Simplest graphs and more

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# Set global options for all code chunks
knitr::opts_chunk$set(
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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"

library(reticulate)
py_config()

## python:         /home/rierasl/miniconda3/envs/phdenv/bin/python
## libpython:      /home/rierasl/miniconda3/envs/phdenv/lib/libpython3.11.so
## pythonhome:     /home/rierasl/miniconda3/envs/phdenv:/home/rierasl/miniconda3/envs/phdenv
## version:        3.11.14 (main, Oct 21 2025, 18:31:21) [GCC 11.2.0]
## numpy:          /home/rierasl/miniconda3/envs/phdenv/lib/python3.11/site-packages/numpy
## numpy_version:  2.4.2
## 
## NOTE: Python version was forced by RETICULATE_PYTHON

capture.output(
  knitr::purl(here::here("functionality1.Rmd"), output = here::here("functionality1.R")),
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)
source(here::here("functionality1.R"))

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1 Part 1

1.1 Interval graph

Below we plot the interval graph \(\Gamma_I = (\mathcal{V}, \mathcal{E})\) with \(\mathcal{V} = \{v_1, v_2\}\) and \(\mathcal{E} = \{e_1\}\), where \(e_1\) is the edge connecting \(v_1\) and \(v_2\).

e1_ini_x <- 0; e1_ini_y <- 0; e1_ini_z <- 0
e1_fin_x <- 1; 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 <- 1; 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

# for the circle
r <- 1/pi
cx <- 1 + r
cy <- 0

# parameter value where the cone will be placed
t0 <- 0

# position on the curve
x0 <- cx + r*cos(t0)
y0 <- cy + r*sin(t0)
z0 <- 0

# tangent vector (derivative)
dx <- -r*sin(t0)
dy <-  r*cos(t0)
dz <- 0

sizeref <- 0.1

n <- 3#333
graph <- gets.graph.interval(n = n)

notes_int <- list(annotations = list(
             list(
               x = 0, y = 0, z = 0,
               text = TeX("v_1"),
               textangle = 0, ax = 0, ay = 35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "gray", arrowsize = 1, arrowwidth = 0.5, arrowhead = 1),
             list(
               x = 0, y = 1, z = 0,
               text = TeX("v_2"),
               textangle = 0, ax = 0, ay = 35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "gray", arrowsize = 1, arrowwidth = 0.5, arrowhead = 1),
             list(
               x = 0, y = 0.5, z = 0,
               text = TeX("e_1"),
               textangle = 0, ax = 0, ay = 35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "rgba(0,0,0,0)")))

layout_int <- 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)),
  camera = list(eye = list(x = 2.3, 
                                       y = 2.3, 
                                       z = 2.3),
                            center = list(x = 0, 
                                          y = 0, 
                                          z = 0)))

p11 <- graph$plot_function(rep(0,nrow(graph$mesh$V)), 
                           vertex_size = gsw,
                           type = "plotly", 
                           line_width = gsw, 
                           edge_width = gsw, 
                           line_color = "black")|>
  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
) |>
  layout(font = list(family = "Palatino"),
         scene =  c(notes_int, layout_int),
         margin = list(l = 0, r = 0, b = 0, t = 0),
         showlegend = FALSE,
         font = list(family = "Palatino"))



p2int <- graph$plot_function(rep(0,nrow(graph$mesh$V)), 
                           vertex_size = gsw,
                           type = "plotly", 
                           line_width = gsw, 
                           edge_width = gsw, 
                           line_color = "black")|>
  config(mathjax = 'cdn') |>
    add_trace(
  type = "cone",
  x = xm2,
  y = ym2,
  z = zm2,
  u = dx2,
  v = dy2,
  w = dz2,
  sizemode = "absolute",
  sizeref = sizeref,
  showscale = FALSE,
  showlegend = FALSE,
  colorscale = list(c(0, 1), c("green", "green")),
  cmin = 0,
  cmax = 1
) |>
  layout(font = list(family = "Palatino"),
         scene =  c(notes_int, layout_int),
         margin = list(l = 0, r = 0, b = 0, t = 0),
         showlegend = FALSE,
         font = list(family = "Palatino"))


save(p11, file = here::here("data_files/interval_graph.Rdata"))
save(p2int, file = here::here("data_files/interval_graph_flip_edge.Rdata"))

