Package 'cycloids'

Calculating coordinate representations of hypocycloids, epicyloids, hypotrochoids, and epitrochoids

Description

Functions for calculating coordinate representations of hypocycloids, epicyloids, hypotrochoids, and epitrochoids (altogether called 'cycloids' here) with different scaling and positioning options. The cycloids can be visualised with any appropriate graphics function in R.

Details

This package has been written for calculating cartesian coordinate representations of hypocycloids, epicyloids, hypotrochoids, and epitrochoids (altogether called 'cycloids' here). These can be easily visualized with any R graphic routine that handles two-dimensional data. All examples shown here use standard R graphics. While there are technical applications, the main purpose of this package is to create mathematical artwork.
Geometrically, cycloids in the sense of this package are generated as follows (Figure 1, 2): Imagine a circle $cfix$, with radius $A$, which is fixed on a plane. Another circle, $cmov$, with radius $a$, is rolling along $cfix$'s circumference at the outside of $cfix$. The figure created by the trace of a point on $cmov$'s circumference is called an epicycloid (Figure 1A). If $cmov$ is rolling not at the outside but at the inside of $cfix$, the trace of a point on $cmov$'s circumference is called a hypocycloid (Figure 2A).
If in both cases the tracepoint is not located on $cmov$'s circumference but at a fixed distance from its midpoint either in- or outside cmov, the resulting figure is an epitrochoid (Figure 1B, C) or a hypotrochoid (Figure 2B, C), respectively. Hypotrochoids and epitrochoids became quite popular through toys like the spirograph.
The most important functions of the package are zykloid, zykloid.scaleA, zykloid.scaleAa, and zykloid.scaleP.

Note

Type demo(cycloids) for seeing some examples.

Author(s)

Peter Biber
Maintainer: Peter Biber <[email protected]>

References

Bronstein IN, Semendjaev KA, Musiol G, Muehlig H (2001): Taschenbuch der Mathematik, 5th Edition, Verlag Harri Deutsch, 1186 p. (103 – 105)

http://en.wikipedia.org/wiki/Epicycloid

http://en.wikipedia.org/wiki/Hypocycloid

http://en.wikipedia.org/wiki/Epitrochoid

http://en.wikipedia.org/wiki/Hypotrochoid

http://en.wikipedia.org/wiki/Spirograph

See Also

zykloid, zykloid.scaleA, zykloid.scaleAa, zykloid.scaleP

Examples

library(cycloids)

# Create and plot a hypocycloid, a hypotrochoid, an epicycloid,
# and an epitrochoid, all of them with radii A = 5 and a = 3
npeaks(5, 3)  # The cycloids will have five peaks
# The hypocycloid
cyc <- zykloid(A = 5, a = 3, lambda = 1, hypo = TRUE)
plot(y ~ x, data = cyc, type = "l", asp = 1, xlim = c(-12, 12),
     ylim = c(-12, 12), main = "A = 5, a = 3")
# The hypotrochoid
cyc <- zykloid(A = 5, a = 3, lambda = 1/2, hypo = TRUE)
lines(y ~ x, data = cyc, type = "l", asp = 1, col = "green")
# The epicycloid
cyc <- zykloid(A = 5, a = 3, lambda = 1, hypo = FALSE)
lines(y ~ x, data = cyc, type = "l", col = "red")
# The epitrochoid
cyc <- zykloid(A = 5, a = 3, lambda = 1/2, hypo = FALSE)
lines(y ~ x, data = cyc, type = "l", col = "blue")
legend("topleft", c("hypocycloid", "hypotrochoid", "epicycloid",
       "epitrochoid"), lty = rep("solid", 4),
       col = c("black", "green", "red", "blue"), bty = "n")



