Arc-length parametrization

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library(MetricGraph)
library(ggplot2)
library(reshape2)
library(dplyr)
library(viridis)
library(plotly)
library(patchwork)
library(slackr)
source("keys.R")
slackr_setup(token = token) # token comes from keys.R

## [1] "Successfully connected to Slack"

capture.output(
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  file = here::here("old/purl_log.txt")
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source(here::here("functionality1.R"))

For example, consider the regular curve \(\gamma(t) = (bt +c,a\cos(t), a\sin(t))\) for \(t\in[0,T]\), where \(a = 1\), \(b=0.5\), \(T=6\pi\), and \(c\) is a constant to be defined later. The curve has arc-length \(\ell = T\sqrt{a^2+b^2}\) and its arc-length parameterization is given by \(\tilde{\gamma}(s) = \gamma\left(s/\sqrt{a^2+b^2}\right)\) for \(s\in[0,\ell]\). Let \(\Gamma = (\mathcal{V}, \mathcal{E})\) denote the one edge graph induced by \(\tilde{\gamma}\), where \(\mathcal{V} = \{v_1 = \tilde{\gamma}(0), v_2 = \tilde{\gamma}(\ell)\}\) and \(\mathcal{E}=\{e = \tilde{\gamma}([0,\ell])\}\). Then we have the identification \(e=[0,\ell]\). For visualization purposes, we let \(c = T(\sqrt{a^2+b^2}-b)/2\). This makes the \(\gamma\) and the interval \([0,\ell]\) in Figure~\(\ref{arc_length_par}\) centered with respect to each other.

1 Helix function

library(plotly)

# Parameters
a <- 1
b <- 0.5
TT <- 6*pi
# Total arc-length
L <- sqrt(a^2 + b^2) * TT

half_L <- L / 2


max_x <- b * (L / sqrt(a^2 + b^2))

half_max_x <- max_x / 2

dist_to_move <- half_L - half_max_x

# Arc-length parametrization (helix around x-axis)
alpha_tilde <- function(s){
  t <- s / sqrt(a^2 + b^2)
  x_shift <- TT*(sqrt(a^2 + b^2) - b) / 2
  x <- b * t + x_shift
  y <- a * cos(t)
  z <- a * sin(t)
  data.frame(x=x, y=y, z=z)
}

a0 = alpha_tilde(0)
aL = alpha_tilde(L)
aLh = alpha_tilde(half_L)

hhhh <- 3
ff <- function(s) 2*abs(sin(hhhh*pi*s/L))
k <- 0:hhhh
s_special <- k*L/hhhh

# Smooth helix
n_smooth <- 500

#s_smooth <- seq(0, L, length.out = n_smooth)
s_smooth <- sort(unique(c(
  seq(0, L, length.out = n_smooth),
  s_special
)))
curve_smooth <- alpha_tilde(s_smooth)


# Points for mapping lines
n_map <- 100
s_map <- seq(0, L, length.out = n_map)
curve_map <- alpha_tilde(s_map)


#s_semi_coarse <- seq(0, L, length.out = n_smooth/2)

# Interval along x-axis
int_map <- data.frame(
  x = s_map,
  y = rep(0, n_map)+4,
  z = rep(0, n_map)
)


ff_eval_coarse <- ff(s_map)

ff_eval <- ff(s_smooth)

curve_smooth$height <- ff_eval

s_semi_coarse <- s_smooth
ff_s_semi_coarse <- ff(s_semi_coarse)
y_semi_coarse <- rep(0, length(s_semi_coarse)) + 4


ff_df <- data.frame(x = s_smooth, y = rep(0, length(s_smooth))+4, z = ff_eval)

ff_df_coarse <- data.frame(x = s_map, y = rep(0, length(s_map))+4, z = ff_eval_coarse)

# Build mapping lines
rows <- lapply(1:n_map, function(i) {
  list(int_map[i, ], curve_map[i, ], data.frame(x = NA, y = NA, z = NA))
})

result <- do.call(rbind, unlist(rows, recursive = FALSE))

# Build mapping lines
rows <- lapply(1:n_map, function(i) {
  list(curve_map[i, ], ff_df_coarse[i, ], data.frame(x = NA, y = NA, z = NA))
})

result2 <- do.call(rbind, unlist(rows, recursive = FALSE))


rows <- lapply(1:n_map, function(i) {
  val <- ff_eval_coarse[i]

  rbind(
    cbind(curve_map[i, ], val = val),
    cbind(ff_df_coarse[i, ], val = val),
    data.frame(x = NA, y = NA, z = NA, val = NA)
  )
})

result2 <- do.call(rbind, rows)

ff_df_coarse_new <- result2$val



zero <- int_map[1, ]
ell <- int_map[n_map, ]

vertex <- rbind(a0, aL, zero, ell)


ttt <- half_L / sqrt(a^2 + b^2)

u <- b
v <- -a*sin(ttt)
w <-  a*cos(ttt)

zmin <- min(ff_eval)
zmax <- max(ff_eval)

# Plot
p <- plot_ly() |>
  
  # Smooth helix
  add_trace(
    data = curve_smooth,
    x = ~x, y = ~y, z = ~z,
    type = "scatter3d",
    mode = "lines",
    line = list(color = ff_eval, 
                colorscale = "Viridis",
                cmin = zmin,
                cmax = zmax,
                width = 7),
    showlegend = FALSE
  ) |>
  
