Intrinsic continuity is equivalent to the Kirchhoff condition for degree-two vertices

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captioner <- function(caption) {
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library(MetricGraph)
library(ggplot2)
library(reshape2)
library(dplyr)
library(viridis)
library(plotly)
library(patchwork)
library(slackr)
source("keys.R")
slackr_setup(token = token) # token comes from keys.R

## [1] "Successfully connected to Slack"

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

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

Let \(\Gamma = (\mathcal{V}, \mathcal{E})\) be a metric graph with vertex set \(\mathcal{V} = \{v_1, v, v_2\}\) and edge set \(\mathcal{E} = \{e_1, e_2\}\). Assume that the edges \(e_1\) and \(e_2\) meet at the vertex \(v\), so that \(v\) is their common endpoint. For later reference, we recall the Kirchhoff vertex conditions \[\begin{equation} \label{eq:kirchhoff_cond} \tag{1} \mathcal{K}(\Gamma) = \left\{f\in C(\Gamma)\;\middle|\; \forall v\in \mathcal{V}:\; \sum_{e\in\mathcal{E}_v}\partial_e f(v)=0 \right\}, \end{equation}\] where \(\mathcal{E}_v\) denotes the set of edges incident to \(v\), and \[\begin{equation} \label{eq:dir_der} \tag{2} \partial_ef(v) = \begin{cases} f'_e(0), \quad &\text{if }v=0,\\ -f'_e(\ell_e), \quad &\text{if }v=\ell_e. \end{cases} \end{equation}\]

1 Compatible orientation

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]

  out <- sapply(seq_len(nrow(VtE)), function(i) f_list[[edge[i]]](s_true[i]))
  return(out)
}

We first assume that the edges are compatibly oriented, namely \(e_1 = [0,\ell_1]\) and \(e_2=[0,\ell_2]\), with the vertex \(v\) corresponding to \(\ell_1\) on \(e_1\) and \(0\) on \(e_2\). With this parametrization, we can glue the two edges into a single interval \([e_1,e_2]\simeq [0,\ell_1+\ell_2]\) and view a function \(f = \{f_{e_1},f_{e_2}\}\in C(\Gamma)\) as a function \(\hat{f}\in C([0,\ell_1+\ell_2])\) so that \(f_{e_1}(\ell_1) = \hat{f}(v) = f_{e_2}(0)\). Consider the function \(f = \{f_{e_1},f_{e_2}\}\) given by \[\begin{equation*} \begin{cases} f_{e_1}(t) = t^4,& t\in[0,\ell_1],\\ f_{e_2}(t) = (t+\ell_1)^4,& t\in[0,\ell_2]. \end{cases} \end{equation*}\] Its edgewise derivatives are given by \[\begin{equation*} \begin{cases} f'_{e_1}(t) = 4t^3,& t\in[0,\ell_1],\\ f'_{e_2}(t) = 4(t+\ell_1)^3,& t\in[0,\ell_2]. \end{cases} \end{equation*}\] Under the above identification, \(f\) corresponds to the function \[\begin{align*} \hat{f}(t) = t^4,\quad t\in[0,\ell_1 + \ell_2], \end{align*}\] whose derivative is \[\begin{align*} \hat{f}(t) = 4t^3,\quad t\in[0,\ell_1 + \ell_2]. \end{align*}\] In particular, \(\hat{f} \in C^1([0,\ell_1+\ell_2])\), and therefore \(f\) has a continuous first derivative across the vertex \(v\). See Figure 2. Moreover, the edgewise derivatives must agree at \(v\), that is, we must have \(f'_{e_1}(\ell_1) = \hat{f}'(v) = f'_{e_2}(0)\), which is the case since \[\begin{align*} f'_{e_1}(\ell_1) = 4\ell_1^3\quad\text{ and }\quad \hat{f}'(v) = \hat{f}'(\ell_1)=4\ell_1^3\quad \text{ and }\quad f'_{e_2}(0) = 4\ell_1^3. \end{align*}\] By using \(\eqref{eq:dir_der}\), we get that \[\begin{align*} \partial_{e_1}f(v) = -f'_{e_1}(\ell_1)=-4\ell_1^3\quad\text{ and }\quad \partial_{e_2}f(v) = f'_{e_2}(0)=4\ell_1^3, \end{align*}\] and therefore, \[\begin{equation} \sum_{e \in \mathcal{E}_v} \partial_e f(v) =\partial_{e_1}f(v) + \partial_{e_2}f(v) = -f'_{e_1}(\ell_1) + f'_{e_2}(0) = -4\ell_1^3 + 4\ell_1^3 = 0, \end{equation}\] which means that \(f\) satisfies the Kirchhoff vertex conditions \(\eqref{eq:kirchhoff_cond}\) at vertex \(v\). This is an instance of the fact that \(f\) satisfies the Kirchhoff vertex conditions \(\eqref{eq:kirchhoff_cond}\) at all vertices \(v\) with \(\deg(v) =2\) if and only if \(f'\) is continuous at at all vertices \(v\) with \(\deg(v) =2\).

