Kirchhoff vertex conditions

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# Start the figure counter
fig_count <- 0
# Define the captioner function
captioner <- function(caption) {
  fig_count <<- fig_count + 1
  paste0("Figure ", fig_count, ": ", caption)
}

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(
  knitr::purl(here::here("functionality1.Rmd"), output = here::here("functionality1.R")),
  file = here::here("old/purl_log.txt")
)
source(here::here("functionality1.R"))

edges_roots_of_unity <- function(n, length = 10) {
  x_coords <- cos(2 * pi * (0:(n-1)) / n)
  y_coords <- sin(2 * pi * (0:(n-1)) / n)
  return(lapply(1:n, function(i) {rbind(c(0, 0), c(length*x_coords[i], length*y_coords[i]))}))
}
n <- 3
my_length <- 2
edges <-edges_roots_of_unity(n, length = my_length)
#edges[[1]] <- edges[[1]][2:1,]
graph <- metric_graph$new(edges = edges)
graph_disc <- graph$clone()

e1_ini_x <- edges[[1]][1,1]; e1_ini_y <- edges[[1]][1,2]; e1_ini_z <- 0
e1_fin_x <- edges[[1]][2,1]; e1_fin_y <- edges[[1]][2,2]; e1_fin_z <- 0

# Midpoint
xm1 <- (e1_fin_x + e1_ini_x)/2
ym1 <- (e1_fin_y + e1_ini_y)/2
zm1 <- (e1_fin_z + e1_ini_z)/2

# Direction vector
dx1 <- e1_fin_x - e1_ini_x
dy1 <- e1_fin_y - e1_ini_y
dz1 <- e1_fin_z - e1_ini_z

e2_ini_x <- edges[[2]][1,1]; e2_ini_y <- edges[[2]][1,2]; e2_ini_z <- 0
e2_fin_x <- edges[[2]][2,1]; e2_fin_y <- edges[[2]][2,2]; e2_fin_z <- 0

# Midpoint
xm2 <- (e2_fin_x + e2_ini_x)/2
ym2 <- (e2_fin_y + e2_ini_y)/2
zm2 <- (e2_fin_z + e2_ini_z)/2

# Direction vector
dx2 <- e2_fin_x - e2_ini_x
dy2 <- e2_fin_y - e2_ini_y
dz2 <- e2_fin_z - e2_ini_z

e3_ini_x <- edges[[3]][1,1]; e3_ini_y <- edges[[3]][1,2]; e3_ini_z <- 0
e3_fin_x <- edges[[3]][2,1]; e3_fin_y <- edges[[3]][2,2]; e3_fin_z <- 0

# Midpoint
xm3 <- (e3_fin_x + e3_ini_x)/2
ym3 <- (e3_fin_y + e3_ini_y)/2
zm3 <- (e3_fin_z + e3_ini_z)/2

# Direction vector
dx3 <- e3_fin_x - e3_ini_x
dy3 <- e3_fin_y - e3_ini_y
dz3 <- e3_fin_z - e3_ini_z

e4_ini_x <- 0; e4_ini_y <- 0; e4_ini_z <- 0
e4_fin_x <- -my_length; e4_fin_y <- 0; e4_fin_z <- 0

# Midpoint
xm4 <- (e4_fin_x + e4_ini_x)/2
ym4 <- (e4_fin_y + e4_ini_y)/2
zm4 <- (e4_fin_z + e4_ini_z)/2

# Direction vector
dx4 <- e4_fin_x - e4_ini_x
dy4 <- e4_fin_y - e4_ini_y
dz4 <- e4_fin_z - e4_ini_z

sizeref <- 0.2

h = 0.02
graph$build_mesh(h = h)
graph_disc$build_mesh(h = h, continuous = FALSE)

apply_edge_functions_fast <- function(graph, f_list) {
  
  if (length(f_list) != graph$nE) {
    stop(sprintf(
      "Number of functions (%d) must equal number of edges (%d).",
      length(f_list), graph$nE
    ))
  }
  VtE <- graph$mesh$VtE
  edge_lengths <- graph$edge_lengths
  
  edge   <- VtE[,1]
  s_norm <- VtE[,2]
  s_true <- s_norm * edge_lengths[edge]
  
  # s_true <- ifelse(graph$edge_orientation[edge] == -1,
  #                edge_lengths[edge] * (1 - s_norm),
  #                edge_lengths[edge] * s_norm)

  
  # out <- numeric(length(edge))
  # for (e in seq_len(graph$nE)) {
  #   idx <- which(edge == e)
  #   out[idx] <- f_list[[e]](s_true[idx])
  # }
  # return(out)
  out <- sapply(seq_len(nrow(VtE)), function(i) f_list[[edge[i]]](s_true[i]))
  return(out)
}

1 Example of a function \(f\in C(\Gamma)\cap \mathcal{K}(\Gamma)\setminus C^1(\Gamma)\)

Let us consider the \(3\)-star graph \(\Gamma = (\mathcal{V}, \mathcal{E})\), where \(\mathcal{V} = \{v, v_1, v_2, v_3\}\) and \(\mathcal{E} = \{e_1, e_2, e_3\}\). Each edge \(e_i\) has common length \(\ell\) and connects the central vertex \(v\) to the outer vertex \(v_i\) for \(i=1,2,3\) in such a way that \(v = \underline{e_1} = \underline{e_2} = \underline{e_3}\). On \(\Gamma\), we define the function \(f = \{f_e\}_{e\in\mathcal{E}}\) via

\[ f_{e_1}(s) = a + b_1 s - \frac{b_1 s^3}{3\ell^2}, \quad f_{e_2}(s) = a + b_2 s - \frac{b_2 s^3}{3\ell^2}, \quad f_{e_3}(s) = a + b_3 s - \frac{b_3 s^3}{3\ell^2}, \]

where \(a\) (to be chosen) is the value of \(f\) at the central vertex and \(b_1, b_2, b_3\) are constants (to be chosen) such that \(b_1 + b_2 + b_3 = 0\). Clearly, \(f\) is a continuous function since

\[ f_{e_1}(0) = f_{e_2}(0) = f_{e_3}(0) = a. \]

The derivative \(f' = \{f'_e\}_{e\in\mathcal{E}}\) of \(f\) on each edge is given by

\[ f_{e_1}'(s) = b_1 - \frac{b_1 s^2}{\ell^2}, \quad f_{e_2}'(s) = b_2 - \frac{b_2 s^2}{\ell^2}, \quad f_{e_3}'(s) = b_3 - \frac{b_3 s^2}{\ell^2}. \] Since \(f_{e_1}'(0) = b_1\), \(f_{e_2}'(0) = b_2\) and \(f_{e_3}'(0) = b_3\), we can choose \(b_1, b_2, b_3\) such that \(f'\) is discontinuous at the central vertex and satisfies the Kirchhoff vertex conditions simultaneously. The Kirchhoff vertex conditions at the central vertex are satisfied since

\[ f_{e_1}'(0) + f_{e_2}'(0) + f_{e_3}'(0) = b_1 + b_2 + b_3 = 0. \]

and at the outer vertices we have

\[ f_{e_1}'(\ell) = f_{e_2}'(\ell) = f_{e_3}'(\ell) = 0. \]

For instance, we can choose \(b_1 = 1\), \(b_2 = 1\) and \(b_3 = -2\). This choice ensures \(f\) to be continuous and satisfy the Kirchhoff vertex conditions while \(f'\) is discontinuous at the central vertex. See Figure 1 and 2.

# General parameters
L <- my_length             # edge length
a <- 1             # value at central vertex
b <- c(1, 1, -2)   # b1+b2+b3 must be 0

f1 <- function(s) a + b[1]*s - b[1]*s^3/(3*L^2)
f2 <- function(s) a + b[2]*s - b[2]*s^3/(3*L^2)
f3 <- function(s) a + b[3]*s - b[3]*s^3/(3*L^2)


df1 <- function(s) b[1] - b[1]*s^2/L^2
df2 <- function(s) b[2] - b[2]*s^2/L^2
df3 <- function(s) b[3] - b[3]*s^2/L^2

ddf1 <- function(s) -2*b[1]*s/L^2
ddf2 <- function(s) -2*b[2]*s/L^2
ddf3 <- function(s) -2*b[3]*s/L^2

dddf1 <- function(s) -2*b[1]/L^2
dddf2 <- function(s) -2*b[2]/L^2
dddf3 <- function(s) -2*b[3]/L^2

f_list <- list(f1, f2, f3)
df_list <- list(df1, df2, df3)
ddf_list <- list(ddf1, ddf2, ddf3)
dddf_list <- list(dddf1, dddf2, dddf3)

f <- apply_edge_functions_fast(graph, f_list)
df <- apply_edge_functions_fast(graph_disc, df_list)
ddf <- apply_edge_functions_fast(graph, ddf_list)
dddf <- apply_edge_functions_fast(graph_disc, dddf_list)

l_mesh <- seq(0, my_length, by = h)
dfl1 <- df1(l_mesh)
dfl2 <- df2(l_mesh)
dfl3 <- df3(l_mesh)

l_mesh_norm <- l_mesh/my_length
PtE1 <- cbind(rep(1, length(l_mesh)), l_mesh_norm)
PtE2 <- cbind(rep(2, length(l_mesh)), l_mesh_norm)
PtE3 <- cbind(rep(3, length(l_mesh)), l_mesh_norm)
XY1 <- graph$coordinates(PtE1)
XY2 <- graph$coordinates(PtE2)
XY3 <- graph$coordinates(PtE3)
DF1 <- data.frame(x = XY1[,1], y = XY1[,2], z = dfl1)
DF2 <- data.frame(x = XY2[,1], y = XY2[,2], z = dfl2)
DF3 <- data.frame(x = XY3[,1], y = XY3[,2], z = dfl3)

DF <- rbind(DF1, rep(NA, 3), DF2, rep(NA, 3), DF3)
DF_NO_NA <- rbind(DF1, DF2, DF3)

x_eye <- 5#8
y_eye <- -1#-2
z_eye <- 2#3

graph_disc$plot_function(X = df, vertex_size = gsw, edge_width = gsw, line_width = gsw, continuous = FALSE, interpolate_plot = TRUE, type = "plotly") |> 
  config(mathjax = 'cdn') |>
  plotly::layout(title = TeX("f'"), 
                 font = list(family = "Palatino"),
                 scene = list(
                   aspectratio = list(x = 2, 
                                 y = 2, 
                                 z = 2),
                   camera = list(eye = list(x = x_eye, 
                                       y = y_eye, 
                                       z = z_eye),
                            center = list(x = 0, 
                                          y = 0, 
                                          z = 0))
                 ))

star3f <- graph$plot_function(X = f, vertex_size = gsw, edge_width = gsw, line_width = gsw, type = "plotly") |> 
  config(mathjax = 'cdn') |>
  add_trace(
  type = "cone",
  x = ym1,
  y = xm1,
  z = zm1,
  u = dy1,
  v = dx1,
  w = dz1,
  sizemode = "absolute",
  sizeref = sizeref,
  showscale = FALSE,
  showlegend = FALSE,
  colorscale = list(c(0, 1), c("red", "red")),
  cmin = 0,
  cmax = 1
) |>
  add_trace(
  type = "cone",
  x = ym2,
  y = xm2,
  z = zm2,
  u = dy2,
  v = dx2,
  w = dz2,
  sizemode = "absolute",
  sizeref = sizeref,
  showscale = FALSE,
  showlegend = FALSE,
  colorscale = list(c(0, 1), c("red", "red")),
  cmin = 0,
  cmax = 1
) |>
  add_trace(
  type = "cone",
  x = ym3,
  y = xm3,
  z = zm3,
  u = dy3,
  v = dx3,
  w = dz3,
  sizemode = "absolute",
  sizeref = sizeref,
  showscale = FALSE,
  showlegend = FALSE,
  colorscale = list(c(0, 1), c("red", "red")),
  cmin = 0,
  cmax = 1
) |>
  plotly::layout(title = list(text = TeX("f"), y = 0.8),
                 font = list(family = "Palatino"),
                 scene = list(xaxis = list(title = list(text = "x", font = list(color = colaxnn)),  tickfont = list(color = colaxnn)),
              yaxis = list(title = list(text = "y", font = list(color = colaxnn)),  tickfont = list(color = colaxnn)),
              zaxis = list(title = list(text = "z", font = list(color = colaxnn)),  tickfont = list(color = colaxnn)),
                   aspectratio = list(x = 2, 
                                 y = 2, 
                                 z = 2),
                   camera = list(eye = list(x = x_eye, 
                                       y = y_eye, 
                                       z = z_eye),
                            center = list(x = 0, 
                                          y = 0, 
                                          z = 0)),
              annotations = list(
                     
