# 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

1 2 | ```
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` |
The radius of the moving circle before re-sizing. Must be an
integer Number > 0. Together with |

`lambda` |
The distance of the tracepoint from the moving circle's ( |

`hypo` |
logical. If TRUE, the resulting figure is a hypocycloid ( |

`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 ( |

### 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

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 | ```
# 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)
``` |