h = 0.01
ell1 <- 1
ell2 <- 0.4
e1 <- rbind(c(-ell1,0), # (x,y) = underline(e_1) 
            c(0,0))  # (x,y) = overline(e_1) 
e2 <- rbind(c(0,0),
            c(ell2,0))

e1_ini_x <- -ell1; e1_ini_y <- 0; e1_ini_z <- 0
e1_fin_x <- 0; e1_fin_y <- 0; 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 <- 0; e2_ini_y <- 0; e2_ini_z <- 0
e2_fin_x <- ell2; e2_fin_y <- 0; 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


graph <- metric_graph$new(edges = list(e1 = e1, e2 = e2))
graph$build_mesh(h = h)

fe1 <- function(t) t^4
fe2 <- function(t) (t + ell1)^4

dfe1 <- function(t) 4*t^3
dfe2 <- function(t) 4*(t + ell1)^3

ddfe1 <- function(t) 12*t^2
ddfe2 <- function(t) 12*(t + ell1)^2

dddfe1 <- function(t) 24*t
dddfe2 <- function(t) 24*(t + ell1)

f_list <- list(fe1, fe2)
df_list <- list(dfe1, dfe2)
ddf_list <- list(ddfe1, ddfe2)
dddf_list <- list(dddfe1, dddfe2)

f_list_aux <- f_list
df_list_aux <- df_list
ddf_list_aux <- ddf_list
dddf_list_aux <- dddf_list

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

sizeref <- 0.1
pf <- graph$plot_function(X = f, vertex_size = gsw, 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("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
) |>
  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 = e1_ini_y, y = e1_ini_x, z = e1_ini_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 = 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 = 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 = 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)"))
                 ))

pdf <- graph$plot_function(X = df, vertex_size = gsw, 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("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
) |>
  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 = e1_ini_y, y = e1_ini_x, z = e1_ini_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 = 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 = 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 = 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)"))
                 ))

pddf <- graph$plot_function(X = ddf, vertex_size = gsw, 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("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
) |>
  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 = e1_ini_y, y = e1_ini_x, z = e1_ini_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 = 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 = 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 = 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)"))
                 ))

pdddf <- graph$plot_function(X = dddf, vertex_size = gsw, 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("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
) |>
  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 = e1_ini_y, y = e1_ini_x, z = e1_ini_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 = 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 = 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 = 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)"))
                 ))

save(pf, file = here::here("data_files/pf.Rdata"))
save(pdf, file = here::here("data_files/pdf.Rdata"))
save(pddf, file = here::here("data_files/pddf.Rdata"))
save(pdddf, file = here::here("data_files/pdddf.Rdata"))

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

Figure 1: Function \(f = \{f_e\}_{e\in\mathcal{E}}\) given by \(f_{e_1}(t) = t^4\) and \(f_{e_2}(t) = (t+\ell_1)^4\).