             list(
               x = 0, y = 0, z = 0,
               text = TeX("v"),
               textangle = 0, ax = 0, ay = 35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "rgba(0,0,0,0)"),
             list(
               x = e1_fin_y, y = e1_fin_x, z = e1_fin_z,
               text = TeX("v_1"),
               textangle = 0, ax = 0, ay = 35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "rgba(0,0,0,0)"),
             list(
               x = e2_fin_y, y = e2_fin_x, z = e2_fin_z,
               text = TeX("v_2"),
               textangle = 0, ax = 0, ay = 35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "rgba(0,0,0,0)"),
             list(
               x = e3_fin_y, y = e3_fin_x, z = e3_fin_z,
               text = TeX("v_3"),
               textangle = 0, ax = 0, ay = 35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "rgba(0,0,0,0)"),
             list(
               x = ym1, y = xm1, z = zm1,
               text = TeX("e_1"),
               textangle = 0, ax = 0, ay = 35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "rgba(0,0,0,0)"),
             list(
               x = ym2, y = xm2, z = zm2,
               text = TeX("e_2"),
               textangle = 0, ax = 0, ay = 35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "rgba(0,0,0,0)"),
             list(
               x = ym3, y = xm3, z = zm3,
               text = TeX("e_3"),
               textangle = 0, ax = 0, ay = 35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "rgba(0,0,0,0)"))
                 ))

star3df <- graph$plot_function(X = f*0, vertex_size = gsw, line_color = "black", edge_width = gsw, line_width = gsw, type = "plotly") |>
  add_trace(
  type = "cone",
  x = ym1,
  y = xm1,
  z = zm1,
  u = dy1,
  v = dx1,
  w = dz1,
  sizemode = "absolute",
  sizeref = sizeref,
  showscale = FALSE,
  showlegend = FALSE,
  colorscale = list(c(0, 1), c("red", "red")),
  cmin = 0,
  cmax = 1
) |>
  add_trace(
  type = "cone",
  x = ym2,
  y = xm2,
  z = zm2,
  u = dy2,
  v = dx2,
  w = dz2,
  sizemode = "absolute",
  sizeref = sizeref,
  showscale = FALSE,
  showlegend = FALSE,
  colorscale = list(c(0, 1), c("red", "red")),
  cmin = 0,
  cmax = 1
) |>
  add_trace(
  type = "cone",
  x = ym3,
  y = xm3,
  z = zm3,
  u = dy3,
  v = dx3,
  w = dz3,
  sizemode = "absolute",
  sizeref = sizeref,
  showscale = FALSE,
  showlegend = FALSE,
  colorscale = list(c(0, 1), c("red", "red")),
  cmin = 0,
  cmax = 1
) |>
    add_trace(data = DF,
            x = ~y, 
            y = ~x, 
            z = ~z, 
            type = "scatter3d",
            mode = "lines",  
            line = list(color = "rgb(0,0,200)", width = gsw),
            showlegend = FALSE) |>
  add_trace(x = rep(DF$y, each = 3), 
            y = rep(DF$x, each = 3), 
            z = unlist(lapply(DF$z, function(zj) c(0, zj, NA))),
            type = "scatter3d", 
            mode = "lines",
            line = list(color = "gray", width = 0.5),
            showlegend = FALSE) |>
  config(mathjax = 'cdn') |>
  plotly::layout(title = list(text = TeX("f'"), y = 0.8),
                 font = list(family = "Palatino"),
                 scene = list(xaxis = list(title = list(text = "x", font = list(color = colaxnn)),  tickfont = list(color = colaxnn)),
              yaxis = list(title = list(text = "y", font = list(color = colaxnn)),  tickfont = list(color = colaxnn)),
              zaxis = list(title = list(text = "z", font = list(color = colaxnn)),  tickfont = list(color = colaxnn)),
                   aspectratio = list(x = 2, 
                                 y = 2, 
                                 z = 2),
                   camera = list(eye = list(x = x_eye, 
                                       y = y_eye, 
                                       z = z_eye),
                            center = list(x = 0, 
                                          y = 0, 
                                          z = 0)),
              annotations = list(
                     
             list(
               x = 0, y = 0, z = 0,
               text = TeX("v"),
               textangle = 0, ax = 0, ay = 35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "rgba(0,0,0,0)"),
             list(
               x = e1_fin_y, y = e1_fin_x, z = e1_fin_z,
               text = TeX("v_1"),
               textangle = 0, ax = 0, ay = 35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "rgba(0,0,0,0)"),
             list(
               x = e2_fin_y, y = e2_fin_x, z = e2_fin_z,
               text = TeX("v_2"),
               textangle = 0, ax = 0, ay = 35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "rgba(0,0,0,0)"),
             list(
               x = e3_fin_y, y = e3_fin_x, z = e3_fin_z,
               text = TeX("v_3"),
               textangle = 0, ax = 0, ay = 35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "rgba(0,0,0,0)"),
             list(
               x = ym1, y = xm1, z = zm1,
               text = TeX("e_1"),
               textangle = 0, ax = 0, ay = 35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "rgba(0,0,0,0)"),
             list(
               x = ym2, y = xm2, z = zm2,
               text = TeX("e_2"),
               textangle = 0, ax = 0, ay = 35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "rgba(0,0,0,0)"),
             list(
               x = ym3, y = xm3, z = zm3,
               text = TeX("e_3"),
               textangle = 0, ax = 0, ay = 35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "rgba(0,0,0,0)"))
                 ))

star3ddf <- graph$plot_function(X = ddf, vertex_size = gsw, edge_width = gsw, line_width = gsw, continuous = TRUE, interpolate_plot = TRUE, type = "plotly") |> 
  add_trace(
  type = "cone",
  x = ym1,
  y = xm1,
  z = zm1,
  u = dy1,
  v = dx1,
  w = dz1,
  sizemode = "absolute",
  sizeref = sizeref,
  showscale = FALSE,
  showlegend = FALSE,
  colorscale = list(c(0, 1), c("red", "red")),
  cmin = 0,
  cmax = 1
) |>
  add_trace(
  type = "cone",
  x = ym2,
  y = xm2,
  z = zm2,
  u = dy2,
  v = dx2,
  w = dz2,
  sizemode = "absolute",
  sizeref = sizeref,
  showscale = FALSE,
  showlegend = FALSE,
  colorscale = list(c(0, 1), c("red", "red")),
  cmin = 0,
  cmax = 1
) |>
  add_trace(
  type = "cone",
  x = ym3,
  y = xm3,
  z = zm3,
  u = dy3,
  v = dx3,
  w = dz3,
  sizemode = "absolute",
  sizeref = sizeref,
  showscale = FALSE,
  showlegend = FALSE,
  colorscale = list(c(0, 1), c("red", "red")),
  cmin = 0,
  cmax = 1
) |>
  config(mathjax = 'cdn') |>
  plotly::layout(title = list(text = TeX("f''"), y = 0.8),
                 font = list(family = "Palatino"),
                 scene = list(xaxis = list(title = list(text = "x", font = list(color = colaxnn)),  tickfont = list(color = colaxnn)),
              yaxis = list(title = list(text = "y", font = list(color = colaxnn)),  tickfont = list(color = colaxnn)),
              zaxis = list(title = list(text = "z", font = list(color = colaxnn)),  tickfont = list(color = colaxnn)),
                   aspectratio = list(x = 2, 
                                 y = 2, 
                                 z = 2),
                   camera = list(eye = list(x = x_eye, 
                                       y = y_eye, 
                                       z = z_eye),
                            center = list(x = 0, 
                                          y = 0, 
                                          z = 0)),
              annotations = list(
                     
             list(
               x = 0, y = 0, z = 0,
               text = TeX("v"),
               textangle = 0, ax = 0, ay = 35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "rgba(0,0,0,0)"),
             list(
               x = e1_fin_y, y = e1_fin_x, z = e1_fin_z,
               text = TeX("v_1"),
               textangle = 0, ax = 0, ay = 35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "rgba(0,0,0,0)"),
             list(
               x = e2_fin_y, y = e2_fin_x, z = e2_fin_z,
               text = TeX("v_2"),
               textangle = 0, ax = 0, ay = 35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "rgba(0,0,0,0)"),
             list(
               x = e3_fin_y, y = e3_fin_x, z = e3_fin_z,
               text = TeX("v_3"),
               textangle = 0, ax = 0, ay = 35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "rgba(0,0,0,0)"),
             list(
               x = ym1, y = xm1, z = zm1,
               text = TeX("e_1"),
               textangle = 0, ax = 0, ay = 35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "rgba(0,0,0,0)"),
             list(
               x = ym2, y = xm2, z = zm2,
               text = TeX("e_2"),
               textangle = 0, ax = 0, ay = 35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "rgba(0,0,0,0)"),
             list(
               x = ym3, y = xm3, z = zm3,
               text = TeX("e_3"),
               textangle = 0, ax = 0, ay = 35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "rgba(0,0,0,0)"))
                 ))

star3dddf <- graph_disc$plot_function(X = dddf, vertex_size = gsw, edge_width = gsw, line_width = gsw, continuous = FALSE, interpolate_plot = TRUE, type = "plotly") |> 
  add_trace(
  type = "cone",
  x = ym1,
  y = xm1,
  z = zm1,
  u = dy1,
  v = dx1,
  w = dz1,
  sizemode = "absolute",
  sizeref = sizeref,
  showscale = FALSE,
  showlegend = FALSE,
  colorscale = list(c(0, 1), c("red", "red")),
  cmin = 0,
  cmax = 1
) |>
  add_trace(
  type = "cone",
  x = ym2,
  y = xm2,
  z = zm2,
  u = dy2,
  v = dx2,
  w = dz2,
  sizemode = "absolute",
  sizeref = sizeref,
  showscale = FALSE,
  showlegend = FALSE,
  colorscale = list(c(0, 1), c("red", "red")),
  cmin = 0,
  cmax = 1
) |>
  add_trace(
  type = "cone",
  x = ym3,
  y = xm3,
  z = zm3,
  u = dy3,
  v = dx3,
  w = dz3,
  sizemode = "absolute",
  sizeref = sizeref,
  showscale = FALSE,
  showlegend = FALSE,
  colorscale = list(c(0, 1), c("red", "red")),
  cmin = 0,
  cmax = 1
) |>
  config(mathjax = 'cdn') |>
  plotly::layout(title = list(text = TeX("f'''"), y = 0.8),
                 font = list(family = "Palatino"),
                 scene = list(xaxis = list(title = list(text = "x", font = list(color = colaxnn)),  tickfont = list(color = colaxnn)),
              yaxis = list(title = list(text = "y", font = list(color = colaxnn)),  tickfont = list(color = colaxnn)),
              zaxis = list(title = list(text = "z", font = list(color = colaxnn)),  tickfont = list(color = colaxnn)),
                   aspectratio = list(x = 2, 
                                 y = 2, 
                                 z = 2),
                   camera = list(eye = list(x = x_eye, 
                                       y = y_eye, 
                                       z = z_eye),
                            center = list(x = 0, 
                                          y = 0, 
                                          z = 0)),
              annotations = list(
                     
             list(
               x = 0, y = 0, z = 0,
               text = TeX("v"),
               textangle = 0, ax = 0, ay = 35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "rgba(0,0,0,0)"),
             list(
               x = e1_fin_y, y = e1_fin_x, z = e1_fin_z,
               text = TeX("v_1"),
               textangle = 0, ax = 0, ay = 35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "rgba(0,0,0,0)"),
             list(
               x = e2_fin_y, y = e2_fin_x, z = e2_fin_z,
               text = TeX("v_2"),
               textangle = 0, ax = 0, ay = 35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "rgba(0,0,0,0)"),
             list(
               x = e3_fin_y, y = e3_fin_x, z = e3_fin_z,
               text = TeX("v_3"),
               textangle = 0, ax = 0, ay = 35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "rgba(0,0,0,0)"),
             list(
               x = ym1, y = xm1, z = zm1,
               text = TeX("e_1"),
               textangle = 0, ax = 0, ay = 35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "rgba(0,0,0,0)"),
             list(
               x = ym2, y = xm2, z = zm2,
               text = TeX("e_2"),
               textangle = 0, ax = 0, ay = 35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "rgba(0,0,0,0)"),
             list(
               x = ym3, y = xm3, z = zm3,
               text = TeX("e_3"),
               textangle = 0, ax = 0, ay = 35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "rgba(0,0,0,0)"))
                 ))


save(star3f, file = here::here("data_files/star3f.Rdata"))
save(star3df, file = here::here("data_files/star3df.Rdata"))
save(star3ddf, file = here::here("data_files/star3ddf.Rdata"))
save(star3dddf, file = here::here("data_files/star3dddf.Rdata"))

For illustration purposes, Figure 3 and 4 shows \(f''\) and \(f'''\).

load(here::here("data_files/star3f.Rdata"))
star3f

Figure 1: Function \(f = \{f_e\}_{e\in\mathcal{E}}\) given by \(f_{e_i}(s) = a + b_i s - \dfrac{b_i s^3}{3\ell^2}\) for \(i=1,2,3\).

load(here::here("data_files/star3df.Rdata"))
star3df

Figure 2: Function \(f' = \{f'_e\}_{e\in\mathcal{E}}\) given by \(f'_{e_i}(s) = b_i - \dfrac{b_i s^2}{\ell^2}\) for \(i=1,2,3\).

load(here::here("data_files/star3ddf.Rdata"))
star3ddf

Figure 3: Function \(f'' = \{f''_e\}_{e\in\mathcal{E}}\) given by \(f''_{e_i}(s) = -\dfrac{2 b_i s}{\ell^2}\) for \(i=1,2,3\).

load(here::here("data_files/star3dddf.Rdata"))
star3dddf

Figure 4: Function \(f''' = \{f'''_e\}_{e\in\mathcal{E}}\) given by \(f'''_{e_i}(s) = -\dfrac{2 b_i}{\ell^2}\) for \(i=1,2,3\).