  # Interval [0,L]
  add_trace(
    data = int_map,
    x = ~x, y = ~y, z = ~z,
    type = "scatter3d",
    mode = "lines",
    line = list(color = "black", width = 7),
    showlegend = FALSE
  ) |>
  add_trace(
  data = result2,
  x = ~x, y = ~y, z = ~z,
  type = "scatter3d",
  mode = "lines",
  line = list(
    color = ff_df_coarse_new,
    colorscale = "Viridis",
    cmin = zmin,
    cmax = zmax,
    width = 1
  ),
  showlegend = FALSE
) |>
  add_trace(data = vertex,
  x = ~x,
  y = ~y,
  z = ~z,
  type = "scatter3d",
  mode = "markers",
  marker = list(size = 5, color = "black"),
    showlegend = FALSE
) |>
  add_trace(
    type = "cone",
    x = aLh$x,
    y = aLh$y,
    z = aLh$z,
    u = u,
    v = v,
    w = w,
    sizemode = "absolute",
    sizeref = 0.4,
    colorscale = list(c(0, 1), c("green", "green")),
    showscale = FALSE
  ) |>
  # function on interval
  add_trace(
    data = ff_df,
    x = ~x, y = ~y, z = ~z,
    type = "scatter3d",
    mode = "lines",
    line = list(
      color = ~z,                 # color based on height
      colorscale = "Viridis",     # choose any colorscale you like
    cmin = zmin,
    cmax = zmax,
      width = 7
    ),
    showlegend = FALSE
  ) |>
    add_trace(x = rep(s_semi_coarse, each = 3), 
            y = rep(y_semi_coarse, each = 3), 
            z = unlist(lapply(ff_s_semi_coarse, function(zj) c(0, zj, NA))),
            type = "scatter3d", 
            mode = "lines",
            line = list(color = "lightgray", width = 0.5),
            showlegend = FALSE)
 
dx <- int_map$x - curve_map$x
dy <- int_map$y - curve_map$y
dz <- int_map$z - curve_map$z

norm <- sqrt(dx^2 + dy^2 + dz^2)

u <- dx / norm
v <- dy / norm
w <- dz / norm

dx2 <- curve_map$x - ff_df_coarse$x
dy2 <- curve_map$y - ff_df_coarse$y
dz2 <- curve_map$z - ff_df_coarse$z

norm2 <- sqrt(dx2^2 + dy2^2 + dz2^2)

u2 <- dx2 / norm2
v2 <- dy2 / norm2
w2 <- dz2 / norm2

mag <- sqrt(u2^2 + v2^2 + w2^2)

scale <- ff_eval_coarse / mag

u_col <- u2 * scale
v_col <- v2 * scale
w_col <- w2 * scale

p <- p |>
add_trace(
  type = "cone",
  x = curve_map$x,
  y = curve_map$y,
  z = curve_map$z,
  u = u_col,
  v = v_col,
  w = w_col,
  colorscale = "Viridis",
  cmin = zmin,
  cmax = zmax,
  sizemode = "absolute",
  sizeref = 0.2,
  anchor = "tip",
  showscale = FALSE
)

rv <- 0.65
p2fhelix <- p |> config(mathjax = 'cdn') |>
            layout(p,
                   font = list(family = "Palatino"),
            margin = list(l = 0, r = 0, b = 0, t = 0),
            scene = list(xaxis = list(title = list(text = "x", font = list(color = colaxnn)),  tickfont = list(color = colaxnn),  range = c(0, L)),
              yaxis = list(title = list(text = "y", font = list(color = colaxnn)),  tickfont = list(color = colaxnn), range = c(-1, 4.1)),
              zaxis = list(title = list(text = "z", font = list(color = colaxnn)),  tickfont = list(color = colaxnn), range = c(-1, max(ff_eval))),
              #aspectmode="data",
              aspectratio = list(x = L/9, y = 5/3, z = (max(ff_eval)+1)/3),
              camera = list(eye = list(x = -3*rv, y = -6*rv, z = 6*rv),
                            center = list(x = 0, y = 0, z = 0)),
              annotations = list(
                list(
               x = zero$x, y = zero$y, z = zero$z,
               text = TeX("0"),
               textangle = 0, ax = 0, ay = -35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "gray", arrowsize = 1, arrowwidth = 0.5, arrowhead = 1),
               list(
               x = ell$x, y = ell$y, z = ell$z,
               text = TeX("\\ell"),
               textangle = 0, ax = 0, ay = -35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "gray", arrowsize = 1, arrowwidth = 0.5, arrowhead = 1),    
                list(
               x = aLh$x, y = aLh$y, z = aLh$z,
               text = TeX("e_1"),
               textangle = 0, ax = 0, ay = -35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "gray", arrowsize = 1, arrowwidth = 0.5, arrowhead = 1),
               list(
               x = aL$x, y = aL$y, z = aL$z,
               text = TeX("v_2"),
               textangle = 0, ax = 0, ay = -35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "gray", arrowsize = 1, arrowwidth = 0.5, arrowhead = 1),      
             list(
               x = a0$x, y = a0$y, z = a0$z,
               text = TeX("v_1"),
               textangle = 0, ax = 0, ay = -35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "gray", arrowsize = 1, arrowwidth = 0.5, arrowhead = 1))
            )
)

save(p2fhelix, file = here::here("data_files/graphs6p2fhelix.Rdata"))

load(here::here("data_files/graphs6p2fhelix.Rdata"))
p2fhelix

2 Helix arc-length parametrization

library(plotly)