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

Figure 2: Function \(f' = \{f'_e\}_{e\in\mathcal{E}}\) given by \(f'_{e_1}(t) = 4t^3\) and \(f'_{e_2}(t) = 4(t+\ell_1)^3\).

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

Figure 3: Function \(f'' = \{f''_e\}_{e\in\mathcal{E}}\) given by \(f''_{e_1}(t) = 12t^2\) and \(f''_{e_2}(t) = 12(t+\ell_1)^2\).

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

Figure 4: Function \(f''' = \{f'''_e\}_{e\in\mathcal{E}}\) given by \(f'''_{e_1}(t) = 24t\) and \(f'''_{e_2}(t) = 24(t+\ell_1)\).

2 Incompatible orientation

my_apply_edge_functions <- function(graph, f_list, ell1, ell2, h){
  l1_mesh <- seq(0, ell1, by = h)
  l2_mesh <- seq(0, ell2, by = h)
  f1 <- f_list[[1]](l1_mesh)
  f2 <- f_list[[2]](l2_mesh)
  l1_mesh_norm <- l1_mesh/ell1
  l2_mesh_norm <- l2_mesh/ell2
  PtE1 <- cbind(rep(1, length(l1_mesh)), l1_mesh_norm)
  PtE2 <- cbind(rep(2, length(l2_mesh)), l2_mesh_norm)
  XY1 <- graph$coordinates(PtE1)
  XY2 <- graph$coordinates(PtE2)
  DF1 <- data.frame(x = XY1[,1], y = XY1[,2], z = f1)
  DF2 <- data.frame(x = XY2[,1], y = XY2[,2], z = f2)
  DF <- rbind(DF1, rep(NA, 3), DF2)
  return(list(DF = DF))
}

Now study the case where the edges are not compatible oriented. For that, we flip edge \(e_2\), that is, \(e_1 = [0,\ell_1]\) and \(e_2=[0,\ell_2]\) with \(v\) corresponding to \(\ell_1\) on \(e_1\) and \(\ell_2\) on \(e_2\). Consider the function \(g = \{g_{e_1},g_{e_2}\}\) given by \[\begin{equation*} \begin{cases} g_{e_1}(\tau) = \tau^4,& \tau\in[0,\ell_1],\\ g_{e_2}(\tau) = (\tau-(\ell_1+\ell_2))^4,& \tau\in[0,\ell_2]. \end{cases} \end{equation*}\] Its edgewise derivatives are given by \[\begin{equation*} \begin{cases} g'_{e_1}(\tau) = 4\tau^3,& \tau\in[0,\ell_1],\\ g'_{e_2}(\tau) = 4(\tau-(\ell_1+\ell_2))^3,& \tau\in[0,\ell_2]. \end{cases} \end{equation*}\] The coordinates derivatives at \(v\) are given by \[\begin{equation} g'_{e_1}(\ell_1)=4\ell_1^3\quad\text{ and }\quad g'_{e_2}(\ell_2)=-4\ell_1^3. \end{equation}\] Let us check the Kirchhoff vertex conditions \(\eqref{eq:kirchhoff_cond}\) at vertex \(v\). By using \(\eqref{eq:dir_der}\), we get that \[\begin{align*} \partial_{e_1}g(v) = -g'_{e_1}(\ell_1)=-4\ell_1^3\quad\text{ and }\quad \partial_{e_2}g(v) = -g'_{e_2}(\ell_2)=4\ell_1^3, \end{align*}\] and therefore, \[\begin{equation} \label{eq:invariant_c} \tag{3} \sum_{e \in \mathcal{E}_v} \partial_e g(v) =\partial_{e_1}g(v) + \partial_{e_2}g(v) = -g'_{e_1}(\ell_1) -g'_{e_2}(\ell_2) = -4\ell_1^3 + 4\ell_1^3 = 0, \end{equation}\] which shows that \(g\) satisfies the Kirchhoff vertex conditions \(\eqref{eq:kirchhoff_cond}\) at vertex \(v\). At first glance, the edgewise derivative, \(g' = \{g'_{e_1},g'_{e_2}\}\) appears discontinuous at \(v\), which might seem contradictory. See Figure 6.