# Same Framework, different shape: A = 17, a = 5
npeaks(17, 5)  # The cycloids will have seventeen peaks
# The hypocycloid
cyc <- zykloid(A = 17, a = 5, lambda = 1, hypo = TRUE)
plot(y ~ x, data = cyc, type = "l", asp = 1, xlim = c(-27, 27),
     ylim = c(-27, 27), main = "A = 17, a = 5")
# The hypotrochoid
cyc <- zykloid(A = 17, a = 5, lambda = 1/2, hypo = TRUE)
lines(y ~ x, data = cyc, type = "l", asp = 1, col = "green")
# The epicycloid
cyc <- zykloid(A = 17, a = 5, lambda = 1, hypo = FALSE)
lines(y ~ x, data = cyc, type = "l", col = "red")
# The epitrochoid
cyc <- zykloid(A = 17, a = 5, lambda = 1/2, hypo = FALSE)
lines(y ~ x, data = cyc, type = "l", col = "blue")
legend("topleft", c("hypocycloid", "hypotrochoid", "epicycloid",
       "epitrochoid"), lty = rep("solid", 4),
       col = c("black", "green", "red", "blue"), bty = "n")



# Pretty - a classic Spirograph pattern with the same settings 
# for A (5) and a (3) as in the first example.
# Varying parameters (here: lambda) within a loop often gives
# nice results.
op <- par(mar = c(0,0,0,0)) # no plot margins
lambdax <- seq(0.85, by = -0.05, length.out = 14)
ccol <- rep(c("blue", "blue", "red", "red"), 4)
plot.new()
plot.window(asp = 1, xlim = c(-4.5, 4.5), ylim = c(-4.5, 4.5))
# draw fourteen hypotrochoids with decreasing lambda
for (i in c(1:14)) {
     z <- zykloid(5, 3, lambdax[i])     
     lines(y ~ x, data = z, type = "l", col = ccol[i])
} # for i
par(op) # set graphics parameters back to original values



# A bit more of the same kind to get the big picture...
op <- par(mar = c(0,0,0,0)) # no plot margins
lambdax <- seq(1, by = -0.05, length.out = 16)
ccol <- rep(c("blue", "blue", "red", "red"), 4)
plot.new()
plot.window(asp = 1, xlim = c(-11, 11), ylim = c(-11, 11))
# first loop: sixteen epitrochoids with decreasing lambda
for (i in 1:16) {
     z <- zykloid(5, 3, lambdax[i], hypo = FALSE) 
     lines(y ~ x, data = z, type = "l", col = ccol[i])
} # for i - first loop
# first loop: sixteen epitrochoids with decreasing lambda
for (i in 1:16) {
     z <- zykloid(5, 3, lambdax[i], hypo = TRUE)     
     lines(y ~ x, data = z, type = "l", col = ccol[i])
} # for i - second loop
par(op) # set graphics parameters back to original values



# Show off with an example for zykloid.scaleP
#    No plot margins, and ... paint it black
op <- par(mar = c(0,0,0,0), bg = "black")
lambdax <- seq(2, 0.0, -0.05) # Note: some lambdas are greater than 1
ccol <- rep(c("lightblue", "lightblue", "yellow", "yellow", "yellow"), 9)
plot.new()
plot.window(asp = 1, xlim = c(-1, 1), ylim = c(-1, 1))
for (ll in c(1:length(lambdax))) {
     z <- zykloid.scaleP(A = 7, a = 5, hypo = TRUE, lambda = lambdax[ll]) 
     lines(y ~ x, data = z, col = ccol[ll])
} # for ll
par(op) # set graphics parameters back to original values



# Spiky Flower with zykloid.scaleA and zykloid
op <- par(mar = c(0,0,0,0), bg = "black")
plot.new()
plot.window(asp = 1, xlim = c(-150, 150), ylim = c(-150, 150))
z <- zykloid.scaleA(A = 90, a = 32, lambda = 1, Radius = 150, hypo = TRUE)
lines(y ~ x, data = z, col = "lightblue")
for (ll in seq(2, 0.8, -0.4)) {
     if (ll == 2) ccol <- "royalblue"
     else         ccol <- "plum"
     z <- zykloid(A = 90, a = 32, lambda = ll, hypo = TRUE, steps = 360, start = pi/2)
     lines(y ~ x, data = z, col = ccol)
} # for ll
par(op)

---

Calculates the greatest common divisor of two natural numbers a and b based on the Euclidean Algorithm

Description

The function ggT calculates the greatest common divisor of two natural numbers. In this package it is called by the function kgV which calculates the least common multiple of two natural numbers. The latter is needed by the function zykloid and by the function npeaks which calculates the number of peaks (or loops) a cycloid has. As the greatest common divisor might be useful for other purposes, the function ggT is accessible to external use in this package.

