Vtyjkyuk

I'm not even sure how to succinctly ask this question in the title, so let me explain.

I occasionally find myself wanting to map a number to a color in RGB space. I usually do this by expressing a circle in 3d as a function of an angle. As an aid in designing this mapping, I decided to put together a shiny app that visualizes the line in RGB space. This started off as a plain old circle in 3d space, using trigonometric functions, but then I realized I can get some fairly crazy curves (and thus interesting color mappings) by fiddling with the phase, radius, wavelength etc.

library(plotly)
library(shiny)

xPhase = 20
yPhase = 120
zPhase = 240
xCenter = 127
yCenter = 127
zCenter = 127
xWavelength = 1
yWavelength = 1
zWavelength = 1
zRadRadius = 0
xRadius = 100
yRadius = 100
zRadius = 100

gcd = function(args)
 if (any(args <= 0))
 
 stop("All arguments to gcd must be positive")
 
 if (length(args) < 2)
 
 stop("Invalid number of arguments")
 
 a = args[1]
 b = args[2]
 r = a %% b
 if (r > 0)
 
 return(gcd(c(b,r,args[-c(1,2)])))
 
 else if (length(args) > 2)
 
 return(gcd(c(b,args[-c(1,2)])))
 
 else
 
 return(b)
 


lcm = function(args)
 if (any(args <= 0))
 
 stop("All arguments to gcd must be positive")
 
 if (length(args) < 2)
 
 stop("Invalid number of arguments")
 
 g = gcd(args[1:2])
 m = prod(args[1:2])/g
 if (length(args)>2)
 
 return(lcm(c(m,args[-c(1,2)])))
 
 else
 
 return(m)
 


maxTheta = lcm(c(xWavelength,yWavelength,zWavelength))*360
theta = 0:maxTheta

x = cos(((theta + xPhase) / xWavelength) / 180 * pi) * xRadius + xCenter
y = cos(((theta + yPhase) / yWavelength) / 180 * pi) * yRadius + yCenter
z = cos(((theta + zPhase) / zWavelength) / 180 * pi) * zRadius + zCenter
colors = rgb(x, y, z, maxColorValue = 255)
data = data.frame(theta = theta, x = x, y = y, z = z, colors = colors, avg = apply(cbind(x,y,z),1,mean))

p <- plot_ly(data, x = ~x, y = ~y, z = ~z, colors = ~colors, type = 'scatter3d', mode = 'lines',
 opacity = 1, line = list(width = 6, reverscale = FALSE, color = ~colors)) %>%
 layout(
 scene = list(
 xaxis = list(range = c(0,255)),
 yaxis = list(range = c(0,255)),
 zaxis = list(range = c(0,255)),
 aspectmode = "cube"
 )
 )

print(p)

The above R code generates a simple cirlce in 3d, but set one of the wavelengths to a higher integer and then it gets more interesting.

Sine/Cos functions are defined in terms of a unit circle, where $x^2 + y^2 = r^2$, so my function maps an angle to a some point in the set of points that is a fixed 2norm distance form the center.

I don't like how the curves tend not to explore the corners of the cube so I want to try above concept based on a higher norm. For example, $x^3 + y^3 = r^3$. Is there a simple way of achieving this?
I can do this for a single loop by parameterizing in terms of a dummy variable (t) and just walking along the curve based on a start location and gradient, but then I won't be able to fiddle with the phase/wavelength etc, or can I?