To clarify, observe that \(f = g\) as functions on the graph. Indeed, by reparametrizing edge \(e_1\) by \(\tau=t\) and edge \(e_2\) by \(\tau = \ell_2-t\), we obtain \(f_{e_1}=g_{e_1}\) and \(f_{e_2}=g_{e_2}\). More precisely, \[\begin{align} \label{eq:eq_of_f_g} \tag{4} g_{e_1}(\tau) = g_{e_1}(t) = t^4 = f_{e_1}(t)\quad\text{ and }\quad g_{e_2}(\tau) = g_{e_2}(\ell_2-t) = (\ell_2-t-(\ell_1+\ell_2))^4 = (t+\ell_1)^4= f_{e_2}(t), \end{align}\] Thus \(f\) and \(g\) coincide on each edge after reparametrization, and define the same function on the graph. The apparent discontinuity of \(g'\) arises solely from the reversed coordinate on \(e_2\). The function itself has not changed, only its coordinate representation.

From \(\eqref{eq:eq_of_f_g}\), by the chain rule, the first derivative transforms as \[\begin{equation} \label{eq:der_inv} \tag{5} g'_{e_2}(\tau) = \dfrac{d}{d\tau} (g_{e_2}(\tau)) = \dfrac{d}{d\tau} (f_{e_2}(t)) = \dfrac{d}{dt} (f_{e_2}(t)) \dfrac{dt}{d\tau} = - \dfrac{d}{dt} (f_{e_2}(t)) = - f'_{e_2}(t) \end{equation}\]

so the coordinate derivative changes sign under reparametrization. Hence, edgewise derivatives are not invariant under orientation changes. By contrast, the outward derivative \(\partial_e g(v)\) is invariant, and is therefore the correct intrinsic quantity to check the Kirchhoff condition.

In the compatible orientation case, \(f'_{e_1}(\ell_1) = f'_{e_2}(0)\), so the coordinate derivatives match. In the flipped orientation case, \(g'_{e_1}(\ell_1) \neq g'_{e_2}(\ell_2)\), but this mismatch is purely a consequence of the reversed parametrization. The correct invariant comparison is \(\eqref{eq:invariant_c}\), which demonstrates that the Kirchhoff condition is intrinsic and independent of edge parametrization.

In conclusion, the apparent contradiction arises only when comparing coordinate derivatives with opposite orientations, which is not geometrically meaningful. Continuity of the derivative at a vertex must be interpreted intrinsically: at a degree-two vertex, this intrinsic continuity is equivalent to the Kirchhoff condition.

e2 <- e2[2:1,]
graph <- metric_graph$new(edges = list(e1 = e1, e2 = e2))
graph$build_mesh(h = h)

e1_ini_x <- -ell1; e1_ini_y <- 0; e1_ini_z <- 0
e1_fin_x <- 0; e1_fin_y <- 0; 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 <- ell2; e2_ini_y <- 0; e2_ini_z <- 0
e2_fin_x <- 0; e2_fin_y <- 0; 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

fe1 <- function(t) t^4
fe2 <- function(t) (t - (ell1 + ell2))^4

dfe1 <- function(t) 4*t^3
dfe2 <- function(t) 4*(t - (ell1 + ell2))^3

ddfe1 <- function(t) 12*t^2
ddfe2 <- function(t) 12*(t - (ell1 + ell2))^2

dddfe1 <- function(t) 24*t
dddfe2 <- function(t) 24*(t - (ell1 + ell2))