Usage

ggT(a, b)

Arguments

a	A natural number (integer value > 0)
b	A natural number (integer value > 0)

Value

A natural number if $a$ and $b$ are natural numbers. In any other case, the function returns NA.

Author(s)

Peter Biber

References

Bronstein IN, Semendjaev KA, Musiol G, Muehlig H (2001): Taschenbuch der Mathematik, 5th Edition, Verlag Harri Deutsch, 1186 p. (p. 333)

http://en.wikipedia.org/wiki/Euclidean_algorithm

See Also

kgV, npeaks

Examples

ggT(18, 6)        # 6
ggT(38, 105)      # 1
ggT(36, 9)        # 9
ggT(12, 9)        # 3
ggT(9, 12)        # 3
ggT(-5, 12)       # NA - only integer numbers > 0 allowed
ggT(3, 0)         # NA - only integer numbers > 0 allowed
ggT(3.2, 12)      # NA - only integer numbers > 0 allowed

---

Calculates the least common multiple of two natural numbers a and b

Description

The function kgV calculates the least common multiple of two natural numbers. In this package it is used by the function zykloid and by the function npeaks which calculates the number of peaks (or loops) a cycloid has. As it might be useful for other purposes, it is externally available in this package.

Usage

kgV(a, b)

Arguments

a	A natural number (integer value > 0)
b	A natural number (integer value > 0)

Value

A natural number if $a$ and $b$ are natural numbers. In any other case, the function returns NA.

Author(s)

Peter Biber

References

Bronstein IN, Semendjaev KA, Musiol G, Muehlig H (2001): Taschenbuch der Mathematik, 5th Edition, Verlag Harri Deutsch, 1186 p. (p. 334)

http://en.wikipedia.org/wiki/Least_common_multiple

See Also

npeaks, ggT, zykloid

Examples

kgV(18, 6)          #   18
kgV(38, 105)        # 3990
kgV(36, 9)          #   36
kgV(12, 9)          #   36
kgV(9, 12)          #   36
kgV(-5, 12)         # NA - only integer numbers > 0 allowed
kgV(3, 0)           # NA - only integer numbers > 0 allowed
kgV(3.2, 12)        # NA - only integer numbers > 0 allowed

---

Calculates the number of a cycloid's peaks based on the radii A (fixed circle) and a (moving circle)

Description

This function may be useful for calculating the number $n$ of peaks a cycloid (zykloid) based on the radii $A$ (fixed circle) and $a$ (moving circle) will have. The equation for $n$ is

$n = kgV(A, a)/a$

where $kgV(A, a)$ is the least common multiple of $A$ and $a$ as implemented in the function kgV

Usage

npeaks(A, a)

Arguments

A	A natural number (integer value > 0)
a	A natural number (integer value > 0)

Value

A natural number if $A$ and $a$ are natural numbers. In any other case, the function returns NA.

Author(s)

Peter Biber

See Also

kgV, ggT, zykloid

Examples

npeaks(18, 6)      #  3
npeaks(38, 105)    # 38
npeaks(36, 9)      #  4
npeaks(12, 9)      #  4
npeaks(9, 12)      #  3
npeaks(-5, 12)     # NA - only integer numbers > 0 allowed
npeaks(3, 0)       # NA - only integer numbers > 0 allowed
npeaks(3.2, 12)    # NA - only integer numbers > 0 allowed

---

Core function for calculating coordinate representations of hypocycloids, epicyloids, hypotrochoids, and epitrochoids (altogether called 'cycloids' here)

Description

This is the package's core function for calculating cycloids. These are represented by a set of two-dimensional point coordinates. Although this function provides the essential mathematics, you may want to use the wrappers zykloid.scaleA, zykloid.scaleAa, and zykloid.scaleP due to their convenient scaling and positioning options.