# Parameters
a <- 1
b <- 0.5
TT <- 6*pi
# Total arc-length
L <- sqrt(a^2 + b^2) * TT

half_L <- L / 2


max_x <- b * (L / sqrt(a^2 + b^2))

half_max_x <- max_x / 2

dist_to_move <- half_L - half_max_x

# Arc-length parametrization (helix around x-axis)
alpha_tilde <- function(s){
  t <- s / sqrt(a^2 + b^2)
  x_shift <- TT*(sqrt(a^2 + b^2) - b) / 2
  x <- b * t + x_shift
  y <- a * cos(t)
  z <- a * sin(t)
  data.frame(x=x, y=y, z=z)
}

a0 = alpha_tilde(0)
aL = alpha_tilde(L)
aLh = alpha_tilde(half_L)

# Smooth helix
n_smooth <- 500
s_smooth <- seq(0, L, length.out = n_smooth)
curve_smooth <- alpha_tilde(s_smooth)

# Points for mapping lines
n_map <- 100
s_map <- seq(0, L, length.out = n_map)
curve_map <- alpha_tilde(s_map)

# Interval along x-axis
int_map <- data.frame(
  x = s_map,
  y = rep(0, n_map)+4,
  z = rep(0, n_map)
)

zero <- int_map[1, ]
ell <- int_map[n_map, ]

vertex <- rbind(a0, aL, zero, ell)


ttt <- half_L / sqrt(a^2 + b^2)

u <- b
v <- -a*sin(ttt)
w <-  a*cos(ttt)

# Plot
p <- plot_ly() |>
  
  # Smooth helix
  add_trace(
    data = curve_smooth,
    x = ~x, y = ~y, z = ~z,
    type = "scatter3d",
    mode = "lines",
    line = list(color = "darkred", width = 7),
    showlegend = FALSE
  ) |>
  
  # Interval [0,L]
  add_trace(
    data = int_map,
    x = ~x, y = ~y, z = ~z,
    type = "scatter3d",
    mode = "lines",
    line = list(color = "#0000C8", width = 7),
    showlegend = FALSE
  ) |>
  add_trace(data = vertex,
  x = ~x,
  y = ~y,
  z = ~z,
  type = "scatter3d",
  mode = "markers",
  marker = list(size = 5, color = "black"),
    showlegend = FALSE
) |>
  add_trace(
    type = "cone",
    x = aLh$x,
    y = aLh$y,
    z = aLh$z,
    u = u,
    v = v,
    w = w,
    sizemode = "absolute",
    sizeref = 0.4,
    colorscale = list(c(0, 1), c("green", "green")),
    showscale = FALSE
  )

pal <- colorRampPalette(c(
  "#0000C8",  # dark navy
  "#0074D9",  # royalblue
  "#7FDBFF",  # cyan
  "#2ECC40",  # green
  "#FFDC00",  # yellow
  "#FF851B",  # orange
  "#FF4136",  # red
  "darkred"
))(100)

gradient_line <- function(p0, p1, n = 20){

  t <- seq(0,1,length.out = n)

  x <- (1-t)*p0$x + t*p1$x
  y <- (1-t)*p0$y + t*p1$y
  z <- (1-t)*p0$z + t*p1$z

  data.frame(x=x,y=y,z=z,t=t)
}


dx <- int_map$x - curve_map$x
dy <- int_map$y - curve_map$y
dz <- int_map$z - curve_map$z

norm <- sqrt(dx^2 + dy^2 + dz^2)

u <- dx / norm
v <- dy / norm
w <- dz / norm


p <- p |>
  add_trace(
    type = "cone",
    x = int_map$x,
    y = int_map$y,
    z = int_map$z,
    u = u,
    v = v,
    w = w,
    sizemode = "absolute",
    sizeref = 0.6,
    anchor = "tip",
    colorscale = list(c(0, "#0000C8"), c(1, "#0000C8")),
    showscale = FALSE
  ) 

for(i in 1:n_map){

  g <- gradient_line(int_map[i,], curve_map[i,])

  for(j in 1:(nrow(g)-1)){

    seg <- g[j:(j+1),]

    col <- pal[ round(seg$t[1]*(length(pal)-1))+1 ]

    p <- p |> add_trace(
      data = seg,
      x = ~x, y = ~y, z = ~z,
      type = "scatter3d",
      mode = "lines",
      line = list(color = col, width = 1),
      showlegend = FALSE
    )
  }
}