f_list <- list(fe1, fe2)
df_list <- list(dfe1, dfe2)
ddf_list <- list(ddfe1, ddfe2)
dddf_list <- list(dddfe1, dddfe2)

f <- my_apply_edge_functions(graph, f_list, ell1, ell2, h)
df <- my_apply_edge_functions(graph, df_list, ell1, ell2, h)
ddf <- my_apply_edge_functions(graph, ddf_list, ell1, ell2, h)
dddf <- my_apply_edge_functions(graph, dddf_list, ell1, ell2, h)

f_aux <- my_apply_edge_functions(graph, f_list_aux, ell1, ell2, h)
df_aux <- my_apply_edge_functions(graph, df_list_aux, ell1, ell2, h)
ddf_aux <- my_apply_edge_functions(graph, ddf_list_aux, ell1, ell2, h)
dddf_aux <- my_apply_edge_functions(graph, dddf_list_aux, ell1, ell2, h)

p_base <- graph$plot_function(X = rep(0, nrow(graph$mesh$VtE)), vertex_size = gsw, line_color = "black", edge_width = gsw, line_width = gsw, type = "plotly")
DF <- f$DF
pf3 <- p_base |>
  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(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("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 = e1_ini_y, y = e1_ini_x, z = e1_ini_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 = 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 = e2_ini_y, y = e2_ini_x, z = e2_ini_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 = 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)"))
                 ))

DF <- df$DF
pdf3 <- p_base |>
  add_trace(
  type = "cone",
  x = ym1,
  y = xm1,
  z = zm1,
  u = dy1,
  v = dx1,
  w = dz1,
  sizemode = "absolute",
  sizeref = sizeref,
  showscale = FALSE,
  showlegend = FALSE,
  colorscale = list(c(0, 1), c("red", "red")),
  cmin = 0,
  cmax = 1
) |>
  add_trace(
  type = "cone",
  x = ym2,
  y = xm2,
  z = zm2,
  u = dy2,
  v = dx2,
  w = dz2,
  sizemode = "absolute",
  sizeref = sizeref,
  showscale = FALSE,
  showlegend = FALSE,
  colorscale = list(c(0, 1), c("red", "red")),
  cmin = 0,
  cmax = 1
) |>
    add_trace(data = DF,
            x = ~y, 
            y = ~x, 
            z = ~z, 
            type = "scatter3d",
            mode = "lines",  
            line = list(color = "rgb(0,0,200)", width = gsw),
            showlegend = FALSE) |>
  add_trace(x = rep(DF$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("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 = e1_ini_y, y = e1_ini_x, z = e1_ini_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 = 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 = e2_ini_y, y = e2_ini_x, z = e2_ini_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 = 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)"))
                 ))

DF <- ddf$DF
pddf3 <- p_base |>
  add_trace(
  type = "cone",
  x = ym1,
  y = xm1,
  z = zm1,
  u = dy1,
  v = dx1,
  w = dz1,
  sizemode = "absolute",
  sizeref = sizeref,
  showscale = FALSE,
  showlegend = FALSE,
  colorscale = list(c(0, 1), c("red", "red")),
  cmin = 0,
  cmax = 1
) |>
  add_trace(
  type = "cone",
  x = ym2,
  y = xm2,
  z = zm2,
  u = dy2,
  v = dx2,
  w = dz2,
  sizemode = "absolute",
  sizeref = sizeref,
  showscale = FALSE,
  showlegend = FALSE,
  colorscale = list(c(0, 1), c("red", "red")),
  cmin = 0,
  cmax = 1
) |>
    add_trace(data = DF,
            x = ~y, 
            y = ~x, 
            z = ~z, 
            type = "scatter3d",
            mode = "lines",  
            line = list(color = "rgb(0,0,200)", width = gsw),
            showlegend = FALSE) |>
  add_trace(x = rep(DF$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("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 = e1_ini_y, y = e1_ini_x, z = e1_ini_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 = 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 = e2_ini_y, y = e2_ini_x, z = e2_ini_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 = 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)"))
                 ))