1.2 Circle graph

Below we plot the circle graph \(\Gamma_C = (\mathcal{V}, \mathcal{E})\) with \(\mathcal{V} = \{v_2\}\) and \(\mathcal{E} = \{e_2\}\), where \(e_2\) is the edge connecting \(v_2\) to itself.

n <- 6#666
graph <- gets.graph.circle(n = n)

notes_cir <- list(annotations = list(
             list(
               x = 0, y = 1, z = 0,
               text = TeX("v_2"),
               textangle = 0, ax = 0, ay = 35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "gray", arrowsize = 1, arrowwidth = 0.5, arrowhead = 1),
             list(
               x = 0, y = 1 + 2/pi, z = 0,
               text = TeX("e_2"),
               textangle = 0, ax = 0, ay = 35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "rgba(0,0,0,0)")))
  
  
p22 <- graph$plot_function(rep(0,nrow(graph$mesh$V)), 
                           vertex_size = gsw,
                           type = "plotly", 
                           line_width = gsw, 
                           edge_width = gsw, 
                           line_color = "black") %>% 
  config(mathjax = 'cdn') %>% 
  add_trace(
    type = "cone",
    x = y0,
    y = x0,
    z = z0,
    u = dy,
    v = dx,
    w = dz,
    sizemode = "absolute",
    sizeref = sizeref,
    showscale = FALSE,
    showlegend = FALSE,
    colorscale = list(c(0,1), c("green","green")),
    cmin = 0,
    cmax = 1
  ) |>
  layout(font = list(family = "Palatino"),
         scene =  c(notes_cir, layout_int),
         margin = list(l = 0, r = 0, b = 0, t = 0),
         showlegend = FALSE,
         font = list(family = "Palatino"))
save(p22, file = here::here("data_files/circle_graph.Rdata"))

1.3 Tadpole graph

Below we plot the tadpole graph \(\Gamma_T = (\mathcal{V}, \mathcal{E})\) with \(\mathcal{V} = \{v_1, v_2\}\) and \(\mathcal{E} = \{e_1, e_2\}\), where \(e_1\) is the edge connecting \(v_1\) and \(v_2\), and \(e_2\) is the edge connecting \(v_2\) to itself.

graph <- gets.graph.tadpole(h = 1/4)

notes_tad_1 <- list(annotations = list(
             list(
               x = 0, y = 0, z = 0,
               text = TeX("v_1"),
               textangle = 0, ax = 0, ay = 35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "gray", arrowsize = 1, arrowwidth = 0.5, arrowhead = 1),
             list(
               x = 0, y = 1, z = 0,
               text = TeX("v_2"),
               textangle = 0, ax = 0, ay = 35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "gray", arrowsize = 1, arrowwidth = 0.5, arrowhead = 1),
             list(
               x = 0, y = 0.5, z = 0,
               text = TeX("e_1"),
               textangle = 0, ax = 0, ay = 35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "rgba(0,0,0,0)"),
             list(
               x = 0, y = 1+2/pi, z = 0,
               text = TeX("e_2"),
               textangle = 0, ax = 0, ay = 35,
               font = list(color = "black", size = gfsize),
               arrowcolor = "rgba(0,0,0,0)")))

p33 <- graph$plot_function(rep(0,nrow(graph$mesh$V)), 
                           vertex_size = gsw,
                           type = "plotly", 
                           line_width = gsw, 
                           edge_width = gsw, 
                           line_color = "black") |>
  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 = y0,
    y = x0,
    z = z0,
    u = dy,
    v = dx,
    w = dz,
    sizemode = "absolute",
    sizeref = sizeref,
    showscale = FALSE,
    showlegend = FALSE,
    colorscale = list(c(0,1), c("green","green")),
    cmin = 0,
    cmax = 1
  ) |>
  layout(font = list(family = "Palatino"),
         scene =  c(notes_tad_1, layout_int),
         margin = list(l = 0, r = 0, b = 0, t = 0),
         showlegend = FALSE,
         font = list(family = "Palatino"))
save(p33, file = here::here("data_files/tadpole_graph.Rdata"))