rv <- 0.65
p2 <- p |> config(mathjax = 'cdn') |>
            layout(p,
                   font = list(family = "Palatino"),
            margin = list(l = 0, r = 0, b = 0, t = 0),
            scene = list(xaxis = list(title = list(text = "x", font = list(color = colaxnn)),  tickfont = list(color = colaxnn),  range = c(0, L)),
              yaxis = list(title = list(text = "y", font = list(color = colaxnn)),  tickfont = list(color = colaxnn), range = c(-1, 4.1)),
              zaxis = list(title = list(text = "z", font = list(color = colaxnn)),  tickfont = list(color = colaxnn), range = c(-1, 2)),
              #aspectmode="data",
              aspectratio = list(x = L/9, y = 5/3, z = (2+1)/3),
              camera = list(eye = list(x = -3*rv, y = -6*rv, z = 6*rv),
                            center = list(x = 0, y = 0, z = 0)),
              annotations = list(
                list(
               x = zero$x, y = zero$y, z = zero$z,
               text = TeX("0"),
               textangle = 0, ax = 0, ay = -35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "gray", arrowsize = 1, arrowwidth = 0.5, arrowhead = 1),
               list(
               x = ell$x, y = ell$y, z = ell$z,
               text = TeX("\\ell"),
               textangle = 0, ax = 0, ay = -35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "gray", arrowsize = 1, arrowwidth = 0.5, arrowhead = 1),   
                list(
               x = aLh$x, y = aLh$y, z = aLh$z,
               text = TeX("e_1"),
               textangle = 0, ax = 0, ay = -35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "gray", arrowsize = 1, arrowwidth = 0.5, arrowhead = 1),
               list(
               x = aL$x, y = aL$y, z = aL$z,
               text = TeX("v_2"),
               textangle = 0, ax = 0, ay = -35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "gray", arrowsize = 1, arrowwidth = 0.5, arrowhead = 1),      
             list(
               x = a0$x, y = a0$y, z = a0$z,
               text = TeX("v_1"),
               textangle = 0, ax = 0, ay = -35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "gray", arrowsize = 1, arrowwidth = 0.5, arrowhead = 1))
            )
)


save(p2, file = here::here("data_files/graphs6p2.Rdata"))

load(here::here("data_files/graphs6p2.Rdata"))
p2

3 Tadpole and function

library(plotly)
n_smooth <- 250

TT <- 3*pi
# parameter
t <- seq(0, TT, length.out = n_smooth)

# speed
speed <- sqrt(1 + cos(t)^2)

# arc-length function
s <- pracma::cumtrapz(t, speed)
L <- max(s)
# s <- cumsum(c(0, diff(t) * (head(speed,-1) + tail(speed,-1))/2))
f1 <- function(t) exp(t/4)


f1_on_curve <- f1(s)
y_up_range <- max(f1_on_curve)

f2 <- function(t) sin(3*pi*t/L) + y_up_range
f1df_int <- data.frame(x = s, y = rep(0, n_smooth), z = f1_on_curve)

half_L <- L / 2
half_TT <- TT / 2
dist_to_move <- abs(half_L - half_TT)

y_shift <- 4
# curve alpha(t)
x <- t + dist_to_move
y <- sin(t) + y_shift
z <- rep(0,length(t))

curve_smooth <- data.frame(x = x, y = y, z = z)
f1_on_curve_smooth <- data.frame(x = x, y = y, z = f1_on_curve)

n_map <- 25
# points where we draw connectors
idx <- seq(1, length(t), length.out = n_map)

f1df_int_map <- f1df_int[idx,]
f1_on_curve_smooth_map <- f1_on_curve_smooth[idx,]

# Build mapping lines
rows <- lapply(1:n_map, function(i) {
  list(f1df_int_map[i, ], f1_on_curve_smooth_map[i, ], data.frame(x = NA, y = NA, z = NA))
})

resultee1 <- do.call(rbind, unlist(rows, recursive = FALSE))





curve_map <- data.frame(x = x[idx], y = y[idx], z = z[idx])
int_map <- data.frame(x = s[idx], y = rep(0, n_map), z = rep(0, n_map))

# 
# # Build mapping lines
# rows <- lapply(1:n_map, function(i) {
#   list(int_map[i, ], curve_map[i, ], data.frame(x = NA, y = NA, z = NA))
# })
# 
# result <- do.call(rbind, unlist(rows, recursive = FALSE))

s_circ <- L
radius <- s_circ / (2*pi)
x_shift_for_circle <- radius + TT + dist_to_move

theta <- seq(from=-pi,to=pi,length.out = n_smooth)
circle_curve <- data.frame(x = radius*cos(theta) + x_shift_for_circle, 
                    y = radius*sin(theta) + y_shift, 
                    z = rep(0, length(theta)))

arclength_circle <- radius * (theta + pi)

dist_to_move_circle <- abs(half_L - y_shift)

f2_on_curve <- f2(arclength_circle)
f2df_int <- data.frame(x = rep(x_shift_for_circle +2*radius, n_smooth),
                       y = arclength_circle-dist_to_move_circle, 
                       z = f2_on_curve)

f2_on_curve_smooth <- data.frame(x = radius*cos(theta) + x_shift_for_circle, 
                                 y = radius*sin(theta) + y_shift,
                                 z = f2_on_curve)

f2df_int_map <- f2df_int[idx,]
f2_on_curve_smooth_map <- f2_on_curve_smooth[idx,]

# Build mapping lines
rows <- lapply(1:n_map, function(i) {
  list(f2df_int_map[i, ], f2_on_curve_smooth_map[i, ], data.frame(x = NA, y = NA, z = NA))
})

resultee2 <- do.call(rbind, unlist(rows, recursive = FALSE))



circle_map <- circle_curve[idx, ]
int_map_circle <- data.frame(x = rep(x_shift_for_circle +2*radius, n_map), 
                             y = s[idx]-dist_to_move_circle, 
                             z = rep(0, n_map))