Usage

zykloid(A, a, lambda, hypo = TRUE, steps = 360, start = pi/2)

Arguments

A	The Radius of the fixed circle $cfix$. Must be an integer Number > 0.
a	The radius of the moving circle $cmov$. Must be an integer Number > 0. Together with $A$, $a$ determines the resulting cycloid's shape and number of peaks which can be calculated with npeaks.
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.
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.

Details

Geometrically, cycloids in the sense of this package are generated as follows (Figure 1, 2): Imagine a circle $cfix$, with radius $A$, which is fixed on a plane. Another circle, $cmov$, with radius $a$, is rolling along $cfix$'s circumference at the outside of $cfix$. The figure created by the trace of a point on $cmov$'s circumference is called an epicycloid (Figure 1A). If $cmov$ is rolling not at the outside but at the inside of $cfix$, the trace of a point on $cmov$'s circumference is called an hypocycloid (Figure 2A).
If in both cases the tracepoint is not located on $cmov$'s circumference but at a fixed distance from its midpoint either in- or outside $cmov$, the resulting figure is an epitrochoid (Figure 1B, C) or a hypotrochoid (Figure 2B, C), respectively.
With the arguments of zykloid as defined above, the centre of $cfix$ in the origin, and $phi$ being the counterclockwise angle of $cmov$'s midpoint against the start position with $cfix$' centre as the pivot, the cartesian coordinates of a point on the cycloid are calculated as follows:

$x = (A + a) * cos(phi + start) - lambda * a * cos((A + a)/a * phi + start)$
$y = (A + a) * sin(phi + start) - lambda * a * sin((A + a)/a * phi + start)$

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.

Author(s)

Peter Biber

References

Bronstein IN, Semendjaev KA, Musiol G, Muehlig H (2001): Taschenbuch der Mathematik, 5th Edition, Verlag Harri Deutsch, 1186 p. (103 - 105)

http://en.wikipedia.org/wiki/Epicycloid

http://en.wikipedia.org/wiki/Hypocycloid

http://en.wikipedia.org/wiki/Epitrochoid

http://en.wikipedia.org/wiki/Hypotrochoid

See Also

zykloid.scaleA, zykloid.scaleAa, zykloid.scaleP

Examples

# Very simple example
cycl <- zykloid(A = 17, a = 9, lambda = 0.9, hypo = TRUE)
plot(y ~ x, data = cycl, asp = 1, type = "l")


# More complex: Looks like a passion flower
op <- par(mar = c(0,0,0,0), bg = "black")
plot.new()
plot.window(asp = 1, xlim = c(-23, 23), ylim = c(-23, 23))
ll   <- seq(2, 0, -0.2)
ccol <- rep(c("lightblue", "lightgreen", "yellow", "yellow",
              "yellow"), 2)
for (i in c(1:length(ll))) {
     z <- zykloid(A = 15, a = 7, lambda = ll[i], hypo = TRUE)
     lines(y ~ x, data = z, col = ccol[i])
} # for i
par(op)


# Dense hypotrochoids
op <- par(mar = c(0,0,0,0), bg = "black")
plot.new()
plot.window(asp = 1, xlim = c(-1.5, 1.5), ylim = c(-1.5, 1.5))
m <- zykloid(A = 90, a = 89, lambda = 0.01)
lines(y ~ x, data = m, col = "grey")
m <- zykloid(A = 90, a = 89, lambda = 0.02)
lines(y ~ x, data = m, col = "red")
m <- zykloid(A = 90, a = 89, lambda = 0.015)
lines(y ~ x, data = m, col = "blue")
par(op)


# Fragile star
op <- par(mar = c(0,0,0,0), bg = "black")
plot.new()
plot.window(asp = 1, xlim = c(-14, 14), ylim = c(-14, 14))
l.max <- 1.6
l.min <- 0.1
ll <- seq(l.max, l.min, by = -1 * (l.max - l.min)/30)
n  <- length(ll)
ccol <- rainbow(n, start = 2/3, end = 1)
for (i in c(1:n)) {
    m <- zykloid(A = 9, a = 8, lambda = ll[i])
    lines(y ~ x, data = m, type = "l", col = ccol[i])
}  # for i
par(op)

---

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)

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.

