# zykloid.scaleAa: Wrapper for 'zykloid' which scales a cycloid by its outer... In cycloids: cycloids

## 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 (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 λ != 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

`zykloid`, `zykloid.scaleA`, `zykloid.scaleP`
 ``` 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) ```