cycloids (version 1.0.2)

zykloid.scaleP: Wrapper for zykloid which scales a cycloid by the circle its peaks are located on and allows free positioning

Description

While zykloid provides the basic functionality for calculating cycloids, this functions allows to re-size a cycloid by freely setting the radius of the circle its peaks are located on. In addition, the cycloid can be re-positioned by locating the fixed circle's midpoint. This function behaves similarly as zykloid.scaleAa. See details. See Figures 1, 2, and zykloid for the geometrical principles of cycloids.

Usage

zykloid.scaleP(A, a, lambda, hypo = TRUE, Cx = 0, Cy = 0,
               RadiusP = 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 fix 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 \(RadiusP\) (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 \(RadiusP\) (see below).

lambda

The distance of the tracepoint from the moving circle's (\(cmov\)) 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 crosses 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 fix circle's midpoint. Default is 0.

Cy

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

RadiusP

The actual radius of the circle the cycloid's peaks are located on. 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

This function scales the radius of the circle the cycloids peaks are located on. For hypocycloids and hypotrochoids it will thus behave the same way as zykloid.scaleAa. For epicycloids and epitrochoids the output will be different. While zykloid.scaleAa scales the outer edge of the figure, zykloid.scaleP always scales the circle where the peaks of the figure are located on. In the case of epicycloids and epitrochoids this is at the inside of the figure (see examples below).
Figure 1 and 2 show the principle behind cycloid construction:

See Also

zykloid, zykloid.scaleA, zykloid.scaleAa

Examples

# Epitrochoids with different lambda scaled to the same radius of
# the peak circle
cycl1 <- zykloid.scaleP(A = 21, a = 11, lambda = 1.2, hypo = FALSE,
                         RadiusP = 10)
cycl2 <- zykloid.scaleP(A = 21, a = 11, lambda = 1.0, hypo = FALSE,
                         RadiusP = 10)
cycl3 <- zykloid.scaleP(A = 21, a = 11, lambda = 0.8, hypo = FALSE,
                         RadiusP = 10)
plot (y ~ x, data = cycl1, col = "red", asp = 1, type = "l",
      main = "A = 21, a = 11, RadiusP = 10")
lines(y ~ x, data = cycl2, col = "green")
lines(y ~ x, data = cycl3, col = "blue")
legend("topleft", c("lambda = 1.2", "lambda =  1.0", "lambda =  0.8"),
       lty = rep("solid", 3), col = c("red", "green", "blue"),
       bty = "n")



# Cool Disk by scaling the start angle with an
# exponential function ...
op <- par(mar = c(0,0,0,0), bg = "black")
plot.new()
plot.window(asp = 1, xlim = c(-11, 11), ylim = c(-11, 11))
n <- 30
ccol <- topo.colors(n)
for(i in c(1:n)) {
    zzz <- zykloid.scaleP(A = 3, a = 1, RadiusP = 6, lambda = 1,
           start = 2*pi/3 * exp(-0.1 * (i - 1)), hypo = FALSE)
    lines(y ~ x, data = zzz, col = ccol[i])
} # for i
par(op)



# ... the free space in the centre could be filled with
# the corresponding hypocycloid ...
op <- par(mar = c(0,0,0,0), bg = "black")
plot.new()
plot.window(asp = 1, xlim = c(-11, 11), ylim = c(-11, 11))
n <- 30
ccol <- topo.colors(n)
for(i in c(1:n)) {
    zzz <- zykloid.scaleP(A = 3, a = 1, RadiusP = 6, lambda = 1,
           start = 2*pi/3 * exp(-0.1 * (i - 1)), hypo = FALSE)
    lines(y ~ x, data = zzz, col = ccol[i])
    zzz <- zykloid.scaleP(A = 3, a = 1, RadiusP = 6, lambda = 1,
           start = 2*pi/3 * exp(-0.1 * (i - 1)), hypo = TRUE)
    lines(y ~ x, data = zzz, col = ccol[i])
} # for i
par(op)



# ... or the same ring again and again.
op <- par(mar = c(0,0,0,0), bg = "black")
plot.new()
plot.window(asp = 1, xlim = c(-11, 11), ylim = c(-11, 11))
n <- 30
ccol <- topo.colors(n)
rad <- 6
for(g in c(1:7)) {
    for(i in c(1:n)) {
        zzz <- zykloid.scaleP(A = 3, a = 1, RadiusP = rad,
               lambda = 1, start = 2*pi/3 * exp(-0.1 * (i - 1)),
               hypo = FALSE)
        lines(y ~ x, data = zzz, col = ccol[i])
    } # for i
    rad <- rad * 3/5
} # for g
par(op)



# Cauliflower pattern. Here, an exponential function is used
# for scaling the radius of the circle the cycloid's loops
# are on.
op <- par(mar = c(0,0,0,0), bg = "black")
plot.new()
plot.window(asp = 1, xlim = c(-22, 22), ylim = c(-22, 22))
n <- 15
dcol <- heat.colors(n)
for(i in c(1:n)) {
    lambdax <- seq(2.0, 2.2, 0.1)
    for(j in c(1:length(lambdax))) {
        zzz <- zykloid.scaleP(A = 11, a = 1,
               RadiusP = 15 * exp(-0.3 * (i - 1)),
               lambda = lambdax[j], hypo = FALSE,
               start = pi/2 + (i - 1)*pi/11)
        if(j/2 == floor(j/2)) { colx <- "blue" }
        else                  { colx <- dcol[n + 1 - i] }
        lines(y ~ x, data = zzz, col = colx)
    } # for j
} # for i
par(op)



# Sparkling star
op <- par(mar = c(0,0,0,0), bg = "black")
plot.new()
plot.window(asp = 1, xlim = c(-15, 15), ylim = c(-15, 15))
llam <- seq(0, 8, 0.2)
ccol <- rainbow(length(llam), start = 2/3, end = 1/3)
for(i in c(1:length(llam))) {
    zzz <- zykloid.scaleP(A = 5, a = 1, RadiusP = 2.1,
           hypo = FALSE, lambda = llam[i], start = pi/5)
    lines(y ~ x, data = zzz, col = ccol[i])
} # for i
par(op)