DF <- dddf$DF
pdddf3 <- p_base |>
  add_trace(
  type = "cone",
  x = ym1,
  y = xm1,
  z = zm1,
  u = dy1,
  v = dx1,
  w = dz1,
  sizemode = "absolute",
  sizeref = sizeref,
  showscale = FALSE,
  showlegend = FALSE,
  colorscale = list(c(0, 1), c("red", "red")),
  cmin = 0,
  cmax = 1
) |>
  add_trace(
  type = "cone",
  x = ym2,
  y = xm2,
  z = zm2,
  u = dy2,
  v = dx2,
  w = dz2,
  sizemode = "absolute",
  sizeref = sizeref,
  showscale = FALSE,
  showlegend = FALSE,
  colorscale = list(c(0, 1), c("red", "red")),
  cmin = 0,
  cmax = 1
) |>
    add_trace(data = DF,
            x = ~y, 
            y = ~x, 
            z = ~z, 
            type = "scatter3d",
            mode = "lines",  
            line = list(color = "rgb(0,0,200)", width = gsw),
            showlegend = FALSE) |>
  add_trace(x = rep(DF$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("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 = e1_ini_y, y = e1_ini_x, z = e1_ini_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 = 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 = e2_ini_y, y = e2_ini_x, z = e2_ini_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 = 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)"))
                 ))

save(pf3, file = here::here("data_files/pf3.Rdata"))
save(pdf3, file = here::here("data_files/pdf3.Rdata"))
save(pddf3, file = here::here("data_files/pddf3.Rdata"))
save(pdddf3, file = here::here("data_files/pdddf3.Rdata"))

load(here::here("data_files/pf3.Rdata"))
pf3

Figure 5: Function \(g = \{g_e\}_{e\in\mathcal{E}}\) given by \(g_{e_1}(\tau) = \tau^4\) and \(g_{\hat{e}_2}(\tau) = (\tau-(\ell_1+\ell_2))^4\).

load(here::here("data_files/pdf3.Rdata"))
pdf3

Figure 6: Function \(g' = \{g'_e\}_{e\in\mathcal{E}}\) given by \(g'_{e_1}(\tau) = 4\tau^3\) and \(g'_{\hat{e}_2}(\tau) = 4(\tau-(\ell_1+\ell_2))^3\).

load(here::here("data_files/pddf3.Rdata"))
pddf3

Figure 7: Function \(g'' = \{g''_e\}_{e\in\mathcal{E}}\) given by \(g''_{e_1}(\tau) = 12\tau^2\) and \(g''_{\hat{e}_2}(\tau) = 12(\tau-(\ell_1+\ell_2))^2\).

load(here::here("data_files/pdddf3.Rdata"))
pdddf3

Figure 8: Function \(g''' = \{g'''_e\}_{e\in\mathcal{E}}\) given by \(g'''_{e_1}(\tau) = 24\tau\) and \(g'''_{\hat{e}_2}(\tau) = 24(\tau-(\ell_1+\ell_2))\).

Figure 6 illustrates precisely the relation expressed in \(\eqref{eq:der_inv}\). That is, \(g'_{e_2}\) can be obtained by multiplying \(f'_{e_2}\) by \(-1\).