# # Build mapping lines for circle
# rows <- lapply(1:n_map, function(i) {
#   list(int_map_circle[i, ], circle_map[i, ], data.frame(x = NA, y = NA, z = NA))
# })
# 
# result_for_circle <- do.call(rbind, unlist(rows, recursive = FALSE))

v1 <- curve_map[1, ]
v2 <- curve_map[n_map, ]
e1 <- curve_map[ceiling(n_map/2), ]
e2 <- circle_map[ceiling(n_map/2), ]
zero1 <- int_map[1, ]
le1 <- int_map[n_map, ]

zero2 <- int_map_circle[1, ]
le2 <- int_map_circle[n_map, ]

vertex <- rbind(v1, v2, zero1, zero2, le1, le2)

zmin <- min(f1_on_curve, f2_on_curve)
zmax <- max(f1_on_curve, f2_on_curve)

# plot vertical lines from edges to curves
vertical_lines1 <- data.frame(x = rep(f1df_int$x, each = 3), 
                             y = rep(f1df_int$y, each = 3), 
                             z = unlist(lapply(f1df_int$z, function(zj) c(0, zj, NA))))

vertical_lines2 <- data.frame(x = rep(f2df_int$x, each = 3), 
                             y = rep(f2df_int$y, each = 3), 
                             z = unlist(lapply(f2df_int$z, function(zj) c(0, zj, NA))))

vertical_lines3 <- data.frame(x = rep(f1_on_curve_smooth$x, each = 3), 
                             y = rep(f1_on_curve_smooth$y, each = 3), 
                             z = unlist(lapply(f1_on_curve_smooth$z, function(zj) c(0, zj, NA))))

vertical_lines4 <- data.frame(x = rep(f2_on_curve_smooth$x, each = 3), 
                             y = rep(f2_on_curve_smooth$y, each = 3), 
                             z = unlist(lapply(f2_on_curve_smooth$z, function(zj) c(0, zj, NA))))

vertical_lines <- rbind(vertical_lines1, vertical_lines2, vertical_lines3, vertical_lines4)

p <- plot_ly() |>
  add_trace(data = rbind(curve_smooth, #smooth sin edge
                         data.frame(x = NA, y = NA, z = NA),
                         circle_curve, # smooth circle edge
                         data.frame(x = NA, y = NA, z = NA),
                         int_map, # interval for sin edge
                         data.frame(x = NA, y = NA, z = NA),
                         int_map_circle), # interval for circle edge
            x = ~x, y = ~y, z = ~z,
            type = "scatter3d", mode = "lines",
            line = list(width = 7, color = "black"),
            showlegend = FALSE) |>
  add_trace(data = vertical_lines,
            x = ~x, y = ~y, z = ~z,
            type = "scatter3d", mode = "lines",
            line = list(color = "lightgray", width = 0.5),
            showlegend = FALSE) |>
  add_trace(data = rbind(f1df_int, 
                         data.frame(x = NA, y = NA, z = NA),
                         f2df_int, 
                         data.frame(x = NA, y = NA, z = NA),
                         f1_on_curve_smooth, 
                         data.frame(x = NA, y = NA, z = NA),
                         f2_on_curve_smooth),
            x = ~x, y = ~y, z = ~z,
            type = "scatter3d", mode = "lines",
            line = list(
              color = ~z,                 # color based on height
              colorscale = "Viridis",     # choose any colorscale you like
              cmin = zmin,
              cmax = zmax,
              width = 7
              ),
            showlegend = FALSE) |>
  add_trace(data = rbind(resultee1, # mapping lines for sin
                         resultee2), # mapping lines for circle
            x = ~x, y = ~y, z = ~z,
            type = "scatter3d", mode = "lines",
            line = list(
              color = ~z,                 # color based on height
              colorscale = "Viridis",     # choose any colorscale you like
              cmin = zmin,
              cmax = zmax,
              width = 1
              ),
    showlegend = FALSE) |>
  add_trace(data = vertex,
            x = ~x,
            y = ~y,
            z = ~z,
            type = "scatter3d",
            mode = "markers",
            marker = list(size = 5, color = "black"),
            showlegend = FALSE) |> 
  add_trace(
    type = "cone",
    x = e1$x,
    y = e1$y,
    z = e1$z,
    u = 1,
    v = 0,
    w = 0,
    sizemode = "absolute",
    sizeref = 0.6,
    colorscale = list(c(0, 1), c("green", "green")),
    showscale = FALSE) |>
  add_trace(
    type = "cone",
    x = e2$x,
    y = e2$y,
    z = e2$z,
    u = 0,
    v = 1,
    w = 0,
    sizemode = "absolute",
    sizeref = 0.6,
    colorscale = list(c(0, 1), c("green", "green")),
    showscale = FALSE
  )