Details

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.

Author(s)

Peter Biber

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)

---

Wrapper for zykloid which scales a cycloid by its outer radius 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 its circumcircle. In addition, the cycloid can be re-positioned by locating the fixed circle's midpoint. This function behaves similarly as zykloid.scaleP. See details. Figures 1 and 2 and zykloid describe the geometrical principles of cycloids.

Usage

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

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 $RadiusAa$ (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 $RadiusAa$ (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 fixed circle's midpoint. Default is 0.
Cy	y-coordinate of the fixed circle's midpoint. Default is 0.
RadiusAa	The actual radius of the cycloids outer 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.

Details

This function scales in either case the radius of the whole cycloid's circumcircle. Thus, for hypocycloids and hypotrochoids it will behave the same way as zykloid.scaleP. For epicycloids and epitrochoids their output will be different. zykloid.scaleAa scales the outer edge of the figure, while 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).
Figure 1 and 2 show the principle behind cycloid construction:

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.

Author(s)

Peter Biber

See Also

zykloid, zykloid.scaleA, zykloid.scaleP

Examples

# Same epicycloid scaled to different maximum radii of the figure
cycl1 <- zykloid.scaleAa(A = 21, a = 11, lambda = 1, hypo = FALSE,
                         RadiusAa = 100)
cycl2 <- zykloid.scaleAa(A = 21, a = 11, lambda = 1, hypo = FALSE,
                         RadiusAa =  70)
cycl3 <- zykloid.scaleAa(A = 21, a = 11, lambda = 1, hypo = FALSE,
                         RadiusAa =  40)
plot (y ~ x, data = cycl1, col = "red", asp = 1, type = "l",
      main = "A = 21, a = 11, lambda = 1")
lines(y ~ x, data = cycl2, col = "green")
lines(y ~ x, data = cycl3, col = "blue")
legend("topleft", c("RadiusAa = 100", "RadiusAa =  70", "RadiusAa =  40"),
       lty = rep("solid", 3), col = c("red", "green", "blue"), bty = "n")
       

# Pentagram by constructing a hypocycloid and an epicycloid
# with the same outer radius and scaling this radius exponentially
op <- par(mar = c(0,0,0,0), bg = "black")
plot.new()
plot.window(asp = 1, xlim = c(-40, 40), ylim = c(-40, 40))
n <- 20
ccol <- heat.colors(n)
for(i in c(1:n)) {
    zzz <- zykloid.scaleAa(A = 5, a = 2,
           RadiusAa = 38*exp(-0.05*(i-1)), hypo = FALSE, lambda = 1)
    lines(y ~ x, data = zzz, col = ccol[i])
    zzz <- zykloid.scaleAa(A = 5, a = 2,
           RadiusAa = 38*exp(-0.05*(i-1)), hypo = TRUE,  lambda = 1)
    lines(y ~ x, data = zzz, col = ccol[i])
} # for i
par(op)



# Psychedelic star by modifying lambda while keeping the outer
# radius constant
op <- par(mar = c(0,0,0,0), bg = "black")
plot.new()
plot.window(asp = 1, xlim = c(-5, 5), ylim = c(-5, 5))
llam <- seq(0, 8, 0.2)
ccol <- terrain.colors(length(llam))
for(i in c(1:length(llam))) {
    zzz <- zykloid.scaleAa(A = 5, a = 1, RadiusAa = 4.5,
           hypo = FALSE, lambda = llam[i])
    lines(y ~ x, data = zzz, col = ccol[i])
} # for i
par(op)

---

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)

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.

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:

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.

Author(s)

Peter Biber

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)

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