Drawing nested Steiner chains

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This is a Steiner chain with its enveloping cyclide:

And these are nested Steiner chains:

With R

We will include the cyclides in the plot. We firstly write a function returning a mesh of a cyclide. It is obtained by applying an inversion to the mesh of a torus.

library(rgl)

torusMesh <- function(R, r, S, s, arc, ...){
  vertices <- matrix(NA_real_, nrow = 3L, ncol = S*s)
  Normals <- matrix(NA_real_, nrow = 3L, ncol = S*s)
  full <- arc == 2*pi
  SS <- ifelse(full, S, S-1)
  indices <- matrix(NA_integer_, nrow = 4L, ncol = SS*s)
  u_ <- if(full){
    seq(0, 2*pi, length.out = S+1)[-1]
  }else{
    seq(0, arc, length.out = S)
  }
  v_ <- seq(0, 2*pi, length.out = s+1)[-1]
  for(i in 1:S){
    cos_ui <- cos(u_[i])
    sin_ui <- sin(u_[i])
    cx <- R * cos_ui
    cy <- R * sin_ui
    for(j in 1:s){
      rcos_vj <- r*cos(v_[j])
      n <- c(rcos_vj*cos_ui, rcos_vj*sin_ui, r*sin(v_[j]))
      Normals[, (i-1)*s+j] <- -n
      vertices[, (i-1)*s+j] <- c(cx,cy,0) + n 
    }
  }
  # quads
  s <- as.integer(s)
  for(i in 1L:SS){
    ip1 <- ifelse(i==S, 1L, i+1L)
    for(j in 1L:s){
      jp1 <- ifelse(j==s, 1L, j+1L)
      indices[,(i-1)*s+j] <- 
        c((i-1L)*s+j, (i-1L)*s+jp1, (ip1-1L)*s+jp1, (ip1-1L)*s+j)   
    }
  }
  qmesh3d(
    vertices = vertices,
    indices = indices,
    homogeneous = FALSE,
    material = list(...),
    normals = t(Normals)
  )
}

# only for mu>c (ring cyclide)
cyclideMesh <- function(mu, a, c, S=128, s=64, arc=2*pi, ...){
  b <- sqrt(a^2-c^2)
  bb <- b*sqrt(mu^2-c^2)
  omega <- (a*mu + bb)/c
  Omega <- c(omega,0,0)
  inversion <- function(M){
    Omega + 1/c(crossprod(M-Omega)) * (M-Omega)
  }
  d <- (a-c)*(mu-c)+bb
  r <- c*c*(mu-c) / ((a+c)*(mu-c)+bb) / d
  R <- c*c*(a-c) / ((a-c)*(mu+c)+bb) / d
  bb2 <- b*b*(mu*mu-c*c)
  denb1 <- c*(a*c-mu*c+c*c-a*mu-bb)
  b1 <- (a*mu*(c-mu)*(a+c)-bb2+c*c+bb*(c*(a-mu+c)-2*a*mu))/denb1
  denb2 <- c*(a*c-mu*c-c*c+a*mu+bb)
  b2 <- (a*mu*(c+mu)*(a-c)+bb2-c*c+bb*(c*(a-mu-c)+2*a*mu))/denb2
  omegaT <- (b1+b2)/2
  tmesh <- torusMesh(R, r, S, s, arc, ...)
  tmesh$normals <- 
    apply(tmesh$normals + tmesh$vb[1:3,] + c(omegaT,0,0), 2, inversion)
  tmesh$vb[1:3,] <- apply(tmesh$vb[1:3,] + c(omegaT,0,0), 2, inversion)
  tmesh$normals <- tmesh$vb[1:3,] - tmesh$normals  
  tmesh
}

Now here is the code which plots the nested Steiner chains:

# make a smooth unit sphere
unitSphere <- subdivision3d(icosahedron3d(), depth=4)
unitSphere$vb[4,] <- 
  apply(unitSphere$vb[1:3,], 2, function(x) sqrt(sum(x^2)))
unitSphere$normals <- unitSphere$vb
# draw a sphere in the x-y plane
drawSphere <- function(x, y, radius, ...){
  shade3d(
    translate3d(
      scale3d(unitSphere, radius, radius, radius), 
      x, y, 0), 
    ...)
}

# draw a cyclide translated by (x,y,0)
drawCyclide <- function(mu, a, c, x, y, ...){
  mesh <- cyclideMesh(mu, a, c)
  shade3d(translate3d(mesh, x, y, 0), ...)
}

# image of circle (center, radius) by the inversion 
#   with center c and power k
iotaCircle <- function(c, k, center, radius){
  r <- sqrt(abs(k)) 
  z1 <- sign(k) * (center-c)/r 
  D1 <- (radius/r)^2 - c(crossprod(z1))
  z2 <- -z1/D1
  R2 <- sqrt(c(crossprod(z2)) + 1/D1)
  list(center = r*z2+c, radius = r*R2)
}