2 Example of a function \(g\in C^1(\Gamma)\cap \mathcal{K}(\Gamma)\)

Let us consider the function \(g = \{g_e\}_{e\in\mathcal{E}}\) defined on the \(3\)-star graph with common edge length \(\ell\) via \[ g_{e_1}(s) = \cos\left(\frac{\pi s}{\ell}\right), \quad g_{e_2}(s) = \cos\left(\frac{\pi s}{\ell}\right), \quad g_{e_3}(s) = \cos\left(\frac{\pi s}{\ell}\right). \] Clearly, \(g\) is a continuous function (see Figure 5) on \(\Gamma\) since \[ g_{e_1}(0) = g_{e_2}(0) = g_{e_3}(0) = 1. \] The derivative \(g' = \{g'_e\}_{e\in\mathcal{E}}\) of \(g\) on each edge is given by \[ g_{e_1}'(s) = -\frac{\pi}{\ell}\sin\left(\frac{\pi s}{\ell}\right), \quad g_{e_2}'(s) = -\frac{\pi}{\ell}\sin\left(\frac{\pi s}{\ell}\right), \quad g_{e_3}'(s) = -\frac{\pi}{\ell}\sin\left(\frac{\pi s}{\ell}\right). \] Since \(g_{e_1}'(0) = g_{e_2}'(0) = g_{e_3}'(0) = 0\), \(g\) satisfies the Kirchhoff vertex conditions \[ g_{e_1}'(0) + g_{e_2}'(0) + g_{e_3}'(0) = 0\text{ and }g_{e_1}'(\ell) = g_{e_2}'(\ell) = g_{e_3}'(\ell) = 0. \] Moreover, \(g'\) is a continuous function. See Figure 6.

g1 <- function(s) cos(pi*s/L)
g2 <- function(s) cos(pi*s/L)
g3 <- function(s) cos(pi*s/L)

dg1 <- function(s) -pi/L*sin(pi*s/L)
dg2 <- function(s) -pi/L*sin(pi*s/L)
dg3 <- function(s) -pi/L*sin(pi*s/L)

ddg1 <- function(s) -pi^2/L^2*cos(pi*s/L)
ddg2 <- function(s) -pi^2/L^2*cos(pi*s/L)
ddg3 <- function(s) -pi^2/L^2*cos(pi*s/L)

dddg1 <- function(s) pi^3/L^3*sin(pi*s/L)
dddg2 <- function(s) pi^3/L^3*sin(pi*s/L)
dddg3 <- function(s) pi^3/L^3*sin(pi*s/L)

g_list <- list(g1, g2, g3)
dg_list <- list(dg1, dg2, dg3)
ddg_list <- list(ddg1, ddg2, ddg3)
dddg_list <- list(dddg1, dddg2, dddg3)

g <- apply_edge_functions_fast(graph, g_list)
dg <- apply_edge_functions_fast(graph, dg_list)
ddg <- apply_edge_functions_fast(graph, ddg_list)
dddg <- apply_edge_functions_fast(graph, dddg_list)

star3g <- graph$plot_function(X = g, vertex_size = gsw, edge_width = gsw, line_width = gsw, type = "plotly") |> 
  config(mathjax = 'cdn') |>
  add_trace(
  type = "cone",
  x = ym1,
  y = xm1,
  z = zm1,
  u = dy1,
  v = dx1,
  w = dz1,
  sizemode = "absolute",
  sizeref = sizeref,
  showscale = FALSE,
  showlegend = FALSE,
  colorscale = list(c(0, 1), c("red", "red")),
  cmin = 0,
  cmax = 1
) |>
  add_trace(
  type = "cone",
  x = ym2,
  y = xm2,
  z = zm2,
  u = dy2,
  v = dx2,
  w = dz2,
  sizemode = "absolute",
  sizeref = sizeref,
  showscale = FALSE,
  showlegend = FALSE,
  colorscale = list(c(0, 1), c("red", "red")),
  cmin = 0,
  cmax = 1
) |>
  add_trace(
  type = "cone",
  x = ym3,
  y = xm3,
  z = zm3,
  u = dy3,
  v = dx3,
  w = dz3,
  sizemode = "absolute",
  sizeref = sizeref,
  showscale = FALSE,
  showlegend = FALSE,
  colorscale = list(c(0, 1), c("red", "red")),
  cmin = 0,
  cmax = 1
) |>
  plotly::layout(title = list(text = TeX("g"), y = 0.8), 
                 font = list(family = "Palatino"),
                 scene = list(xaxis = list(title = list(text = "x", font = list(color = colaxnn)),  tickfont = list(color = colaxnn)),
              yaxis = list(title = list(text = "y", font = list(color = colaxnn)),  tickfont = list(color = colaxnn)),
              zaxis = list(title = list(text = "z", font = list(color = colaxnn)),  tickfont = list(color = colaxnn)),
                   aspectratio = list(x = 2, 
                                 y = 2, 
                                 z = 2),
                   camera = list(eye = list(x = x_eye, 
                                       y = y_eye, 
                                       z = z_eye),
                            center = list(x = 0, 
                                          y = 0, 
                                          z = 0)),
              annotations = list(
                     
             list(
               x = 0, y = 0, z = 0,
               text = TeX("v"),
               textangle = 0, ax = 0, ay = 35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "rgba(0,0,0,0)"),
             list(
               x = e1_fin_y, y = e1_fin_x, z = e1_fin_z,
               text = TeX("v_1"),
               textangle = 0, ax = 0, ay = 35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "rgba(0,0,0,0)"),
             list(
               x = e2_fin_y, y = e2_fin_x, z = e2_fin_z,
               text = TeX("v_2"),
               textangle = 0, ax = 0, ay = 35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "rgba(0,0,0,0)"),
             list(
               x = e3_fin_y, y = e3_fin_x, z = e3_fin_z,
               text = TeX("v_3"),
               textangle = 0, ax = 0, ay = 35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "rgba(0,0,0,0)"),
             list(
               x = ym1, y = xm1, z = zm1,
               text = TeX("e_1"),
               textangle = 0, ax = 0, ay = 35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "rgba(0,0,0,0)"),
             list(
               x = ym2, y = xm2, z = zm2,
               text = TeX("e_2"),
               textangle = 0, ax = 0, ay = 35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "rgba(0,0,0,0)"),
             list(
               x = ym3, y = xm3, z = zm3,
               text = TeX("e_3"),
               textangle = 0, ax = 0, ay = 35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "rgba(0,0,0,0)"))
                 ))

star3dg <- graph$plot_function(X = dg, vertex_size = gsw, edge_width = gsw, line_width = gsw, type = "plotly") |> 
  config(mathjax = 'cdn') |>
  add_trace(
  type = "cone",
  x = ym1,
  y = xm1,
  z = zm1,
  u = dy1,
  v = dx1,
  w = dz1,
  sizemode = "absolute",
  sizeref = sizeref,
  showscale = FALSE,
  showlegend = FALSE,
  colorscale = list(c(0, 1), c("red", "red")),
  cmin = 0,
  cmax = 1
) |>
  add_trace(
  type = "cone",
  x = ym2,
  y = xm2,
  z = zm2,
  u = dy2,
  v = dx2,
  w = dz2,
  sizemode = "absolute",
  sizeref = sizeref,
  showscale = FALSE,
  showlegend = FALSE,
  colorscale = list(c(0, 1), c("red", "red")),
  cmin = 0,
  cmax = 1
) |>
  add_trace(
  type = "cone",
  x = ym3,
  y = xm3,
  z = zm3,
  u = dy3,
  v = dx3,
  w = dz3,
  sizemode = "absolute",
  sizeref = sizeref,
  showscale = FALSE,
  showlegend = FALSE,
  colorscale = list(c(0, 1), c("red", "red")),
  cmin = 0,
  cmax = 1
) |>
  plotly::layout(title = list(text = TeX("g'"), y = 0.8),
                 font = list(family = "Palatino"),
                 scene = list(xaxis = list(title = list(text = "x", font = list(color = colaxnn)),  tickfont = list(color = colaxnn)),
              yaxis = list(title = list(text = "y", font = list(color = colaxnn)),  tickfont = list(color = colaxnn)),
              zaxis = list(title = list(text = "z", font = list(color = colaxnn)),  tickfont = list(color = colaxnn)),
                   aspectratio = list(x = 2, 
                                 y = 2, 
                                 z = 2),
                   camera = list(eye = list(x = x_eye, 
                                       y = y_eye, 
                                       z = z_eye),
                            center = list(x = 0, 
                                          y = 0, 
                                          z = 0)),
              annotations = list(
                     
             list(
               x = 0, y = 0, z = 0,
               text = TeX("v"),
               textangle = 0, ax = 0, ay = 35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "rgba(0,0,0,0)"),
             list(
               x = e1_fin_y, y = e1_fin_x, z = e1_fin_z,
               text = TeX("v_1"),
               textangle = 0, ax = 0, ay = 35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "rgba(0,0,0,0)"),
             list(
               x = e2_fin_y, y = e2_fin_x, z = e2_fin_z,
               text = TeX("v_2"),
               textangle = 0, ax = 0, ay = 35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "rgba(0,0,0,0)"),
             list(
               x = e3_fin_y, y = e3_fin_x, z = e3_fin_z,
               text = TeX("v_3"),
               textangle = 0, ax = 0, ay = 35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "rgba(0,0,0,0)"),
             list(
               x = ym1, y = xm1, z = zm1,
               text = TeX("e_1"),
               textangle = 0, ax = 0, ay = 35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "rgba(0,0,0,0)"),
             list(
               x = ym2, y = xm2, z = zm2,
               text = TeX("e_2"),
               textangle = 0, ax = 0, ay = 35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "rgba(0,0,0,0)"),
             list(
               x = ym3, y = xm3, z = zm3,
               text = TeX("e_3"),
               textangle = 0, ax = 0, ay = 35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "rgba(0,0,0,0)"))
                 ))

star3ddg <- graph$plot_function(X = ddg, vertex_size = gsw, edge_width = gsw, line_width = gsw, type = "plotly") |> 
  config(mathjax = 'cdn') |>
  add_trace(
  type = "cone",
  x = ym1,
  y = xm1,
  z = zm1,
  u = dy1,
  v = dx1,
  w = dz1,
  sizemode = "absolute",
  sizeref = sizeref,
  showscale = FALSE,
  showlegend = FALSE,
  colorscale = list(c(0, 1), c("red", "red")),
  cmin = 0,
  cmax = 1
) |>
  add_trace(
  type = "cone",
  x = ym2,
  y = xm2,
  z = zm2,
  u = dy2,
  v = dx2,
  w = dz2,
  sizemode = "absolute",
  sizeref = sizeref,
  showscale = FALSE,
  showlegend = FALSE,
  colorscale = list(c(0, 1), c("red", "red")),
  cmin = 0,
  cmax = 1
) |>
  add_trace(
  type = "cone",
  x = ym3,
  y = xm3,
  z = zm3,
  u = dy3,
  v = dx3,
  w = dz3,
  sizemode = "absolute",
  sizeref = sizeref,
  showscale = FALSE,
  showlegend = FALSE,
  colorscale = list(c(0, 1), c("red", "red")),
  cmin = 0,
  cmax = 1
) |>
  plotly::layout(title = list(text = TeX("g''"), y = 0.8),
                 font = list(family = "Palatino"),
                 scene = list(xaxis = list(title = list(text = "x", font = list(color = colaxnn)),  tickfont = list(color = colaxnn)),
              yaxis = list(title = list(text = "y", font = list(color = colaxnn)),  tickfont = list(color = colaxnn)),
              zaxis = list(title = list(text = "z", font = list(color = colaxnn)),  tickfont = list(color = colaxnn)),
                   aspectratio = list(x = 2, 
                                 y = 2, 
                                 z = 2),
                   camera = list(eye = list(x = x_eye, 
                                       y = y_eye, 
                                       z = z_eye),
                            center = list(x = 0, 
                                          y = 0, 
                                          z = 0)),
              annotations = list(
                     
