cycloids (version 1.0.2)

zykloid.scaleA: Wrapper for zykloid which allows to scale and position a cycloid by the radius A of the fixed circle and its midpoint

Description

While zykloid provides the basic functionality for calculating cycloids, this functions allows to re-size a cycloid by freely setting the radius on the fixed circle. In addition, the cycloid can be re-positioned by locating the fix circle's midpoint. See Figures 1 and 2 and zykloid for the geometrical principles of cycloids.

Usage

zykloid.scaleA(A, a, lambda, hypo = TRUE, Cx = 0, Cy = 0,
               RadiusA = 1, steps = 360, start = pi/2)

Value

A dataframe with the columns \(x\) and \(y\). Each row represents a tracepoint position. The positions are ordered along the trace with the last and the first point being identical in order to warrant a closed figure when plotting the data.

Arguments

A

The Radius of the fixed circle before re-sizing. Must be an integer Number > 0. Together with \(a\) (see below), \(A\) is only determining the cycloid's shape and number of peaks (see npeaks), while its actual size is defined by the argument \(RadiusA\) (see below).

a

The radius of the moving circle before re-sizing. Must be an integer Number > 0. Together with \(A\), \(a\) only determines the cycloid's shape and number of peaks (see npeaks), while its actual size is defined via the argument \(RadiusA\) (see below).

lambda

The distance of the tracepoint from the moving circle's (c\(mov\)) centre in relative units of its radius \(a\). \(lambda = 1\) means that the tracepoint is located on \(cmov\)'s circumference. For \(lambda < 1\), the tracepoint is on \(cmov\)'s area, e.g. if \(lambda = 0.5\), it is halfway between \(cmov\)'s centre and its circumference. If \(lambda > 1\) the tracepoint is outside \(cmov\)'s area, you might imagine it being attached to a rod which is attached to \(cmov\) and originates from its centre. E.g. \(lambda = 2\) would mean that the tracepoint's distance from cmov's centre equals \(2*a\). \(lambda = 0\) produces a circle because the tracepoint is identical with \(cmov\)'s centre.

hypo

logical. If TRUE, the resulting figure is a hypocycloid (\(lambda = 1\)) or a hypotrochoid (\(lambda != 1\)), because \(cmov\) is rolling along the inner side of the fixed circle (\(cfix\)). If FALSE, an epicycloid (\(lambda = 1\)) or an epitrochoid \(lambda != 1\) is generated, as \(cmov\) is rolling at the outside of \(cfix\)'s circumference.

Cx

x-coordinate of the fixed circle's midpoint. Default is 0.

Cy

y-coordinate of the fixed circle's midpoint. Default is 0.

RadiusA

The actual radius of the fixed circle. Default is 1.

steps

positive integer. The number of steps per circuit of the moving circle (\(cmov\)) for which tracepoint positions are calculated. The default, 360, means steps of 1 degree for the movement of cmov. Analogously, steps = 720 would mean steps of 0.5 degrees.

start

Start angle (radians) of the moving circle's (\(cmov\)) centre counterclockwise to the horizontal with the fixed circle's (\(cfix\)) centre as the pivot. The tracepoint will start at a peak.

Author

Peter Biber

Details

See Also

zykloid, zykloid.scaleAa, zykloid.scaleP

Examples

# Same hypotrochoid scaled to different radii of the fix circle
cycl1 <- zykloid.scaleA(A = 7, a = 3, lambda = 2/3, RadiusA = 1.3)
cycl2 <- zykloid.scaleA(A = 7, a = 3, lambda = 2/3, RadiusA = 1.0)
cycl3 <- zykloid.scaleA(A = 7, a = 3, lambda = 2/3, RadiusA = 0.7)
plot (y ~ x, data = cycl1, asp = 1, col = "red", type = "l",
      main = "A = 7, a = 3, lambda = 2/3")
lines(y ~ x, data = cycl2, asp = 1, col = "green")
lines(y ~ x, data = cycl3, asp = 1, col = "blue")
legend("topleft", c("RadiusA = 1.3", "RadiusA = 1.0", "RadiusA = 0.7"),
       lty = rep("solid", 3), col = c("red", "green", "blue"), bty = "n")
       


# In this example, RadiusA depends on the cosine of the x-coordinate
# of the fixed circle's centre
op <- par(mar = c(0,0,0,0), bg = "black")
ctrx <- seq(-2*pi, 2*pi, pi/10)
ccol <- rainbow(length(ctrx))
plot.new()
plot.window(asp = 1, xlim = c(-8, 8), ylim = c(-0.5, 0.5))
for(i in c(1:length(ctrx))) {
    zzz <- zykloid.scaleA(A = 9, a = 7, hypo = TRUE, Cx = ctrx[i],
                          Cy = -ctrx[i], lambda = 0.9,
                          RadiusA = 1.5 + cos(ctrx[i]), start = -pi/4)
    lines(y ~ x, data = zzz, col = ccol[i])
} # for i
par(op)



# Geometric degression of RadiusA makes a nice star
op <- par(mar = c(0,0,0,0), bg = "black")
plot.new()
plot.window(asp = 1, xlim = c(-10, 10), ylim = c(-10, 10))
rad <- 10
n <- 60
ccol <- heat.colors(n)
for(i in c(1:n)) {
    if (i/2 != floor(i/2)) { sstart = pi/2 }
    else                   { sstart = pi/4 }
    zzz <- zykloid.scaleA(A = 4, a = 3, RadiusA = rad, lambda = 1,
                          start = sstart)
    lines(y ~ x, data = zzz, col = ccol[i])
    rad <- rad * 0.9
} # for i
par(op)



# A windmill
op <- par(mar = c(0,0,0,0), bg = "black")
plot.new()
plot.window(asp = 1, xlim = c(-1.4, 1.4), ylim = c(-1.4, 1.4))
rrad <- sqrt(seq(0.1, 2, 0.1))
n    <- length(rrad)
ccol <- rainbow(n, start = 0, end = 0.3)
for(i in c(1:n)) {
    zzz <- zykloid.scaleA(A = 7, a = 3, RadiusA = rrad[i],
           hypo = TRUE, lambda = 1.1,
           start = pi/2 - (1*pi/7 - (i - 1) * 2*pi/(7 * n)))
    lines(y ~ x, data = zzz, col = ccol[n + 1 - i])
} # for i
par(op)



# Advanced Example: A series of cycloids with their centres
# located on a logarithmic spiral
op <- par(mar = c(0,0,0,0), bg = "black")
plot.new()
plot.window(asp = 1, xlim = c(-50, 50), ylim = c(-50, 50))
a     <- 1/32     # spiral's scaling constant
alpha <- pi/20    # spiral's slope angle
sphi  <- seq(0, 18 * pi, pi/25)   # series of angles for cycloid centres
rad  <- a * exp(tan(alpha)*sphi)  # corresponding spiral radii
spx  <- rad * cos(sphi)           # corresponding x-coordinates
spy  <- rad *sin(sphi)            # corresponding y-coordinates
n    <- length(sphi)
ccol <- rainbow(n, start = 2/3, end = 1/2)
for (i in c(1:n)) {
     czc <- zykloid.scaleA(A = 3, a = 1, lambda = 1.5,
            Cx = spx[i], Cy = spy[i],
            RadiusA = rad[i]/2.5, # cycloid radii depends on spiral radii
            start = pi + sphi[i]) # angle cycloid towards spiral centre
     lines(y ~ x, data = czc, col = ccol[i])
} # for i
par(op)