# n: vector, the numbers of spheres at each step
# -1 < phi < 1, phi != 0
steiner <- function(n, phi, color="red", shift=0, 
                    Center=c(0,0), radius=2, epsilon = 0.005){
  depth <- length(n)
  invphi <- 1/phi 
  I <- c(radius*invphi, 0) + Center
  k <- radius*radius*(1-invphi*invphi)
  m <- n[1]
  sine <- sin(pi/m)
  Coef <- 1/(1+sine)
  O1x <- 2*radius*invphi
  CRadius <- Coef*radius
  CSide <- CRadius*sine
  if(depth == 1){
    circle <- iotaCircle(I-Center, k, c(0,0), CRadius - CSide)
    mu <- (radius - circle$radius)/2;
    a <- (radius + circle$radius)/2;
    c <- (circle$center[1] - O1x)/2;
    pt <- Center + circle$center/2;
    drawCyclide(mu, a, c, pt[1]-O1x/2, pt[2], 
                color = "yellow", alpha = 0.2)
  }
  for(i in 1:m){
    beta <- (i+shift)*2*pi/m 
    pti <- c(CRadius*cos(beta), CRadius*sin(beta)) + Center
    cc <- iotaCircle(I, k, pti, CSide)
    center <- cc$center - c(O1x,0)
    r <- cc$radius
    if(depth == 1){
      drawSphere(center[1], center[2], r-epsilon, color = color)
    }
    if(depth>1){
      steiner(n[-1], phi, color=color, 
              Center=center, radius=r, shift = -shift)
    }
  }
  return(invisible())
}

# background color
bgColor <- rgb(54, 57, 64, maxColorValue = 255)

# plot
open3d(windowRect = c(50,50,550,550))
bg3d(bgColor)
view3d(0, -40, zoom = 0.75)
steiner(n = c(3,3,4), phi = 0.3, color = "#B12A90FF", shift = 0.25)

Here is how to make a gif animation the nested Steiner chains:

# "bounding box"
bBox <- function(){
  r <- 2
  lines3d(
    rbind(
      c(r, r, 0), c(-r, r, 0), c(-r, -r, 0), c(r, -r, 0)
    ),
    color = bgColor, alpha = 0
  )
}

shifts <- seq(0, 1, length.out = 46)[-1]
open3d(windowRect = c(50,50,550,550))
bg3d(bgColor)
view3d(0, -40, zoom = 0.75)
for(i in seq_along(shifts)){
  steiner(n = c(3,3,5), phi = 0.3, color = "#B12A90FF", 
          shift = shifts[i])
  bBox()
  snapshot3d(sprintf("img-%03d.png", i))
  clear3d()
}

pngs <- list.files(pattern = "^img-.*png$")
gifski::gifski(pngs, "SteinerChain_R.gif", 
               width = 500, height = 500, delay = 0.04)
file.remove(pngs)

With POV-Ray

#version 3.7;
global_settings { assumed_gamma 1 }
#include "colors.inc"
#include "textures.inc"
#include "glass.inc"

// image of the circle (Center,Radius) by inversion pole c power k ---
#macro invertedCircle(c, k, Center, Radius)
  #local r = sqrt(abs(k));
  #local sign = (k>0 ? 1 : -1);
  #local z1 = sign * (Center-c)/r;
  #local D1 = Radius*Radius/r/r - vdot(z1,z1);
  #local z2 = -z1/D1;
  #local R2 = sqrt(vdot(z2,z2) + 1/D1);
  #local z3 = r*z2+c;
  
#end

// delete first element of an array ----------------------------------
#macro tail(Array)
  #local l = dimension_size(Array, 1);
  #local out = array[l-1];
  #for(i, 0, l-2)
    #local out[i] = Array[i+1];
  #end
  out
#end