rv <- 2
p4tadpole_arclength <- p |>
  config(mathjax = 'cdn') |> 
  layout(p,
         font = list(family = "Palatino"),
            margin = list(l = 0, r = 0, b = 0, t = 0),
            scene = list(xaxis = list(title = list(text = "x", font = list(color = colaxnn)),  tickfont = list(color = colaxnn),  range = c(0, x_shift_for_circle +2*radius)*1.01),
              yaxis = list(title = list(text = "y", font = list(color = colaxnn)),  tickfont = list(color = colaxnn), range = range(int_map_circle$y)),
              zaxis = list(title = list(text = "z", font = list(color = colaxnn)),  tickfont = list(color = colaxnn), range = c(0, y_up_range+1)),
    aspectratio = list(x = x_shift_for_circle +2*radius, y = L, z = 4),
              camera = list(eye = list(x = 7*rv, y = -15*rv, z = 6*rv),
                            center = list(x = 0, y = 0, z = 0)),
    annotations = list(
      list(
               x = zero1$x, y = zero1$y, z = zero1$z,
               text = TeX("0"),
               textangle = 0, ax = 0, ay = -35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "gray", arrowsize = 1, arrowwidth = 0.5, arrowhead = 1),
      list(
               x = zero2$x, y = zero2$y, z = zero2$z,
               text = TeX("0"),
               textangle = 0, ax = 0, ay = -35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "gray", arrowsize = 1, arrowwidth = 0.5, arrowhead = 1),
      list(
               x = le1$x, y = le1$y, z = le1$z,
               text = TeX("\\ell_{e_1}"),
               textangle = 0, ax = 0, ay = -35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "gray", arrowsize = 1, arrowwidth = 0.5, arrowhead = 1),
      list(
               x = le2$x, y = le2$y, z = le2$z,
               text = TeX("\\ell_{e_2}"),
               textangle = 0, ax = 0, ay = -35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "gray", arrowsize = 1, arrowwidth = 0.5, arrowhead = 1),
                list(
               x = v1$x, y = v1$y, z = v1$z,
               text = TeX("v_1"),
               textangle = 0, ax = 0, ay = -35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "gray", arrowsize = 1, arrowwidth = 0.5, arrowhead = 1),
               list(
               x = v2$x, y = v2$y, z = v2$z,
               text = TeX("v_2"),
               textangle = 0, ax = 0, ay = -35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "gray", arrowsize = 1, arrowwidth = 0.5, arrowhead = 1),
               list(
               x = e1$x, y = e1$y, z = e1$z,
               text = TeX("e_1"),
               textangle = 0, ax = 0, ay = -35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "gray", arrowsize = 1, arrowwidth = 0.5, arrowhead = 1),
               list(
               x = e2$x, y = e2$y, z = e2$z,
               text = TeX("e_2"),
               textangle = 0, ax = 0, ay = -35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "gray", arrowsize = 1, arrowwidth = 0.5, arrowhead = 1))

  )
)

save(p4tadpole_arclength, file = here::here("data_files/graphs6p4tadpole_arclength.Rdata"))

load(here::here("data_files/graphs6p4tadpole_arclength.Rdata"))
p4tadpole_arclength

4 Tadpole arc-length parametrization

library(plotly)
n_smooth <- 250

TT <- 3*pi
# parameter
t <- seq(0, TT, length.out = n_smooth)

# speed
speed <- sqrt(1 + cos(t)^2)

# arc-length function
s <- pracma::cumtrapz(t, speed)
# s <- cumsum(c(0, diff(t) * (head(speed,-1) + tail(speed,-1))/2))

L <- max(s)

half_L <- L / 2
half_TT <- TT / 2
dist_to_move <- abs(half_L - half_TT)

y_shift <- 4
# curve alpha(t)
x <- t + dist_to_move
y <- sin(t) + y_shift
z <- rep(0,length(t))

curve_smooth <- data.frame(x = x, y = y, z = z)

# the following lines are just to get the range of f1 on the curve, so that we can set the range so both plots are similar
f1 <- function(t) exp(t/4)
f1_on_curve <- f1(s)
y_up_range <- max(f1_on_curve)


n_map <- 50
# points where we draw connectors
idx <- seq(1, length(t), length.out = n_map)


curve_map <- data.frame(x = x[idx], y = y[idx], z = z[idx])
int_map <- data.frame(x = s[idx], y = rep(0, n_map), z = rep(0, n_map))



# Build mapping lines
rows <- lapply(1:n_map, function(i) {
  list(int_map[i, ], curve_map[i, ], data.frame(x = NA, y = NA, z = NA))
})

result <- do.call(rbind, unlist(rows, recursive = FALSE))

s_circ <- L
radius <- s_circ / (2*pi)
x_shift_for_circle <- radius + TT + dist_to_move

theta <- seq(from=-pi,to=pi,length.out = n_smooth)
circle_curve <- data.frame(x = radius*cos(theta) + x_shift_for_circle, 
                    y = radius*sin(theta) + y_shift, 
                    z = rep(0, length(theta)))


circle_map <- circle_curve[idx, ]
dist_to_move_circle <- abs(half_L - y_shift)
int_map_circle <- data.frame(x = rep(x_shift_for_circle +2*radius, n_map), y = s[idx]-dist_to_move_circle, z = rep(0, n_map))