             list(
               x = 0, y = 0, z = 0,
               text = TeX("v"),
               textangle = 0, ax = 0, ay = 35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "rgba(0,0,0,0)"),
             list(
               x = e1_fin_y, y = e1_fin_x, z = e1_fin_z,
               text = TeX("v_1"),
               textangle = 0, ax = 0, ay = 35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "rgba(0,0,0,0)"),
             list(
               x = e2_fin_y, y = e2_fin_x, z = e2_fin_z,
               text = TeX("v_2"),
               textangle = 0, ax = 0, ay = 35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "rgba(0,0,0,0)"),
             list(
               x = e3_fin_y, y = e3_fin_x, z = e3_fin_z,
               text = TeX("v_3"),
               textangle = 0, ax = 0, ay = 35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "rgba(0,0,0,0)"),
             list(
               x = ym1, y = xm1, z = zm1,
               text = TeX("e_1"),
               textangle = 0, ax = 0, ay = 35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "rgba(0,0,0,0)"),
             list(
               x = ym2, y = xm2, z = zm2,
               text = TeX("e_2"),
               textangle = 0, ax = 0, ay = 35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "rgba(0,0,0,0)"),
             list(
               x = ym3, y = xm3, z = zm3,
               text = TeX("e_3"),
               textangle = 0, ax = 0, ay = 35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "rgba(0,0,0,0)"))
                 ))

star3dddg <- graph$plot_function(X = dddg, vertex_size = gsw, edge_width = gsw, line_width = gsw, type = "plotly") |> 
  config(mathjax = 'cdn') |>
  add_trace(
  type = "cone",
  x = ym1,
  y = xm1,
  z = zm1,
  u = dy1,
  v = dx1,
  w = dz1,
  sizemode = "absolute",
  sizeref = sizeref,
  showscale = FALSE,
  showlegend = FALSE,
  colorscale = list(c(0, 1), c("red", "red")),
  cmin = 0,
  cmax = 1
) |>
  add_trace(
  type = "cone",
  x = ym2,
  y = xm2,
  z = zm2,
  u = dy2,
  v = dx2,
  w = dz2,
  sizemode = "absolute",
  sizeref = sizeref,
  showscale = FALSE,
  showlegend = FALSE,
  colorscale = list(c(0, 1), c("red", "red")),
  cmin = 0,
  cmax = 1
) |>
  add_trace(
  type = "cone",
  x = ym3,
  y = xm3,
  z = zm3,
  u = dy3,
  v = dx3,
  w = dz3,
  sizemode = "absolute",
  sizeref = sizeref,
  showscale = FALSE,
  showlegend = FALSE,
  colorscale = list(c(0, 1), c("red", "red")),
  cmin = 0,
  cmax = 1
) |>
  plotly::layout(title = list(text = TeX("g'''"), y = 0.8),
                 font = list(family = "Palatino"),
                 scene = list(xaxis = list(title = list(text = "x", font = list(color = colaxnn)),  tickfont = list(color = colaxnn)),
              yaxis = list(title = list(text = "y", font = list(color = colaxnn)),  tickfont = list(color = colaxnn)),
              zaxis = list(title = list(text = "z", font = list(color = colaxnn)),  tickfont = list(color = colaxnn)),
                   aspectratio = list(x = 2, 
                                 y = 2, 
                                 z = 2),
                   camera = list(eye = list(x = x_eye, 
                                       y = y_eye, 
                                       z = z_eye),
                            center = list(x = 0, 
                                          y = 0, 
                                          z = 0)),
              annotations = list(
                     
             list(
               x = 0, y = 0, z = 0,
               text = TeX("v"),
               textangle = 0, ax = 0, ay = 35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "rgba(0,0,0,0)"),
             list(
               x = e1_fin_y, y = e1_fin_x, z = e1_fin_z,
               text = TeX("v_1"),
               textangle = 0, ax = 0, ay = 35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "rgba(0,0,0,0)"),
             list(
               x = e2_fin_y, y = e2_fin_x, z = e2_fin_z,
               text = TeX("v_2"),
               textangle = 0, ax = 0, ay = 35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "rgba(0,0,0,0)"),
             list(
               x = e3_fin_y, y = e3_fin_x, z = e3_fin_z,
               text = TeX("v_3"),
               textangle = 0, ax = 0, ay = 35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "rgba(0,0,0,0)"),
             list(
               x = ym1, y = xm1, z = zm1,
               text = TeX("e_1"),
               textangle = 0, ax = 0, ay = 35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "rgba(0,0,0,0)"),
             list(
               x = ym2, y = xm2, z = zm2,
               text = TeX("e_2"),
               textangle = 0, ax = 0, ay = 35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "rgba(0,0,0,0)"),
             list(
               x = ym3, y = xm3, z = zm3,
               text = TeX("e_3"),
               textangle = 0, ax = 0, ay = 35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "rgba(0,0,0,0)"))
                 ))





save(star3g, file = here::here("data_files/star3g.Rdata"))
save(star3dg, file = here::here("data_files/star3dg.Rdata"))
save(star3ddg, file = here::here("data_files/star3ddg.Rdata"))
save(star3dddg, file = here::here("data_files/star3dddg.Rdata"))

dg1 <- function(s) -pi/L*sin(pi*s/L)
dg2 <- function(s) -pi/L*sin(pi*s/L)*0
dg3 <- function(s) -pi/L*sin(pi*s/L)*(-1)

dg_list <- list(dg1, dg2, dg3)
dg <- apply_edge_functions_fast(graph, dg_list)


star3dg_path1 <- graph$plot_function(X = dg, vertex_size = gsw, edge_width = gsw, line_width = gsw, type = "plotly") |> 
  config(mathjax = 'cdn') |>
  add_trace(
  type = "cone",
  x = ym1,
  y = xm1,
  z = zm1,
  u = dy1,
  v = dx1,
  w = dz1,
  sizemode = "absolute",
  sizeref = sizeref,
  showscale = FALSE,
  showlegend = FALSE,
  colorscale = list(c(0, 1), c("green", "green")),
  cmin = 0,
  cmax = 1
) |>
  add_trace(
  type = "cone",
  x = ym2,
  y = xm2,
  z = zm2,
  u = dy2,
  v = dx2,
  w = dz2,
  sizemode = "absolute",
  sizeref = sizeref,
  showscale = FALSE,
  showlegend = FALSE,
  colorscale = list(c(0, 1), c("red", "red")),
  cmin = 0,
  cmax = 1
) |>
  add_trace(
  type = "cone",
  x = ym3,
  y = xm3,
  z = zm3,
  u = -dy3,
  v = -dx3,
  w = -dz3,
  sizemode = "absolute",
  sizeref = sizeref,
  showscale = FALSE,
  showlegend = FALSE,
  colorscale = list(c(0, 1), c("green", "green")),
  cmin = 0,
  cmax = 1
) |>
  add_trace(
    x = c(e2_ini_y, e2_fin_y),
    y = c(e2_ini_x, e2_fin_x),
    z = c(e2_ini_z, e2_fin_z),
    type = "scatter3d",
    mode = "lines",
    line = list(color = "black", width = 7),
    showlegend = FALSE
  ) |>
  add_trace(
    x = c(e3_fin_y, 0, e1_fin_y),
    y = c(e3_fin_x, 0, e1_fin_x),
    z = c(e3_fin_z, 0, e1_fin_z),
    type = "scatter3d",
    mode = "lines",
    line = list(color = "green", width = 7),
    showlegend = FALSE
  ) |>
  plotly::layout(title = list(text = TeX("(g|_{p_{\\hat{e}_3e_1}})'"), y = 0.8),
                 font = list(family = "Palatino"),
                 scene = list(xaxis = list(title = list(text = "x", font = list(color = colaxnn)),  tickfont = list(color = colaxnn)),
              yaxis = list(title = list(text = "y", font = list(color = colaxnn)),  tickfont = list(color = colaxnn)),
              zaxis = list(title = list(text = "z", font = list(color = colaxnn)),  tickfont = list(color = colaxnn)),
                   aspectratio = list(x = 2, 
                                 y = 2, 
                                 z = 2),
                   camera = list(eye = list(x = x_eye, 
                                       y = y_eye, 
                                       z = z_eye),
                            center = list(x = 0, 
                                          y = 0, 
                                          z = 0)),
              annotations = list(
                     
             list(
               x = 0, y = 0, z = 0,
               text = TeX("v"),
               textangle = 0, ax = 0, ay = 35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "rgba(0,0,0,0)"),
             list(
               x = e1_fin_y, y = e1_fin_x, z = e1_fin_z,
               text = TeX("v_1"),
               textangle = 0, ax = 0, ay = 35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "rgba(0,0,0,0)"),
             list(
               x = e2_fin_y, y = e2_fin_x, z = e2_fin_z,
               text = TeX("v_2"),
               textangle = 0, ax = 0, ay = 35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "rgba(0,0,0,0)"),
             list(
               x = e3_fin_y, y = e3_fin_x, z = e3_fin_z,
               text = TeX("v_3"),
               textangle = 0, ax = 0, ay = 35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "rgba(0,0,0,0)"),
             list(
               x = ym1, y = xm1, z = zm1,
               text = TeX("e_1"),
               textangle = 0, ax = 0, ay = 35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "rgba(0,0,0,0)"),
             list(
               x = ym2, y = xm2, z = zm2,
               text = TeX("e_2"),
               textangle = 0, ax = 0, ay = 35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "rgba(0,0,0,0)"),
             list(
               x = ym3, y = xm3, z = zm3,
               text = TeX("\\hat{e}_3"),
               textangle = 0, ax = 0, ay = 35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "rgba(0,0,0,0)"))
                 ))



dg1 <- function(s) -pi/L*sin(pi*s/L)*0
dg2 <- function(s) -pi/L*sin(pi*s/L)
dg3 <- function(s) -pi/L*sin(pi*s/L)*(-1)

dg_list <- list(dg1, dg2, dg3)
dg <- apply_edge_functions_fast(graph, dg_list)


star3dg_path2 <- graph$plot_function(X = dg, vertex_size = gsw, edge_width = gsw, line_width = gsw, type = "plotly") |> 
  config(mathjax = 'cdn') |>
  add_trace(
  type = "cone",
  x = ym1,
  y = xm1,
  z = zm1,
  u = dy1,
  v = dx1,
  w = dz1,
  sizemode = "absolute",
  sizeref = sizeref,
  showscale = FALSE,
  showlegend = FALSE,
  colorscale = list(c(0, 1), c("red", "red")),
  cmin = 0,
  cmax = 1
) |>
  add_trace(
  type = "cone",
  x = ym2,
  y = xm2,
  z = zm2,
  u = dy2,
  v = dx2,
  w = dz2,
  sizemode = "absolute",
  sizeref = sizeref,
  showscale = FALSE,
  showlegend = FALSE,
  colorscale = list(c(0, 1), c("green", "green")),
  cmin = 0,
  cmax = 1
) |>
  add_trace(
  type = "cone",
  x = ym3,
  y = xm3,
  z = zm3,
  u = -dy3,
  v = -dx3,
  w = -dz3,
  sizemode = "absolute",
  sizeref = sizeref,
  showscale = FALSE,
  showlegend = FALSE,
  colorscale = list(c(0, 1), c("green", "green")),
  cmin = 0,
  cmax = 1
) |>
  add_trace(
    x = c(e1_ini_y, e1_fin_y),
    y = c(e1_ini_x, e1_fin_x),
    z = c(e1_ini_z, e1_fin_z),
    type = "scatter3d",
    mode = "lines",
    line = list(color = "black", width = 7),
    showlegend = FALSE
  ) |>
  add_trace(
    x = c(e3_fin_y, 0, e2_fin_y),
    y = c(e3_fin_x, 0, e2_fin_x),
    z = c(e3_fin_z, 0, e2_fin_z),
    type = "scatter3d",
    mode = "lines",
    line = list(color = "green", width = 7),
    showlegend = FALSE
  ) |>
  plotly::layout(title = list(text = TeX("(g|_{p_{\\hat{e}_3e_2}})'"), y = 0.8),
                 font = list(family = "Palatino"),
                 scene = list(xaxis = list(title = list(text = "x", font = list(color = colaxnn)),  tickfont = list(color = colaxnn)),
              yaxis = list(title = list(text = "y", font = list(color = colaxnn)),  tickfont = list(color = colaxnn)),
              zaxis = list(title = list(text = "z", font = list(color = colaxnn)),  tickfont = list(color = colaxnn)),
                   aspectratio = list(x = 2, 
                                 y = 2, 
                                 z = 2),
                   camera = list(eye = list(x = x_eye, 
                                       y = y_eye, 
                                       z = z_eye),
                            center = list(x = 0, 
                                          y = 0, 
                                          z = 0)),
              annotations = list(
                     