3 Even order derivatives are orientation-independent

We again apply the chain rule in \(\eqref{eq:der_inv}\) to obtain \[\begin{equation} g''_{e_2}(\tau) = \dfrac{d}{d\tau} ( - f'_{e_2}(t)) = - \dfrac{d}{dt} (f'_{e_2}(t)) \dfrac{dt}{d\tau} = f''_{e_2}(t), \end{equation}\] so that the second derivative does not change sign under a flip. See Figure 7. This means that even order derivatives are orientation-independent. In general, for the \(k\)-th derivative, we have \[\begin{equation} \dfrac{d^k}{d\tau^k}(g_{e_2}(\tau)) = (-1)^k\dfrac{d^k}{dt^k}(f_{e_2}(t)). \end{equation}\]

4 Additional plots

DF <- f_aux$DF
pf_aux <- p_base |>
  add_trace(
  type = "cone",
  x = ym1,
  y = xm1,
  z = zm1,
  u = dy1,
  v = dx1,
  w = dz1,
  sizemode = "absolute",
  sizeref = sizeref,
  showscale = FALSE,
  showlegend = FALSE,
  colorscale = list(c(0, 1), c("red", "red")),
  cmin = 0,
  cmax = 1
) |>
  add_trace(
  type = "cone",
  x = ym2,
  y = xm2,
  z = zm2,
  u = dy2,
  v = dx2,
  w = dz2,
  sizemode = "absolute",
  sizeref = sizeref,
  showscale = FALSE,
  showlegend = FALSE,
  colorscale = list(c(0, 1), c("red", "red")),
  cmin = 0,
  cmax = 1
) |>
    add_trace(data = DF,
            x = ~y, 
            y = ~x, 
            z = ~z, 
            type = "scatter3d",
            mode = "lines",  
            line = list(color = "rgb(0,0,200)", width = gsw),
            showlegend = FALSE) |>
  add_trace(x = rep(DF$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("\\hat{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 = e1_ini_y, y = e1_ini_x, z = e1_ini_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 = 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 = e2_ini_y, y = e2_ini_x, z = e2_ini_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 = 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)"))
                 ))
DF <- df_aux$DF
pdf_aux <- p_base |>
  add_trace(
  type = "cone",
  x = ym1,
  y = xm1,
  z = zm1,
  u = dy1,
  v = dx1,
  w = dz1,
  sizemode = "absolute",
  sizeref = sizeref,
  showscale = FALSE,
  showlegend = FALSE,
  colorscale = list(c(0, 1), c("red", "red")),
  cmin = 0,
  cmax = 1
) |>
  add_trace(
  type = "cone",
  x = ym2,
  y = xm2,
  z = zm2,
  u = dy2,
  v = dx2,
  w = dz2,
  sizemode = "absolute",
  sizeref = sizeref,
  showscale = FALSE,
  showlegend = FALSE,
  colorscale = list(c(0, 1), c("red", "red")),
  cmin = 0,
  cmax = 1
) |>
    add_trace(data = DF,
            x = ~y, 
            y = ~x, 
            z = ~z, 
            type = "scatter3d",
            mode = "lines",  
            line = list(color = "rgb(0,0,200)", width = gsw),
            showlegend = FALSE) |>
  add_trace(x = rep(DF$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("\\hat{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 = e1_ini_y, y = e1_ini_x, z = e1_ini_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 = 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 = e2_ini_y, y = e2_ini_x, z = e2_ini_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 = 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)"))
                 ))

DF <- ddf_aux$DF
pddf_aux <- p_base |>
  add_trace(
  type = "cone",
  x = ym1,
  y = xm1,
  z = zm1,
  u = dy1,
  v = dx1,
  w = dz1,
  sizemode = "absolute",
  sizeref = sizeref,
  showscale = FALSE,
  showlegend = FALSE,
  colorscale = list(c(0, 1), c("red", "red")),
  cmin = 0,
  cmax = 1
) |>
  add_trace(
  type = "cone",
  x = ym2,
  y = xm2,
  z = zm2,
  u = dy2,
  v = dx2,
  w = dz2,
  sizemode = "absolute",
  sizeref = sizeref,
  showscale = FALSE,
  showlegend = FALSE,
  colorscale = list(c(0, 1), c("red", "red")),
  cmin = 0,
  cmax = 1
) |>
    add_trace(data = DF,
            x = ~y, 
            y = ~x, 
            z = ~z, 
            type = "scatter3d",
            mode = "lines",  
            line = list(color = "rgb(0,0,200)", width = gsw),
            showlegend = FALSE) |>
  add_trace(x = rep(DF$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("\\hat{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 = e1_ini_y, y = e1_ini_x, z = e1_ini_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 = 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 = e2_ini_y, y = e2_ini_x, z = e2_ini_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 = 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)"))
                 ))