# Build mapping lines for circle
rows <- lapply(1:n_map, function(i) {
  list(int_map_circle[i, ], circle_map[i, ], data.frame(x = NA, y = NA, z = NA))
})

result_for_circle <- do.call(rbind, unlist(rows, recursive = FALSE))

v1 <- curve_map[1, ]
v2 <- curve_map[n_map, ]
e1 <- curve_map[ceiling(n_map/2), ]
e2 <- circle_map[ceiling(n_map/2), ]
zero1 <- int_map[1, ]
le1 <- int_map[n_map, ]

zero2 <- int_map_circle[1, ]
le2 <- int_map_circle[n_map, ]

vertex <- rbind(v1, v2, zero1, zero2, le1, le2)

p <- plot_ly() |>
  # smooth sin curve
  add_trace(data = rbind(curve_smooth, # smooth sin curve
                         data.frame(x = NA, y = NA, z = NA),
                         circle_curve), # smooth circle curve
  x = ~x, y = ~y, z = ~z,
  type = "scatter3d",
  mode = "lines",
  line = list(width = 7, color = "darkred"),
    showlegend = FALSE
) |>
add_trace(data = rbind(int_map, # interval for sin
                       data.frame(x = NA, y = NA, z = NA),
                       int_map_circle), # interval for circle
  x = ~x,
  y = ~y,
  z = ~z,
  type = "scatter3d",
  mode = "lines",
  line = list(width = 7, color = "#0000C8"),
    showlegend = FALSE
) |>
  add_trace(data = vertex,
  x = ~x,
  y = ~y,
  z = ~z,
  type = "scatter3d",
  mode = "markers",
  marker = list(size = 5, color = "black"),
    showlegend = FALSE
) |> add_trace(
    type = "cone",
    x = e1$x,
    y = e1$y,
    z = e1$z,
    u = 1,
    v = 0,
    w = 0,
    sizemode = "absolute",
    sizeref = 0.6,
    colorscale = list(c(0, 1), c("green", "green")),
    showscale = FALSE
  ) |> add_trace(
    type = "cone",
    x = e2$x,
    y = e2$y,
    z = e2$z,
    u = 0,
    v = 1,
    w = 0,
    sizemode = "absolute",
    sizeref = 0.6,
    colorscale = list(c(0, 1), c("green", "green")),
    showscale = FALSE
  )

pal <- colorRampPalette(c(
  "#0000C8",  # dark navy
  "#0074D9",  # royalblue
  "#7FDBFF",  # cyan
  "#2ECC40",  # green
  "#FFDC00",  # yellow
  "#FF851B",  # orange
  "#FF4136",  # red
  "darkred"
))(100)

gradient_line <- function(p0, p1, n = 20){

  t <- seq(0,1,length.out = n)

  x <- (1-t)*p0$x + t*p1$x
  y <- (1-t)*p0$y + t*p1$y
  z <- (1-t)*p0$z + t*p1$z

  data.frame(x=x,y=y,z=z,t=t)
}


dx <- int_map$x - curve_map$x
dy <- int_map$y - curve_map$y
dz <- int_map$z - curve_map$z

norm <- sqrt(dx^2 + dy^2 + dz^2)

u <- dx / norm
v <- dy / norm
w <- dz / norm

dxc <- int_map_circle$x - circle_map$x
dyc <- int_map_circle$y - circle_map$y
dzc <- int_map_circle$z - circle_map$z

normc <- sqrt(dxc^2 + dyc^2 + dzc^2)

uc <- dxc / normc
vc <- dyc / normc
wc <- dzc / normc

p <- p |>
  add_trace(
    type = "cone",
    x = int_map$x,
    y = int_map$y,
    z = int_map$z,
    u = u,
    v = v,
    w = w,
    sizemode = "absolute",
    sizeref = 0.7,
    anchor = "tip",
    colorscale = list(c(0, "#0000C8"), c(1, "#0000C8")),
    showscale = FALSE
  ) |> add_trace(
    type = "cone",
    x = int_map_circle$x,
    y = int_map_circle$y,
    z = int_map_circle$z,
    u = uc,
    v = vc,
    w = wc,
    sizemode = "absolute",
    sizeref = 0.4,
    anchor = "tip",
    colorscale = list(c(0, "#0000C8"), c(1, "#0000C8")),
    showscale = FALSE
  )

for(i in 1:n_map){

  g <- gradient_line(int_map[i,], curve_map[i,])

  for(j in 1:(nrow(g)-1)){

    seg <- g[j:(j+1),]

    col <- pal[ round(seg$t[1]*(length(pal)-1))+1 ]

    p <- p |> add_trace(
      data = seg,
      x = ~x, y = ~y, z = ~z,
      type = "scatter3d",
      mode = "lines",
      line = list(color = col, width = 1),
      showlegend = FALSE
    )
  }
}

for(i in 1:n_map){

  g <- gradient_line(int_map_circle[i,], circle_map[i,])

  for(j in 1:(nrow(g)-1)){

    seg <- g[j:(j+1),]

    col <- pal[ round(seg$t[1]*(length(pal)-1))+1 ]

    p <- p |> add_trace(
      data = seg,
      x = ~x, y = ~y, z = ~z,
      type = "scatter3d",
      mode = "lines",
      line = list(color = col, width = 1),
      showlegend = FALSE
    )
  }
}