             list(
               x = 0, y = 0, z = 0,
               text = TeX("v"),
               textangle = 0, ax = 0, ay = 35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "rgba(0,0,0,0)"),
             list(
               x = e1_fin_y, y = e1_fin_x, z = e1_fin_z,
               text = TeX("v_1"),
               textangle = 0, ax = 0, ay = 35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "rgba(0,0,0,0)"),
             list(
               x = e2_fin_y, y = e2_fin_x, z = e2_fin_z,
               text = TeX("v_2"),
               textangle = 0, ax = 0, ay = 35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "rgba(0,0,0,0)"),
             list(
               x = e3_fin_y, y = e3_fin_x, z = e3_fin_z,
               text = TeX("v_3"),
               textangle = 0, ax = 0, ay = 35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "rgba(0,0,0,0)"),
             list(
               x = ym1, y = xm1, z = zm1,
               text = TeX("e_1"),
               textangle = 0, ax = 0, ay = 35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "rgba(0,0,0,0)"),
             list(
               x = ym2, y = xm2, z = zm2,
               text = TeX("e_2"),
               textangle = 0, ax = 0, ay = 35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "rgba(0,0,0,0)"),
             list(
               x = ym3, y = xm3, z = zm3,
               text = TeX("\\hat{e}_3"),
               textangle = 0, ax = 0, ay = 35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "rgba(0,0,0,0)"))
                 ))



dg1 <- function(s) -pi/L*sin(pi*s/L)
dg2 <- function(s) -pi/L*sin(pi*s/L)*(-1)
dg3 <- function(s) -pi/L*sin(pi*s/L)*0

dg_list <- list(dg1, dg2, dg3)
dg <- apply_edge_functions_fast(graph, dg_list)


star3dg_path3 <- graph$plot_function(X = dg, vertex_size = gsw, edge_width = gsw, line_width = gsw, type = "plotly") |> 
  config(mathjax = 'cdn') |>
  add_trace(
  type = "cone",
  x = ym1,
  y = xm1,
  z = zm1,
  u = dy1,
  v = dx1,
  w = dz1,
  sizemode = "absolute",
  sizeref = sizeref,
  showscale = FALSE,
  showlegend = FALSE,
  colorscale = list(c(0, 1), c("green", "green")),
  cmin = 0,
  cmax = 1
) |>
  add_trace(
  type = "cone",
  x = ym2,
  y = xm2,
  z = zm2,
  u = -dy2,
  v = -dx2,
  w = -dz2,
  sizemode = "absolute",
  sizeref = sizeref,
  showscale = FALSE,
  showlegend = FALSE,
  colorscale = list(c(0, 1), c("green", "green")),
  cmin = 0,
  cmax = 1
) |>
  add_trace(
  type = "cone",
  x = ym3,
  y = xm3,
  z = zm3,
  u = dy3,
  v = dx3,
  w = dz3,
  sizemode = "absolute",
  sizeref = sizeref,
  showscale = FALSE,
  showlegend = FALSE,
  colorscale = list(c(0, 1), c("red", "red")),
  cmin = 0,
  cmax = 1
) |>
  add_trace(
    x = c(e3_ini_y, e3_fin_y),
    y = c(e3_ini_x, e3_fin_x),
    z = c(e3_ini_z, e3_fin_z),
    type = "scatter3d",
    mode = "lines",
    line = list(color = "black", width = 7),
    showlegend = FALSE
  ) |>
  add_trace(
    x = c(e1_fin_y, 0, e2_fin_y),
    y = c(e1_fin_x, 0, e2_fin_x),
    z = c(e1_fin_z, 0, e2_fin_z),
    type = "scatter3d",
    mode = "lines",
    line = list(color = "green", width = 7),
    showlegend = FALSE
  ) |>
  plotly::layout(title = list(text = TeX("(g|_{p_{\\hat{e}_2e_1}})'"), y = 0.8),
                 font = list(family = "Palatino"),
                 scene = list(xaxis = list(title = list(text = "x", font = list(color = colaxnn)),  tickfont = list(color = colaxnn)),
              yaxis = list(title = list(text = "y", font = list(color = colaxnn)),  tickfont = list(color = colaxnn)),
              zaxis = list(title = list(text = "z", font = list(color = colaxnn)),  tickfont = list(color = colaxnn)),
                   aspectratio = list(x = 2, 
                                 y = 2, 
                                 z = 2),
                   camera = list(eye = list(x = x_eye, 
                                       y = y_eye, 
                                       z = z_eye),
                            center = list(x = 0, 
                                          y = 0, 
                                          z = 0)),
              annotations = list(
                     
             list(
               x = 0, y = 0, z = 0,
               text = TeX("v"),
               textangle = 0, ax = 0, ay = 35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "rgba(0,0,0,0)"),
             list(
               x = e1_fin_y, y = e1_fin_x, z = e1_fin_z,
               text = TeX("v_1"),
               textangle = 0, ax = 0, ay = 35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "rgba(0,0,0,0)"),
             list(
               x = e2_fin_y, y = e2_fin_x, z = e2_fin_z,
               text = TeX("v_2"),
               textangle = 0, ax = 0, ay = 35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "rgba(0,0,0,0)"),
             list(
               x = e3_fin_y, y = e3_fin_x, z = e3_fin_z,
               text = TeX("v_3"),
               textangle = 0, ax = 0, ay = 35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "rgba(0,0,0,0)"),
             list(
               x = ym1, y = xm1, z = zm1,
               text = TeX("e_1"),
               textangle = 0, ax = 0, ay = 35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "rgba(0,0,0,0)"),
             list(
               x = ym2, y = xm2, z = zm2,
               text = TeX("\\hat{e}_2"),
               textangle = 0, ax = 0, ay = 35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "rgba(0,0,0,0)"),
             list(
               x = ym3, y = xm3, z = zm3,
               text = TeX("e_3"),
               textangle = 0, ax = 0, ay = 35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "rgba(0,0,0,0)"))
                 ))











dg1 <- function(s) -pi/L*sin(pi*s/L)
dg2 <- function(s) -pi/L*sin(pi*s/L)
dg3 <- function(s) -pi/L*sin(pi*s/L)*(-1)

dg_list <- list(dg1, dg2, dg3)
dg <- apply_edge_functions_fast(graph, dg_list)


star3dg_path_ok <- graph$plot_function(X = dg, vertex_size = gsw, edge_width = gsw, line_width = gsw, type = "plotly") |> 
  config(mathjax = 'cdn') |>
  add_trace(
  type = "cone",
  x = ym1,
  y = xm1,
  z = zm1,
  u = dy1,
  v = dx1,
  w = dz1,
  sizemode = "absolute",
  sizeref = sizeref,
  showscale = FALSE,
  showlegend = FALSE,
  colorscale = list(c(0, 1), c("red", "red")),
  cmin = 0,
  cmax = 1
) |>
  add_trace(
  type = "cone",
  x = ym2,
  y = xm2,
  z = zm2,
  u = dy2,
  v = dx2,
  w = dz2,
  sizemode = "absolute",
  sizeref = sizeref,
  showscale = FALSE,
  showlegend = FALSE,
  colorscale = list(c(0, 1), c("red", "red")),
  cmin = 0,
  cmax = 1
) |>
  add_trace(
  type = "cone",
  x = ym3,
  y = xm3,
  z = zm3,
  u = -dy3,
  v = -dx3,
  w = -dz3,
  sizemode = "absolute",
  sizeref = sizeref,
  showscale = FALSE,
  showlegend = FALSE,
  colorscale = list(c(0, 1), c("green", "green")),
  cmin = 0,
  cmax = 1
) |>
  plotly::layout(title = list(text = TeX("g'"), y = 0.8),
                 font = list(family = "Palatino"),
                 scene = list(xaxis = list(title = list(text = "x", font = list(color = colaxnn)),  tickfont = list(color = colaxnn)),
              yaxis = list(title = list(text = "y", font = list(color = colaxnn)),  tickfont = list(color = colaxnn)),
              zaxis = list(title = list(text = "z", font = list(color = colaxnn)),  tickfont = list(color = colaxnn)),
                   aspectratio = list(x = 2, 
                                 y = 2, 
                                 z = 2),
                   camera = list(eye = list(x = x_eye, 
                                       y = y_eye, 
                                       z = z_eye),
                            center = list(x = 0, 
                                          y = 0, 
                                          z = 0)),
              annotations = list(
                     
             list(
               x = 0, y = 0, z = 0,
               text = TeX("v"),
               textangle = 0, ax = 0, ay = 35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "rgba(0,0,0,0)"),
             list(
               x = e1_fin_y, y = e1_fin_x, z = e1_fin_z,
               text = TeX("v_1"),
               textangle = 0, ax = 0, ay = 35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "rgba(0,0,0,0)"),
             list(
               x = e2_fin_y, y = e2_fin_x, z = e2_fin_z,
               text = TeX("v_2"),
               textangle = 0, ax = 0, ay = 35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "rgba(0,0,0,0)"),
             list(
               x = e3_fin_y, y = e3_fin_x, z = e3_fin_z,
               text = TeX("v_3"),
               textangle = 0, ax = 0, ay = 35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "rgba(0,0,0,0)"),
             list(
               x = ym1, y = xm1, z = zm1,
               text = TeX("e_1"),
               textangle = 0, ax = 0, ay = 35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "rgba(0,0,0,0)"),
             list(
               x = ym2, y = xm2, z = zm2,
               text = TeX("e_2"),
               textangle = 0, ax = 0, ay = 35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "rgba(0,0,0,0)"),
             list(
               x = ym3, y = xm3, z = zm3,
               text = TeX("\\hat{e}_3"),
               textangle = 0, ax = 0, ay = 35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "rgba(0,0,0,0)"))
                 ))



save(star3dg_path1, file = here::here("data_files/star3dg_path1.Rdata"))
save(star3dg_path2, file = here::here("data_files/star3dg_path2.Rdata"))
save(star3dg_path3, file = here::here("data_files/star3dg_path3.Rdata"))
save(star3dg_path_ok, file = here::here("data_files/star3dg_path_ok.Rdata"))

ddg1 <- function(s) -pi^2/L^2*cos(pi*s/L)
ddg2 <- function(s) -pi^2/L^2*cos(pi*s/L)*0
ddg3 <- function(s) -pi^2/L^2*cos(pi*s/L)*(-1)



ddg_list <- list(ddg1, ddg2, ddg3)
ddg <- apply_edge_functions_fast(graph, ddg_list)

l_mesh <- seq(0, my_length, by = h)
dfl1 <- ddg1(l_mesh)
dfl2 <- ddg2(l_mesh)
dfl3 <- ddg3(l_mesh)

l_mesh_norm <- l_mesh/my_length
PtE1 <- cbind(rep(1, length(l_mesh)), l_mesh_norm)
PtE2 <- cbind(rep(2, length(l_mesh)), l_mesh_norm)
PtE3 <- cbind(rep(3, length(l_mesh)), l_mesh_norm)
XY1 <- graph$coordinates(PtE1)
XY2 <- graph$coordinates(PtE2)
XY3 <- graph$coordinates(PtE3)
DF1 <- data.frame(x = XY1[,1], y = XY1[,2], z = dfl1)
DF2 <- data.frame(x = XY2[,1], y = XY2[,2], z = dfl2)
DF3 <- data.frame(x = XY3[,1], y = XY3[,2], z = dfl3)