DF <- dddf_aux$DF
pdddf_aux <- p_base |>
  add_trace(
  type = "cone",
  x = ym1,
  y = xm1,
  z = zm1,
  u = dy1,
  v = dx1,
  w = dz1,
  sizemode = "absolute",
  sizeref = sizeref,
  showscale = FALSE,
  showlegend = FALSE,
  colorscale = list(c(0, 1), c("red", "red")),
  cmin = 0,
  cmax = 1
) |>
  add_trace(
  type = "cone",
  x = ym2,
  y = xm2,
  z = zm2,
  u = dy2,
  v = dx2,
  w = dz2,
  sizemode = "absolute",
  sizeref = sizeref,
  showscale = FALSE,
  showlegend = FALSE,
  colorscale = list(c(0, 1), c("red", "red")),
  cmin = 0,
  cmax = 1
) |>
    add_trace(data = DF,
            x = ~y, 
            y = ~x, 
            z = ~z, 
            type = "scatter3d",
            mode = "lines",  
            line = list(color = "rgb(0,0,200)", width = gsw),
            showlegend = FALSE) |>
  add_trace(x = rep(DF$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("\\hat{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 = e1_ini_y, y = e1_ini_x, z = e1_ini_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 = 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 = e2_ini_y, y = e2_ini_x, z = e2_ini_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 = 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)"))
                 ))



save(pf_aux, file = here::here("data_files/pf_aux.Rdata"))
save(pdf_aux, file = here::here("data_files/pdf_aux.Rdata"))
save(pddf_aux, file = here::here("data_files/pddf_aux.Rdata"))
save(pdddf_aux, file = here::here("data_files/pdddf_aux.Rdata"))

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

Figure 9: Function \(f = \{f_e\}_{e\in\mathcal{E}}\) given by \(f_{e_1}(t) = t^4\) and \(f_{e_2}(t) = (t+\ell_1)^4\).

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

Figure 10: Function \(\hat{f} = \{f_e\}_{e\in\mathcal{E}}\) given by \(f_{e_1}(t) = t^4\) and \(f_{\hat{e}_2}(t) = (t+\ell_1)^4\).

load(here::here("data_files/pf3.Rdata"))
pf3

Figure 11: Function \(g = \{g_e\}_{e\in\mathcal{E}}\) given by \(g_{e_1}(\tau) = \tau^4\) and \(g_{\hat{e}_2}(\tau) = (\tau-(\ell_1+\ell_2))^4\).

5 For \(\hat{f}\)

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

Figure 12: Function \(\hat{f} = \{f_e\}_{e\in\mathcal{E}}\) given by \(f_{e_1}(t) = t^4\) and \(f_{\hat{e}_2}(t) = (t+\ell_1)^4\).

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

Figure 13: Function \(\hat{f}' = \{f'_e\}_{e\in\mathcal{E}}\) given by \(f'_{e_1}(t) = 4t^3\) and \(f'_{\hat{e}_2}(t) = 4(t+\ell_1)^3\).

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

Figure 14: Function \(\hat{f}'' = \{f''_e\}_{e\in\mathcal{E}}\) given by \(f''_{e_1}(t) = 12t^2\) and \(f''_{\hat{e}_2}(t) = 12(t+\ell_1)^2\).

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

Figure 15: Function \(\hat{f}''' = \{f'''_e\}_{e\in\mathcal{E}}\) given by \(f'''_{e_1}(t) = 24t\) and \(f'''_{\hat{e}_2}(t) = 24(t+\ell_1)\).

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), 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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