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

Package: cycloids
Type: Package
Version: 1.0
Date: 2013-10-24
License: GPL-3

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 <castor.fiber@gmx.de>

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

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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)