DF <- rbind(DF1, rep(NA, 3), DF2, rep(NA, 3), DF3)
DF_NO_NA <- rbind(DF1, DF2, DF3)



star3ddg_path1 <- graph$plot_function(X = ddg*0, vertex_size = gsw, line_color = "black", edge_width = gsw, line_width = gsw, type = "plotly") |>
  config(mathjax = 'cdn') |>
  add_trace(
  type = "cone",
  x = ym1,
  y = xm1,
  z = zm1,
  u = dy1,
  v = dx1,
  w = dz1,
  sizemode = "absolute",
  sizeref = sizeref,
  showscale = FALSE,
  showlegend = FALSE,
  colorscale = list(c(0, 1), c("green", "green")),
  cmin = 0,
  cmax = 1
) |>
  add_trace(
  type = "cone",
  x = ym2,
  y = xm2,
  z = zm2,
  u = dy2,
  v = dx2,
  w = dz2,
  sizemode = "absolute",
  sizeref = sizeref,
  showscale = FALSE,
  showlegend = FALSE,
  colorscale = list(c(0, 1), c("red", "red")),
  cmin = 0,
  cmax = 1
) |>
  add_trace(
  type = "cone",
  x = ym3,
  y = xm3,
  z = zm3,
  u = -dy3,
  v = -dx3,
  w = -dz3,
  sizemode = "absolute",
  sizeref = sizeref,
  showscale = FALSE,
  showlegend = FALSE,
  colorscale = list(c(0, 1), c("green", "green")),
  cmin = 0,
  cmax = 1
) |>
  add_trace(data = DF,
            x = ~y, 
            y = ~x, 
            z = ~z, 
            type = "scatter3d",
            mode = "lines",  
            line = list(color = "rgb(0,0,200)", width = gsw),
            showlegend = FALSE) |>
  add_trace(x = rep(DF$y, each = 3), 
            y = rep(DF$x, each = 3), 
            z = unlist(lapply(DF$z, function(zj) c(0, zj, NA))),
            type = "scatter3d", 
            mode = "lines",
            line = list(color = "gray", width = 0.5),
            showlegend = FALSE) |>
  add_trace(
    x = c(e2_ini_y, e2_fin_y),
    y = c(e2_ini_x, e2_fin_x),
    z = c(e2_ini_z, e2_fin_z),
    type = "scatter3d",
    mode = "lines",
    line = list(color = "black", width = 7),
    showlegend = FALSE
  ) |>
  add_trace(
    x = c(e3_fin_y, 0, e1_fin_y),
    y = c(e3_fin_x, 0, e1_fin_x),
    z = c(e3_fin_z, 0, e1_fin_z),
    type = "scatter3d",
    mode = "lines",
    line = list(color = "green", width = 7),
    showlegend = FALSE
  ) |>
  plotly::layout(title = list(text = TeX("(g'|_{p_{\\hat{e}_3e_1}})'"), y = 0.8),
                 font = list(family = "Palatino"),
                 scene = list(xaxis = list(title = list(text = "x", font = list(color = colaxnn)),  tickfont = list(color = colaxnn)),
              yaxis = list(title = list(text = "y", font = list(color = colaxnn)),  tickfont = list(color = colaxnn)),
              zaxis = list(title = list(text = "z", font = list(color = colaxnn)),  tickfont = list(color = colaxnn)),
                   aspectratio = list(x = 2, 
                                 y = 2, 
                                 z = 2),
                   camera = list(eye = list(x = x_eye, 
                                       y = y_eye, 
                                       z = z_eye),
                            center = list(x = 0, 
                                          y = 0, 
                                          z = 0)),
              annotations = list(
                     
             list(
               x = 0, y = 0, z = 0,
               text = TeX("v"),
               textangle = 0, ax = 0, ay = 35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "rgba(0,0,0,0)"),
             list(
               x = e1_fin_y, y = e1_fin_x, z = e1_fin_z,
               text = TeX("v_1"),
               textangle = 0, ax = 0, ay = 35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "rgba(0,0,0,0)"),
             list(
               x = e2_fin_y, y = e2_fin_x, z = e2_fin_z,
               text = TeX("v_2"),
               textangle = 0, ax = 0, ay = 35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "rgba(0,0,0,0)"),
             list(
               x = e3_fin_y, y = e3_fin_x, z = e3_fin_z,
               text = TeX("v_3"),
               textangle = 0, ax = 0, ay = 35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "rgba(0,0,0,0)"),
             list(
               x = ym1, y = xm1, z = zm1,
               text = TeX("e_1"),
               textangle = 0, ax = 0, ay = 35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "rgba(0,0,0,0)"),
             list(
               x = ym2, y = xm2, z = zm2,
               text = TeX("e_2"),
               textangle = 0, ax = 0, ay = 35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "rgba(0,0,0,0)"),
             list(
               x = ym3, y = xm3, z = zm3,
               text = TeX("\\hat{e}_3"),
               textangle = 0, ax = 0, ay = 35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "rgba(0,0,0,0)"))
                 ))


ddg1 <- function(s) -pi^2/L^2*cos(pi*s/L)*0
ddg2 <- function(s) -pi^2/L^2*cos(pi*s/L)
ddg3 <- function(s) -pi^2/L^2*cos(pi*s/L)*(-1)



ddg_list <- list(ddg1, ddg2, ddg3)
ddg <- apply_edge_functions_fast(graph, ddg_list)


l_mesh <- seq(0, my_length, by = h)
dfl1 <- ddg1(l_mesh)
dfl2 <- ddg2(l_mesh)
dfl3 <- ddg3(l_mesh)

l_mesh_norm <- l_mesh/my_length
PtE1 <- cbind(rep(1, length(l_mesh)), l_mesh_norm)
PtE2 <- cbind(rep(2, length(l_mesh)), l_mesh_norm)
PtE3 <- cbind(rep(3, length(l_mesh)), l_mesh_norm)
XY1 <- graph$coordinates(PtE1)
XY2 <- graph$coordinates(PtE2)
XY3 <- graph$coordinates(PtE3)
DF1 <- data.frame(x = XY1[,1], y = XY1[,2], z = dfl1)
DF2 <- data.frame(x = XY2[,1], y = XY2[,2], z = dfl2)
DF3 <- data.frame(x = XY3[,1], y = XY3[,2], z = dfl3)

DF <- rbind(DF1, rep(NA, 3), DF2, rep(NA, 3), DF3)
DF_NO_NA <- rbind(DF1, DF2, DF3)



star3ddg_path2 <- graph$plot_function(X = ddg*0, vertex_size = gsw, line_color = "black", edge_width = gsw, line_width = gsw, type = "plotly") |>
  config(mathjax = 'cdn') |>
  add_trace(
  type = "cone",
  x = ym1,
  y = xm1,
  z = zm1,
  u = dy1,
  v = dx1,
  w = dz1,
  sizemode = "absolute",
  sizeref = sizeref,
  showscale = FALSE,
  showlegend = FALSE,
  colorscale = list(c(0, 1), c("red", "red")),
  cmin = 0,
  cmax = 1
) |>
  add_trace(
  type = "cone",
  x = ym2,
  y = xm2,
  z = zm2,
  u = dy2,
  v = dx2,
  w = dz2,
  sizemode = "absolute",
  sizeref = sizeref,
  showscale = FALSE,
  showlegend = FALSE,
  colorscale = list(c(0, 1), c("green", "green")),
  cmin = 0,
  cmax = 1
) |>
  add_trace(
  type = "cone",
  x = ym3,
  y = xm3,
  z = zm3,
  u = -dy3,
  v = -dx3,
  w = -dz3,
  sizemode = "absolute",
  sizeref = sizeref,
  showscale = FALSE,
  showlegend = FALSE,
  colorscale = list(c(0, 1), c("green", "green")),
  cmin = 0,
  cmax = 1
) |>
  add_trace(data = DF,
            x = ~y, 
            y = ~x, 
            z = ~z, 
            type = "scatter3d",
            mode = "lines",  
            line = list(color = "rgb(0,0,200)", width = gsw),
            showlegend = FALSE) |>
  add_trace(x = rep(DF$y, each = 3), 
            y = rep(DF$x, each = 3), 
            z = unlist(lapply(DF$z, function(zj) c(0, zj, NA))),
            type = "scatter3d", 
            mode = "lines",
            line = list(color = "gray", width = 0.5),
            showlegend = FALSE) |>
  add_trace(
    x = c(e1_ini_y, e1_fin_y),
    y = c(e1_ini_x, e1_fin_x),
    z = c(e1_ini_z, e1_fin_z),
    type = "scatter3d",
    mode = "lines",
    line = list(color = "black", width = 7),
    showlegend = FALSE
  ) |>
  add_trace(
    x = c(e3_fin_y, 0, e2_fin_y),
    y = c(e3_fin_x, 0, e2_fin_x),
    z = c(e3_fin_z, 0, e2_fin_z),
    type = "scatter3d",
    mode = "lines",
    line = list(color = "green", width = 7),
    showlegend = FALSE
  ) |>
  plotly::layout(title = list(text = TeX("(g'|_{p_{\\hat{e}_3e_2}})'"), y = 0.8),
                 font = list(family = "Palatino"),
                 scene = list(xaxis = list(title = list(text = "x", font = list(color = colaxnn)),  tickfont = list(color = colaxnn)),
              yaxis = list(title = list(text = "y", font = list(color = colaxnn)),  tickfont = list(color = colaxnn)),
              zaxis = list(title = list(text = "z", font = list(color = colaxnn)),  tickfont = list(color = colaxnn)),
                   aspectratio = list(x = 2, 
                                 y = 2, 
                                 z = 2),
                   camera = list(eye = list(x = x_eye, 
                                       y = y_eye, 
                                       z = z_eye),
                            center = list(x = 0, 
                                          y = 0, 
                                          z = 0)),
              annotations = list(
                     
             list(
               x = 0, y = 0, z = 0,
               text = TeX("v"),
               textangle = 0, ax = 0, ay = 35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "rgba(0,0,0,0)"),
             list(
               x = e1_fin_y, y = e1_fin_x, z = e1_fin_z,
               text = TeX("v_1"),
               textangle = 0, ax = 0, ay = 35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "rgba(0,0,0,0)"),
             list(
               x = e2_fin_y, y = e2_fin_x, z = e2_fin_z,
               text = TeX("v_2"),
               textangle = 0, ax = 0, ay = 35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "rgba(0,0,0,0)"),
             list(
               x = e3_fin_y, y = e3_fin_x, z = e3_fin_z,
               text = TeX("v_3"),
               textangle = 0, ax = 0, ay = 35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "rgba(0,0,0,0)"),
             list(
               x = ym1, y = xm1, z = zm1,
               text = TeX("e_1"),
               textangle = 0, ax = 0, ay = 35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "rgba(0,0,0,0)"),
             list(
               x = ym2, y = xm2, z = zm2,
               text = TeX("e_2"),
               textangle = 0, ax = 0, ay = 35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "rgba(0,0,0,0)"),
             list(
               x = ym3, y = xm3, z = zm3,
               text = TeX("\\hat{e}_3"),
               textangle = 0, ax = 0, ay = 35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "rgba(0,0,0,0)"))
                 ))

ddg1 <- function(s) -pi^2/L^2*cos(pi*s/L)
ddg2 <- function(s) -pi^2/L^2*cos(pi*s/L)*(-1)
ddg3 <- function(s) -pi^2/L^2*cos(pi*s/L)*0



ddg_list <- list(ddg1, ddg2, ddg3)
ddg <- apply_edge_functions_fast(graph, ddg_list)


l_mesh <- seq(0, my_length, by = h)
dfl1 <- ddg1(l_mesh)
dfl2 <- ddg2(l_mesh)
dfl3 <- ddg3(l_mesh)

l_mesh_norm <- l_mesh/my_length
PtE1 <- cbind(rep(1, length(l_mesh)), l_mesh_norm)
PtE2 <- cbind(rep(2, length(l_mesh)), l_mesh_norm)
PtE3 <- cbind(rep(3, length(l_mesh)), l_mesh_norm)
XY1 <- graph$coordinates(PtE1)
XY2 <- graph$coordinates(PtE2)
XY3 <- graph$coordinates(PtE3)
DF1 <- data.frame(x = XY1[,1], y = XY1[,2], z = dfl1)
DF2 <- data.frame(x = XY2[,1], y = XY2[,2], z = dfl2)
DF3 <- data.frame(x = XY3[,1], y = XY3[,2], z = dfl3)