// main macro --------------------------------------------------------
// n: array of integers >1, the numbers of spheres at each step
// phi: number -1 < phi < 1, phi /= 0 
// shift: number 0 <= shift < 1
#macro Steiner3D(n, phi, shift, Center, Radius, epsilon)
  #local Depth = dimension_size(n, 1);
  #local m = n[0];
  #local sine = sin(pi/m);
  #local Side = Radius*sine; // side length of the m-gon 
  #local Coef = 1/(1+sine); // Radius/(Radius+Side)
  #local CSide = Coef*Side;
  #local CRadius = Coef*Radius;
  #local invphi = 1/phi;
  #local I =  + Center; // inversion pole
  #local k = Radius*Radius*(1-invphi*invphi); // inversion power
  #local O1 = <2*invphi*Radius,0,0>; // center of exterior sphere          
  // -----------------------------------------------------------------
  union { 
    // cyclides ------------------------------------------------------
    #if(Depth=1)
      #local circle = invertedCircle(I-Center, k, <0,0>, CRadius-CSide);
      #local r = circle.z;
      #local center = ;
      #local mu = (Radius - r)/2;
      #local a = (Radius + r)/2;
      #local c = (circle.x - O1.x)/2;
      #local b = sqrt(a*a-c*c);
      #local pt = Center + center/2;
      #local O = ;
      polynomial {4 
        xyz(2,0,0): -2*mu*mu+2*b*b-4*a*a,
        xyz(1,0,0): 8*a*c*mu,
        xyz(0,0,0): -4*c*c*mu*mu+mu*mu*mu*mu+b*b*b*b-2*mu*mu*b*b,
        xyz(0,2,0): -2*mu*mu-2*b*b,
        xyz(0,0,2): -2*mu*mu+2*b*b,
        xyz(2,2,0): 2,
        xyz(2,0,2): 2,
        xyz(0,2,2): 2,
        xyz(4,0,0): 1,
        xyz(0,4,0): 1,
        xyz(0,0,4): 1
        texture { 
          Dark_Green_Glass 
          finish {
            reflection 0
          }
        }
        translate O
      }
    #end
    // spheres -------------------------------------------------------
    #local i=1;
    #while(i<=m)
      #local beta = 2*(i+shift)*pi/m; 
      #local pti =  + Center;
      #local circle = invertedCircle(I, k, pti, CSide);
      #local center =  - O1;
      #local r = circle.z;
      #if(Depth=1)
        sphere {
          center, r-epsilon
          texture {
            Chrome_Metal
            finish {
              ambient 0.05
              diffuse 2
              reflection 0
              brilliance 1
              specular 1.08
              roughness 0.01
            }
          }
        }
      #else
        Steiner3D(tail(n), phi, -shift, center, r, epsilon)
      #end
      #local i = i+1;
    #end
  } // end of union
#end

// ----------------------------- SCENE ---------------------------- //
// sky ---------------------------------------------------------------
#declare D = .5;
sky_sphere {
  pigment {
    crackle
    color_map {
      [pow(0.5, D) color Black]
      [pow(0.6, D) color White*10]
    }
    scale .005/D
  }
}

// ----------------------------- plot ----------------------------- //
#declare Center = <0,0>; // arbitrary 
#declare Radius = 3; // arbitrary >0

// camera and light source -------------------------------------------
camera {
  location <0, 0, -12>
  look_at 
  angle 40
  rotate <0,0,0>
}
light_source { <0, 0, -60> White }

//
#declare n = array[3] {3,4,6};
#declare phi = 0.3;
#declare nframes = 45;
#declare shift = frame_number/nframes; 
object {
  Steiner3D(n, phi, shift, Center, Radius, 0.005)
  rotate <50, 0, 0>
  translate <0, 0, 5>
  scale 1.85
}

/* ini file ----------------------------------------------------------
Antialias = On
Antialias_Threshold = 0.3
Antialias_Depth = 3
Input_File_Name = SteinerChain.pov
Initial_Frame = 1
Final_Frame = 45
Cyclic_Animation = on
Pause_when_Done = off
------------------------------------------------------------------- */


We get a nice result when we map a picture to the spheres:

#if(Depth=1)
  sphere {
    center, r-epsilon
    pigment{
      image_map {
        png "R-Ladies.png"
        interpolate 2
      }
      scale 1.1*sqrt(r)
    }
  }
#else

With Asymptote

This Asymptote program produces the frames of the animation:

settings.render = 4;
settings.outformat = "eps";

// -------------------------------------------------------------------
import solids;

// files to create
string[] files = {
    "SC-000", "SC-001", "SC-002", "SC-003", "SC-004", "SC-005",
    "SC-006", "SC-007", "SC-008", "SC-009", "SC-010", "SC-011",
    "SC-012", "SC-013", "SC-014", "SC-015", "SC-016", "SC-017",
    "SC-018", "SC-019", "SC-020", "SC-021", "SC-022", "SC-023",
    "SC-024", "SC-025", "SC-026", "SC-027", "SC-028", "SC-029",
    "SC-030", "SC-031", "SC-032", "SC-033", "SC-034", "SC-035",
    "SC-036", "SC-037", "SC-038", "SC-039", "SC-040", "SC-041",
    "SC-042", "SC-043", "SC-044"};

// camera and light --------------------------------------------------
size(10cm);
currentprojection = orthographic(2,2,6);
currentlight = (20,20,60);
currentlight.background = black;
viewportmargin = (10,10);