rv <- 2
p4 <- p |>
  config(mathjax = 'cdn') |> 
  layout(p,
         font = list(family = "Palatino"),
            margin = list(l = 0, r = 0, b = 0, t = 0),
            scene = list(xaxis = list(title = list(text = "x", font = list(color = colaxnn)),  tickfont = list(color = colaxnn),  range = c(0, x_shift_for_circle +2*radius)*1.01),
              yaxis = list(title = list(text = "y", font = list(color = colaxnn)),  tickfont = list(color = colaxnn), range = range(int_map_circle$y)),
              zaxis = list(title = list(text = "z", font = list(color = colaxnn)),  tickfont = list(color = colaxnn), range = c(0, y_up_range+1)),
    aspectratio = list(x = x_shift_for_circle +2*radius, y = L, z = 4),
              camera = list(eye = list(x = 7*rv, y = -15*rv, z = 6*rv),
                            center = list(x = 0, y = 0, z = 0)),
    annotations = list(
      list(
               x = zero1$x, y = zero1$y, z = zero1$z,
               text = TeX("0"),
               textangle = 0, ax = 0, ay = -35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "gray", arrowsize = 1, arrowwidth = 0.5, arrowhead = 1),
      list(
               x = zero2$x, y = zero2$y, z = zero2$z,
               text = TeX("0"),
               textangle = 0, ax = 0, ay = -35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "gray", arrowsize = 1, arrowwidth = 0.5, arrowhead = 1),
      list(
               x = le1$x, y = le1$y, z = le1$z,
               text = TeX("\\ell_{e_1}"),
               textangle = 0, ax = 0, ay = -35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "gray", arrowsize = 1, arrowwidth = 0.5, arrowhead = 1),
      list(
               x = le2$x, y = le2$y, z = le2$z,
               text = TeX("\\ell_{e_2}"),
               textangle = 0, ax = 0, ay = -35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "gray", arrowsize = 1, arrowwidth = 0.5, arrowhead = 1),
                list(
               x = v1$x, y = v1$y, z = v1$z,
               text = TeX("v_1"),
               textangle = 0, ax = 0, ay = -35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "gray", arrowsize = 1, arrowwidth = 0.5, arrowhead = 1),
               list(
               x = v2$x, y = v2$y, z = v2$z,
               text = TeX("v_2"),
               textangle = 0, ax = 0, ay = -35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "gray", arrowsize = 1, arrowwidth = 0.5, arrowhead = 1),
               list(
               x = e1$x, y = e1$y, z = e1$z,
               text = TeX("e_1"),
               textangle = 0, ax = 0, ay = -35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "gray", arrowsize = 1, arrowwidth = 0.5, arrowhead = 1),
               list(
               x = e2$x, y = e2$y, z = e2$z,
               text = TeX("e_2"),
               textangle = 0, ax = 0, ay = -35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "gray", arrowsize = 1, arrowwidth = 0.5, arrowhead = 1))
  )
)

save(p4, file = here::here("data_files/graphs6p4.Rdata"))

load(here::here("data_files/graphs6p4.Rdata"))
p4

5 References

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We used R version 4.5.2 (R Core Team 2025a) and the following R packages: cowplot v. 1.2.0 (Wilke 2025), ggmap v. 4.0.2 (Kahle and Wickham 2013), ggpubr v. 0.6.3 (Kassambara 2026), ggtext v. 0.1.2 (Wilke and Wiernik 2022), glue v. 1.8.0 (Hester and Bryan 2024), grid v. 4.5.2 (R Core Team 2025b), here v. 1.0.1 (Müller 2020), htmltools v. 0.5.8.1 (Cheng et al. 2024), INLA v. 25.11.22 (Rue, Martino, and Chopin 2009; Lindgren, Rue, and Lindström 2011; Martins et al. 2013; Lindgren and Rue 2015; De Coninck et al. 2016; Rue et al. 2017; Verbosio et al. 2017; Bakka et al. 2018; Kourounis, Fuchs, and Schenk 2018), inlabru v. 2.13.0 (Yuan et al. 2017; Bachl et al. 2019), knitr v. 1.50 (Xie 2014, 2015, 2025), latex2exp v. 0.9.8 (Meschiari 2026), Matrix v. 1.7.3 (Bates, Maechler, and Jagan 2025), MetricGraph v. 1.5.0.9000 (Bolin, Simas, and Wallin 2023a, 2023b, 2024, 2025; Bolin et al. 2024), OpenStreetMap v. 0.4.1 (Fellows and Stotz 2025), patchwork v. 1.3.1 (Pedersen 2025), plotly v. 4.11.0 (Sievert 2020), plotrix v. 3.8.14 (J 2006), pracma v. 2.4.4 (Borchers 2023), renv v. 1.1.7 (Ushey and Wickham 2026), reshape2 v. 1.4.4 (Wickham 2007), reticulate v. 1.44.1 (Ushey, Allaire, and Tang 2025), rmarkdown v. 2.30 (Xie, Allaire, and Grolemund 2018; Xie, Dervieux, and Riederer 2020; Allaire et al. 2025), rSPDE v. 2.5.2.9000 (Bolin and Kirchner 2020; Bolin and Simas 2023; Bolin, Simas, and Xiong 2024), scales v. 1.4.0 (Wickham, Pedersen, and Seidel 2025), sf v. 1.1.0 (E. Pebesma 2018; E. Pebesma and Bivand 2023), slackr v. 3.4.0 (Kaye et al. 2025), sp v. 2.2.1 (E. J. Pebesma and Bivand 2005; Bivand, Pebesma, and Gomez-Rubio 2013), tidyverse v. 2.0.0 (Wickham et al. 2019), tikzDevice v. 0.12.6 (Sharpsteen and Bracken 2023), viridis v. 0.6.5 (Garnier et al. 2024), xaringanExtra v. 0.8.0 (Aden-Buie and Warkentin 2024).

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