DF <- rbind(DF1, rep(NA, 3), DF2, rep(NA, 3), DF3)
DF_NO_NA <- rbind(DF1, DF2, DF3)



star3ddg_path3 <- graph$plot_function(X = ddg*0, vertex_size = gsw, line_color = "black", edge_width = gsw, line_width = gsw, type = "plotly") |>
  config(mathjax = 'cdn') |>
  add_trace(
  type = "cone",
  x = ym1,
  y = xm1,
  z = zm1,
  u = dy1,
  v = dx1,
  w = dz1,
  sizemode = "absolute",
  sizeref = sizeref,
  showscale = FALSE,
  showlegend = FALSE,
  colorscale = list(c(0, 1), c("green", "green")),
  cmin = 0,
  cmax = 1
) |>
  add_trace(
  type = "cone",
  x = ym2,
  y = xm2,
  z = zm2,
  u = -dy2,
  v = -dx2,
  w = -dz2,
  sizemode = "absolute",
  sizeref = sizeref,
  showscale = FALSE,
  showlegend = FALSE,
  colorscale = list(c(0, 1), c("green", "green")),
  cmin = 0,
  cmax = 1
) |>
  add_trace(
  type = "cone",
  x = ym3,
  y = xm3,
  z = zm3,
  u = dy3,
  v = dx3,
  w = dz3,
  sizemode = "absolute",
  sizeref = sizeref,
  showscale = FALSE,
  showlegend = FALSE,
  colorscale = list(c(0, 1), c("red", "red")),
  cmin = 0,
  cmax = 1
) |>
  add_trace(data = DF,
            x = ~y, 
            y = ~x, 
            z = ~z, 
            type = "scatter3d",
            mode = "lines",  
            line = list(color = "rgb(0,0,200)", width = gsw),
            showlegend = FALSE) |>
  add_trace(x = rep(DF$y, each = 3), 
            y = rep(DF$x, each = 3), 
            z = unlist(lapply(DF$z, function(zj) c(0, zj, NA))),
            type = "scatter3d", 
            mode = "lines",
            line = list(color = "gray", width = 0.5),
            showlegend = FALSE) |>
  add_trace(
    x = c(e3_ini_y, e3_fin_y),
    y = c(e3_ini_x, e3_fin_x),
    z = c(e3_ini_z, e3_fin_z),
    type = "scatter3d",
    mode = "lines",
    line = list(color = "black", width = 7),
    showlegend = FALSE
  ) |>
  add_trace(
    x = c(e1_fin_y, 0, e2_fin_y),
    y = c(e1_fin_x, 0, e2_fin_x),
    z = c(e1_fin_z, 0, e2_fin_z),
    type = "scatter3d",
    mode = "lines",
    line = list(color = "green", width = 7),
    showlegend = FALSE
  ) |>
  plotly::layout(title = list(text = TeX("(g'|_{p_{\\hat{e}_2e_1}})'"), y = 0.8),
                 font = list(family = "Palatino"),
                 scene = list(xaxis = list(title = list(text = "x", font = list(color = colaxnn)),  tickfont = list(color = colaxnn)),
              yaxis = list(title = list(text = "y", font = list(color = colaxnn)),  tickfont = list(color = colaxnn)),
              zaxis = list(title = list(text = "z", font = list(color = colaxnn)),  tickfont = list(color = colaxnn)),
                   aspectratio = list(x = 2, 
                                 y = 2, 
                                 z = 2),
                   camera = list(eye = list(x = x_eye, 
                                       y = y_eye, 
                                       z = z_eye),
                            center = list(x = 0, 
                                          y = 0, 
                                          z = 0)),
              annotations = list(
                     
             list(
               x = 0, y = 0, z = 0,
               text = TeX("v"),
               textangle = 0, ax = 0, ay = 35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "rgba(0,0,0,0)"),
             list(
               x = e1_fin_y, y = e1_fin_x, z = e1_fin_z,
               text = TeX("v_1"),
               textangle = 0, ax = 0, ay = 35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "rgba(0,0,0,0)"),
             list(
               x = e2_fin_y, y = e2_fin_x, z = e2_fin_z,
               text = TeX("v_2"),
               textangle = 0, ax = 0, ay = 35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "rgba(0,0,0,0)"),
             list(
               x = e3_fin_y, y = e3_fin_x, z = e3_fin_z,
               text = TeX("v_3"),
               textangle = 0, ax = 0, ay = 35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "rgba(0,0,0,0)"),
             list(
               x = ym1, y = xm1, z = zm1,
               text = TeX("e_1"),
               textangle = 0, ax = 0, ay = 35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "rgba(0,0,0,0)"),
             list(
               x = ym2, y = xm2, z = zm2,
               text = TeX("\\hat{e}_2"),
               textangle = 0, ax = 0, ay = 35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "rgba(0,0,0,0)"),
             list(
               x = ym3, y = xm3, z = zm3,
               text = TeX("e_3"),
               textangle = 0, ax = 0, ay = 35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "rgba(0,0,0,0)"))
                 ))


save(star3ddg_path1, file = here::here("data_files/star3ddg_path1.Rdata"))
save(star3ddg_path2, file = here::here("data_files/star3ddg_path2.Rdata"))
save(star3ddg_path3, file = here::here("data_files/star3ddg_path3.Rdata"))

For illustration purposes, Figure 7 and 8 shows \(g''\) and \(g'''\).

load(here::here("data_files/star3g.Rdata"))
star3g

Figure 5: Function \(g = \{g_e\}_{e\in\mathcal{E}}\) given by \(g_{e_i}(s) = \cos(\dfrac{\pi s}{\ell})\) for \(i=1,2,3\).

load(here::here("data_files/star3dg.Rdata"))
star3dg

Figure 6: Function \(g' = \{g'_e\}_{e\in\mathcal{E}}\) given by \(g'_{e_i}(s) = -\dfrac{\pi}{\ell}\sin(\dfrac{\pi s}{\ell})\) for \(i=1,2,3\).

load(here::here("data_files/star3ddg.Rdata"))
star3ddg

Figure 7: Function \(g'' = \{g''_e\}_{e\in\mathcal{E}}\) given by \(g''_{e_i}(s) = -\dfrac{\pi^2}{\ell^2}\cos(\dfrac{\pi s}{\ell})\) for \(i=1,2,3\).

load(here::here("data_files/star3dddg.Rdata"))
star3dddg

Figure 8: Function \(g''' = \{g'''_e\}_{e\in\mathcal{E}}\) given by \(g'''_{e_i}(s) = \dfrac{\pi^3}{\ell^3}\sin(\dfrac{\pi s}{\ell})\) for \(i=1,2,3\).

3 Example of a function \(h\in C^1(\Gamma)\cap \mathcal{K}(\Gamma)\) where \(h'\) looks discontinuous

Let us consider the \(2\)-star graph \(\Gamma = (\mathcal{V}, \mathcal{E})\), where \(\mathcal{V} = \{v, v_1, v_2\}\) and \(\mathcal{E} = \{e_1, e_2\}\). Each edge \(e_i\) has common length \(\ell\) and connects the central vertex \(v\) to the outer vertex \(v_i\) for \(i=1,2\) in such a way that \(v = \underline{e_1} = \underline{e_2}\). On \(\Gamma\), we define the function \(h = \{h_e\}_{e\in\mathcal{E}}\) via

\[ h_{e_1}(s) = a + b_1 s - \frac{b_1 s^3}{3\ell^2}, \quad h_{e_2}(s) = a + b_2 s - \frac{b_2 s^3}{3\ell^2}. \] Clearly, \(h\) is a continuous function (see Figure 9) since \[ h_{e_1}(0) = h_{e_2}(s) = a. \] The derivative \(h' = \{h'_e\}_{e\in\mathcal{E}}\) of \(h\) on each edge is given by

\[ h_{e_1}'(s) = b_1 - \frac{b_1 s^2}{\ell^2}, \quad h_{e_2}'(s) = b_2 - \frac{b_2 s^2}{\ell^2}. \] By choosing \(b_1=1\) and \(b_2=-1\), we have that \[ h_{e_1}'(0)+h_{e_2}'(0) = b_1+b_2=0\quad\text{ and }\quad h_{e_1}'(\ell) =h_{e_2}'(\ell) = 0, \] so that \(h\) satisfies the Kirchhoff conditions at every vertex. This means that \(h'\) should be a continuous function. However, Figure 10 shows that \(h'\) is discontinuous. This is actually an artefact due to the fact that the edges are not compatible oriented. For a discussion about compatible orientation, go to this page.

n <- 2
my_length <- 2
edges <-edges_roots_of_unity(n, length = my_length)
edges[[1]] <- rbind(c(0, 0), c(my_length, 0))
edges[[2]] <- rbind(c(0, 0), c(-my_length, 0))
graph <- metric_graph$new(edges = edges)
graph_disc <- graph$clone()
graph$build_mesh(h = h)
graph_disc$build_mesh(h = h, continuous = FALSE)

# General parameters
L <- my_length             # edge length
a <- 1             # value at central vertex
b <- c(1, -1)   # b1+b2+b3 must be 0

f1 <- function(s) a + b[1]*s - b[1]*s^3/(3*L^2)
f2 <- function(s) a + b[2]*s - b[2]*s^3/(3*L^2)


df1 <- function(s) b[1] - b[1]*s^2/L^2
df2 <- function(s) b[2] - b[2]*s^2/L^2

ddf1 <- function(s) -2*b[1]*s/L^2
ddf2 <- function(s) -2*b[2]*s/L^2

dddf1 <- function(s) -2*b[1]/L^2
dddf2 <- function(s) -2*b[2]/L^2

f_list <- list(f1, f2)
df_list <- list(df1, df2)
ddf_list <- list(ddf1, ddf2)
dddf_list <- list(dddf1, dddf2)

f <- apply_edge_functions_fast(graph, f_list)
df <- apply_edge_functions_fast(graph_disc, df_list)
ddf <- apply_edge_functions_fast(graph, ddf_list)
dddf <- apply_edge_functions_fast(graph_disc, dddf_list)

l_mesh <- seq(0, my_length, by = h)
dfl1 <- df1(l_mesh)
dfl2 <- df2(l_mesh)

l_mesh_norm <- l_mesh/my_length
PtE1 <- cbind(rep(1, length(l_mesh)), l_mesh_norm)
PtE2 <- cbind(rep(2, length(l_mesh)), l_mesh_norm)
XY1 <- graph$coordinates(PtE1)
XY2 <- graph$coordinates(PtE2)
DF1 <- data.frame(x = XY1[,1], y = XY1[,2], z = dfl1)
DF2 <- data.frame(x = XY2[,1], y = XY2[,2], z = dfl2)

DF <- rbind(DF1, rep(NA, 3), DF2)
DF_NO_NA <- rbind(DF1, DF2)

star2f <- graph$plot_function(X = f, vertex_size = gsw, edge_width = gsw, line_width = gsw, type = "plotly") |> 
  config(mathjax = 'cdn') |>
  add_trace(
  type = "cone",
  x = ym1,
  y = xm1,
  z = zm1,
  u = dy1,
  v = dx1,
  w = dz1,
  sizemode = "absolute",
  sizeref = sizeref,
  showscale = FALSE,
  showlegend = FALSE,
  colorscale = list(c(0, 1), c("red", "red")),
  cmin = 0,
  cmax = 1
) |>
  add_trace(
  type = "cone",
  x = ym4,
  y = xm4,
  z = zm4,
  u = dy4,
  v = dx4,
  w = dz4,
  sizemode = "absolute",
  sizeref = sizeref,
  showscale = FALSE,
  showlegend = FALSE,
  colorscale = list(c(0, 1), c("red", "red")),
  cmin = 0,
  cmax = 1
) |>
  plotly::layout(title = list(text = TeX("h"), y = 0.8), 
                 font = list(family = "Palatino"),
                 scene = list(xaxis = list(title = list(text = "x", font = list(color = colaxnn)),  tickfont = list(color = colaxnn)),
              yaxis = list(title = list(text = "y", font = list(color = colaxnn)),  tickfont = list(color = colaxnn)),
              zaxis = list(title = list(text = "z", font = list(color = colaxnn)),  tickfont = list(color = colaxnn)),
                   aspectratio = list(x = 2, 
                                 y = 2, 
                                 z = 2),
                   camera = list(eye = list(x = x_eye, 
                                       y = y_eye, 
                                       z = z_eye),
                            center = list(x = 0, 
                                          y = 0, 
                                          z = 0)),
              annotations = list(
                     
             list(
               x = 0, y = 0, z = 0,
               text = TeX("v"),
               textangle = 0, ax = 0, ay = 35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "rgba(0,0,0,0)"),
             list(
               x = e1_fin_y, y = e1_fin_x, z = e1_fin_z,
               text = TeX("v_1"),
               textangle = 0, ax = 0, ay = 35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "rgba(0,0,0,0)"),
             list(
               x = e4_fin_y, y = e4_fin_x, z = e4_fin_z,
               text = TeX("v_2"),
               textangle = 0, ax = 0, ay = 35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "rgba(0,0,0,0)"),
             list(
               x = ym1, y = xm1, z = zm1,
               text = TeX("e_1"),
               textangle = 0, ax = 0, ay = 35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "rgba(0,0,0,0)"),
             list(
               x = ym4, y = xm4, z = zm4,
               text = TeX("e_2"),
               textangle = 0, ax = 0, ay = 35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "rgba(0,0,0,0)"))
                 ))