// image of the circle (Center,Radius) by inversion pole c power k ---
struct Circle {
  pair center;
  real radius;
}
Circle invertedCircle(pair c, real k, pair Center, real Radius){
    real r = sqrt(abs(k));
    pair z1 = sgn(k) * (Center-c)/r;
    real D1 = Radius*Radius/r/r - dot(z1,z1);
    pair z2 = -z1/D1;
    real R2 = sqrt(dot(z2,z2) + 1/D1);
    Circle out;
    out.center = r*z2 + c;
    out.radius = r*R2;
    return out;
}

// -------------------------------------------------------------------
struct Sphere {
    triple center;
    real radius;
}

// -------------------------------------------------------------------
struct Cyclide {
  real mu;
  real a;
  real c;
  triple shift;
}

// -------------------------------------------------------------------
// n: array of length depth, the numbers of spheres for each step
// phi: number -1 < phi < 1, phi /= 0 
// shift: number 0 <= shift < 1
void Steiner3D(Sphere[] spheres, Cyclide[] cyclides, int[] n, real phi, 
                int depth, real shift, pair Center = (0,0), real Radius = 2){
    real m = n[0];
    real sine = sin(pi/m);
    real Side = Radius*sine; // side length of the m-gon 
    real Coef = 1/(1+sine); // Radius/(Radius+Side)
    real CSide = Coef*Side;
    real CRadius = Coef*Radius;
    real invphi = 1/phi;
    pair I = (Radius*invphi,0) + Center; // inversion pole
    real k = Radius*Radius*(1-invphi*invphi); // inversion power
    real O1x = 2*invphi*Radius; // (O1x,0,0) center of exterior sphere 
    // -----------------------------------------------------------------
    if(depth == 1){
        Cyclide newcyclide;
        Circle circle = invertedCircle(I-Center, k, (0,0), CRadius-CSide);
        newcyclide.mu = (Radius - circle.radius)/2;
        newcyclide.a = (Radius + circle.radius)/2;
        newcyclide.c = (circle.center.x - O1x)/2;
        pair pt = Center + circle.center/2;
        newcyclide.shift = (pt.x - O1x/2, pt.y, 0);
        cyclides.push(newcyclide);
    } 
    // -----------------------------------------------------------------
    for(int i = 0; i < m; ++i){
        real beta = 2*(i+shift)*pi/m;
        pair pti = (CRadius*cos(beta), CRadius*sin(beta)) + Center;
        Circle circle = invertedCircle(I, k, pti, CSide);
        pair center = (circle.center.x - O1x, circle.center.y);
        real r = circle.radius;
        if(depth == 1){
            Sphere newsphere;
            newsphere.center = (center.x, center.y, 0);
            newsphere.radius = r;
            spheres.push(newsphere);
        }else{
            Steiner3D(spheres, cyclides, n[1:], phi, depth-1, -shift, 
                center, r);
        }
    }
}

// -------------------------------------------------------------------
int[] n = {3,4,5};
real phi = 0.4;
int depth = n.length;

path3 boundingbox = circle(c=O, r=2.1, normal=Z);

real epsilon = 0.005;

for(int k = 0; k < files.length; ++k){
    // compute spheres and cyclides ------------------------------------
    Sphere[] spheres = new Sphere[0];
    Cyclide[] cyclides = new Cyclide[0];
    real shift = k/files.length;
    Steiner3D(spheres, cyclides, n, phi, depth, shift);
    //
    picture pic;
    draw(pic, boundingbox, black+opacity(0));
    // draw the spheres ------------------------------------------------
    for(int i = 0; i < spheres.length; ++i){
        Sphere s = spheres[i];
        draw(pic, surface(sphere(s.center, s.radius-epsilon)), 
          rgb("8c2981ff"));
    }
    // draw the cyclides -----------------------------------------------
    for(int i = 0; i < cyclides.length; ++i){
        Cyclide c = cyclides[i];
        real b = sqrt(c.a*c.a-c.c*c.c);
        triple F(pair uv){
            real h = c.a-c.c*cos(uv.x)*cos(uv.y);
            real x = (c.mu*(c.c-c.a*cos(uv.x)*cos(uv.y))+b*b*cos(uv.x))/h;
            real y = (b*sin(uv.x)*(c.a-c.mu*cos(uv.y)))/h;
            real z = b*sin(uv.y)*(c.c*cos(uv.x)-c.mu)/h;
            return (x,y,z);
        }
        surface s = surface(F, (0,0), (2pi,2pi), 40, 40, Spline);
        draw(pic, shift(c.shift)*s, yellow+opacity(0.2));
    }
    // -----------------------------------------------------------------
    add(pic);
    shipout(files[k], bbox(black, FillDraw(black)));
    erase();
}

With three.js

Here is a variant with three.js. I replaced the spheres with “Barth polyedra”.

To leave a comment for the author, please follow the link and comment on their blog: Saturn Elephant.

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