinates (\(\alpha\), \(\beta\), \(y\)) are projected onto the 2D canvas using the persp transformation matrix and used for drawing the lines and polygons.

References

See cusp-package, cusp3d

See Also

persp, plot.cusp

Examples

if (FALSE) {
p = cusp3d.surface(chroma=40,lcol=1,surf.alpha=.95,phi=30,theta=150,
	bcol="surface",axes=TRUE,main="Cusp Equilibrium Surface")
lines(trans3d(c(5,5), c(3,3), c(-5,4), p), lty=3) # replot some of the box outlines 
lines(trans3d(c(-5,5), c(3,3), c(4,4), p), lty=3)
}