star2df <- graph$plot_function(X = f*0, vertex_size = gsw, line_color = "black", edge_width = gsw, line_width = gsw, type = "plotly") |>
  add_trace(
  type = "cone",
  x = ym1,
  y = xm1,
  z = zm1,
  u = dy1,
  v = dx1,
  w = dz1,
  sizemode = "absolute",
  sizeref = sizeref,
  showscale = FALSE,
  showlegend = FALSE,
  colorscale = list(c(0, 1), c("red", "red")),
  cmin = 0,
  cmax = 1
) |>
  add_trace(
  type = "cone",
  x = ym4,
  y = xm4,
  z = zm4,
  u = dy4,
  v = dx4,
  w = dz4,
  sizemode = "absolute",
  sizeref = sizeref,
  showscale = FALSE,
  showlegend = FALSE,
  colorscale = list(c(0, 1), c("red", "red")),
  cmin = 0,
  cmax = 1
) |>
  add_trace(data = DF,
            x = ~y, 
            y = ~x, 
            z = ~z, 
            type = "scatter3d",
            mode = "lines",  
            line = list(color = "rgb(0,0,200)", width = gsw),
            showlegend = FALSE) |>
  add_trace(x = rep(DF_NO_NA$y, each = 3), 
            y = rep(DF_NO_NA$x, each = 3), 
            z = unlist(lapply(DF_NO_NA$z, function(zj) c(0, zj, NA))),
            type = "scatter3d", 
            mode = "lines",
            line = list(color = "gray", width = 0.5),
            showlegend = FALSE) |>
  config(mathjax = 'cdn') |>
  plotly::layout(title = list(text = TeX("h'"), y = 0.8), 
                 font = list(family = "Palatino"),
                 scene = list(xaxis = list(title = list(text = "x", font = list(color = colaxnn)),  tickfont = list(color = colaxnn)),
              yaxis = list(title = list(text = "y", font = list(color = colaxnn)),  tickfont = list(color = colaxnn)),
              zaxis = list(title = list(text = "z", font = list(color = colaxnn)),  tickfont = list(color = colaxnn)),
                   aspectratio = list(x = 2, 
                                 y = 2, 
                                 z = 2),
                   camera = list(eye = list(x = x_eye, 
                                       y = y_eye, 
                                       z = z_eye),
                            center = list(x = 0, 
                                          y = 0, 
                                          z = 0)),
              annotations = list(
                     
             list(
               x = 0, y = 0, z = 0,
               text = TeX("v"),
               textangle = 0, ax = 0, ay = 35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "rgba(0,0,0,0)"),
             list(
               x = e1_fin_y, y = e1_fin_x, z = e1_fin_z,
               text = TeX("v_1"),
               textangle = 0, ax = 0, ay = 35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "rgba(0,0,0,0)"),
             list(
               x = e4_fin_y, y = e4_fin_x, z = e4_fin_z,
               text = TeX("v_2"),
               textangle = 0, ax = 0, ay = 35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "rgba(0,0,0,0)"),
             list(
               x = ym1, y = xm1, z = zm1,
               text = TeX("e_1"),
               textangle = 0, ax = 0, ay = 35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "rgba(0,0,0,0)"),
             list(
               x = ym4, y = xm4, z = zm4,
               text = TeX("e_2"),
               textangle = 0, ax = 0, ay = 35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "rgba(0,0,0,0)"))
                 ))

star2ddf <- graph$plot_function(X = ddf, vertex_size = gsw, edge_width = gsw, line_width = gsw, type = "plotly") |> 
  config(mathjax = 'cdn') |>
  add_trace(
  type = "cone",
  x = ym1,
  y = xm1,
  z = zm1,
  u = dy1,
  v = dx1,
  w = dz1,
  sizemode = "absolute",
  sizeref = sizeref,
  showscale = FALSE,
  showlegend = FALSE,
  colorscale = list(c(0, 1), c("red", "red")),
  cmin = 0,
  cmax = 1
) |>
  add_trace(
  type = "cone",
  x = ym4,
  y = xm4,
  z = zm4,
  u = dy4,
  v = dx4,
  w = dz4,
  sizemode = "absolute",
  sizeref = sizeref,
  showscale = FALSE,
  showlegend = FALSE,
  colorscale = list(c(0, 1), c("red", "red")),
  cmin = 0,
  cmax = 1
) |>
  plotly::layout(title = list(text = TeX("h''"), y = 0.8), 
                 font = list(family = "Palatino"),
                 scene = list(xaxis = list(title = list(text = "x", font = list(color = colaxnn)),  tickfont = list(color = colaxnn)),
              yaxis = list(title = list(text = "y", font = list(color = colaxnn)),  tickfont = list(color = colaxnn)),
              zaxis = list(title = list(text = "z", font = list(color = colaxnn)),  tickfont = list(color = colaxnn)),
                   aspectratio = list(x = 2, 
                                 y = 2, 
                                 z = 2),
                   camera = list(eye = list(x = x_eye, 
                                       y = y_eye, 
                                       z = z_eye),
                            center = list(x = 0, 
                                          y = 0, 
                                          z = 0)),
              annotations = list(
                     
             list(
               x = 0, y = 0, z = 0,
               text = TeX("v"),
               textangle = 0, ax = 0, ay = 35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "rgba(0,0,0,0)"),
             list(
               x = e1_fin_y, y = e1_fin_x, z = e1_fin_z,
               text = TeX("v_1"),
               textangle = 0, ax = 0, ay = 35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "rgba(0,0,0,0)"),
             list(
               x = e4_fin_y, y = e4_fin_x, z = e4_fin_z,
               text = TeX("v_2"),
               textangle = 0, ax = 0, ay = 35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "rgba(0,0,0,0)"),
             list(
               x = ym1, y = xm1, z = zm1,
               text = TeX("e_1"),
               textangle = 0, ax = 0, ay = 35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "rgba(0,0,0,0)"),
             list(
               x = ym4, y = xm4, z = zm4,
               text = TeX("e_2"),
               textangle = 0, ax = 0, ay = 35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "rgba(0,0,0,0)"))
                 ))

star2dddf <- graph_disc$plot_function(X = dddf, vertex_size = gsw, edge_width = gsw, line_width = gsw, continuous = FALSE, interpolate_plot = TRUE, type = "plotly") |> 
  add_trace(
  type = "cone",
  x = ym1,
  y = xm1,
  z = zm1,
  u = dy1,
  v = dx1,
  w = dz1,
  sizemode = "absolute",
  sizeref = sizeref,
  showscale = FALSE,
  showlegend = FALSE,
  colorscale = list(c(0, 1), c("red", "red")),
  cmin = 0,
  cmax = 1
) |>
  add_trace(
  type = "cone",
  x = ym4,
  y = xm4,
  z = zm4,
  u = dy4,
  v = dx4,
  w = dz4,
  sizemode = "absolute",
  sizeref = sizeref,
  showscale = FALSE,
  showlegend = FALSE,
  colorscale = list(c(0, 1), c("red", "red")),
  cmin = 0,
  cmax = 1
) |>
  config(mathjax = 'cdn') |>
  plotly::layout(title = list(text = TeX("h'''"), y = 0.8),
                 font = list(family = "Palatino"),
                 scene = list(xaxis = list(title = list(text = "x", font = list(color = colaxnn)),  tickfont = list(color = colaxnn)),
              yaxis = list(title = list(text = "y", font = list(color = colaxnn)),  tickfont = list(color = colaxnn)),
              zaxis = list(title = list(text = "z", font = list(color = colaxnn)),  tickfont = list(color = colaxnn)),
                   aspectratio = list(x = 2, 
                                 y = 2, 
                                 z = 2),
                   camera = list(eye = list(x = x_eye, 
                                       y = y_eye, 
                                       z = z_eye),
                            center = list(x = 0, 
                                          y = 0, 
                                          z = 0)),
              annotations = list(
                     
             list(
               x = 0, y = 0, z = 0,
               text = TeX("v"),
               textangle = 0, ax = 0, ay = 35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "rgba(0,0,0,0)"),
             list(
               x = e1_fin_y, y = e1_fin_x, z = e1_fin_z,
               text = TeX("v_1"),
               textangle = 0, ax = 0, ay = 35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "rgba(0,0,0,0)"),
             list(
               x = e4_fin_y, y = e4_fin_x, z = e4_fin_z,
               text = TeX("v_2"),
               textangle = 0, ax = 0, ay = 35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "rgba(0,0,0,0)"),
             list(
               x = ym1, y = xm1, z = zm1,
               text = TeX("e_1"),
               textangle = 0, ax = 0, ay = 35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "rgba(0,0,0,0)"),
             list(
               x = ym4, y = xm4, z = zm4,
               text = TeX("e_2"),
               textangle = 0, ax = 0, ay = 35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "rgba(0,0,0,0)"))
                 ))


save(star2f, file = here::here("data_files/star2f.Rdata"))
save(star2df, file = here::here("data_files/star2df.Rdata"))
save(star2ddf, file = here::here("data_files/star2ddf.Rdata"))
save(star2dddf, file = here::here("data_files/star2dddf.Rdata"))

For illustration purposes, Figure 11 and 12 shows \(h''\) and \(h'''\).

load(here::here("data_files/star2f.Rdata"))
star2f

Figure 9: Function \(h = \{h_e\}_{e\in\mathcal{E}}\) given by \(h_{e_i}(s) = a + b_i s - \dfrac{b_i s^3}{3\ell^2}\) for \(i=1,2\).

load(here::here("data_files/star2df.Rdata"))
star2df

Figure 10: Function \(h' = \{h'_e\}_{e\in\mathcal{E}}\) given by \(h'_{e_i}(s) = b_i - \dfrac{b_i s^2}{\ell^2}\) for \(i=1,2\).

load(here::here("data_files/star2ddf.Rdata"))
star2ddf

Figure 11: Function \(h'' = \{h''_e\}_{e\in\mathcal{E}}\) given by \(h''_{e_i}(s) = - \dfrac{2 b_i s}{\ell^2}\) for \(i=1,2\).

load(here::here("data_files/star2dddf.Rdata"))
star2dddf

Figure 12: Function \(h''' = \{h'''_e\}_{e\in\mathcal{E}}\) given by \(h'''_{e_i}(s) = - \dfrac{2 b_i}{\ell^2}\) for \(i=1,2\).

4 Path derivatives

load(here::here("data_files/star3dg_path1.Rdata"))
star3dg_path1

Figure 13: Function \(h = \{h_e\}_{e\in\mathcal{E}}\) given by \(h_{e_i}(s) = a + b_i s - \dfrac{b_i s^3}{3\ell^2}\) for \(i=1,2\).

load(here::here("data_files/star3dg_path2.Rdata"))
star3dg_path2

Figure 14: Function \(h' = \{h'_e\}_{e\in\mathcal{E}}\) given by \(h'_{e_i}(s) = b_i - \dfrac{b_i s^2}{\ell^2}\) for \(i=1,2\).

load(here::here("data_files/star3dg_path3.Rdata"))
star3dg_path3

Figure 15: Function \(h'' = \{h''_e\}_{e\in\mathcal{E}}\) given by \(h''_{e_i}(s) = - \dfrac{2 b_i s}{\ell^2}\) for \(i=1,2\).

load(here::here("data_files/star3dg_path_ok.Rdata"))
star3dg_path_ok

Figure 16: Function \(h'' = \{h''_e\}_{e\in\mathcal{E}}\) given by \(h''_{e_i}(s) = - \dfrac{2 b_i s}{\ell^2}\) for \(i=1,2\).

5 Path derivatives 2

load(here::here("data_files/star3ddg_path1.Rdata"))
star3ddg_path1

Figure 17: Function \(h = \{h_e\}_{e\in\mathcal{E}}\) given by \(h_{e_i}(s) = a + b_i s - \dfrac{b_i s^3}{3\ell^2}\) for \(i=1,2\).

load(here::here("data_files/star3ddg_path2.Rdata"))
star3ddg_path2

Figure 18: Function \(h' = \{h'_e\}_{e\in\mathcal{E}}\) given by \(h'_{e_i}(s) = b_i - \dfrac{b_i s^2}{\ell^2}\) for \(i=1,2\).

load(here::here("data_files/star3ddg_path3.Rdata"))
star3ddg_path3

Figure 19: Function \(h'' = \{h''_e\}_{e\in\mathcal{E}}\) given by \(h''_{e_i}(s) = - \dfrac{2 b_i s}{\ell^2}\) for \(i=1,2\).

6 References

grateful::cite_packages(output = "paragraph", out.dir = ".")

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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