2 Part 2

2.1 Functions on a graph and geodesic distance illustration

Below we plot the function \(f = \gamma\big|_\Gamma\) where \(\gamma(x,y) = e^{-x^2-y^2}\), together with a path on the graph that illustrates the geodesic distance between points \(s_1\) and \(s_2\).

graph_to_get_loc <- gets.graph.tadpole(h = 1/40)
loc <- graph_to_get_loc$get_mesh_locations()

A <- as.matrix(graph$fem_basis(loc))
A <- apply(A, 2, function(x) plotting.order(x, graph_to_get_loc))
A <- rbind(A, rep(NA, ncol(A))) # Add a row of NAs for the plotting

x_g <- graph_to_get_loc$mesh$V[, 1]
y_g <- graph_to_get_loc$mesh$V[, 2]
f_g <- exp(-x_g^2 - y_g^2)


x <- c(plotting.order(x_g, graph_to_get_loc), NA)
y <- c(plotting.order(y_g, graph_to_get_loc), NA)
f <- c(plotting.order(f_g, graph_to_get_loc), NA)


x_range <- range(x, na.rm = TRUE)*1.001
y_range <- range(y, na.rm = TRUE)*1.001
z_range <- c(0,1)

notes1 <- list(annotations = list(
                   list(
                     x = 0, y = 0, z = 0,
                     text = TeX("v_1"),
                     textangle = 0, ax = 0, ay = 35,
                     font = list(color = "black", size = gfsize),
                     arrowcolor = "gray", arrowsize = 1, arrowwidth = 0.5, arrowhead = 1),
                   list(
                     x = 1, y = 0, z = 0,
                     text = TeX("v_2"),
                     textangle = 0, ax = 0, ay = 35,
                     font = list(color = "black", size = gfsize),
                     arrowcolor = "gray", arrowsize = 1, arrowwidth = 0.5, arrowhead = 1),
                   list(
                     x = 0.5, y = 0, z = 0,
                     text = TeX("e_1"),
                     textangle = 0, ax = 0, ay = 35,
                     font = list(color = "black", size = gfsize),
                     arrowcolor = "rgba(0,0,0,0)"),
                   list(
                     x = 1+2/pi, y = 0, z = 0,
                     text = TeX("e_2"),
                     textangle = 0, ax = 0, ay = 35,
                     font = list(color = "black", size = gfsize),
                     arrowcolor = "rgba(0,0,0,0)"),
                   list(
                     x = 0.25, y = 0, z = 0,
                     text = TeX("s_1"),
                     textangle = 0, ax = 0, ay = 35,
                     font = list(color = "black", size = gfsize),
                     arrowcolor = "gray", arrowsize = 1, arrowwidth = 0.5, arrowhead = 1),
                   list(
                     x = 1+1/pi, y = 1/pi, z = 0,
                     text = TeX("s_2"),
                     textangle = 0, ax = 0, ay = 35,
                     font = list(color = "black", size = gfsize),
                     arrowcolor = "gray", arrowsize = 1, arrowwidth = 0.5, arrowhead = 1),
                   list(
                     x = 0.25, y = 0, z = 1,
                     text = TeX("f=\\gamma\\big|_\\Gamma"),
                     textangle = 0, ax = -40, ay = 0,
                     font = list(color = "rgb(0,0,200)", size = gfsize),
                     arrowcolor = "rgba(0,0,0,0)")))
    
# to plot the path 
theta <- seq(from = pi/2, to = pi, length.out = 100)
SSTHETA <- seq(from = -pi, to = pi/2, length.out = 300)
SS1 <- data.frame(x = c(0, 0.25), 
                  y = c(0,0), 
                  z = c(0,0))
SS2 <- data.frame(x = 1+1/pi+cos(SSTHETA)/pi, 
                  y = sin(SSTHETA)/pi, 
                  z = rep(0, length(SSTHETA)))
df1 <- data.frame(x = c(0.25,1), 
                  y = c(0,0), 
                  z = c(0,0))
df2 <- data.frame(x = 1+1/pi+cos(theta)/pi, 
                  y = sin(theta)/pi, 
                  z = rep(0, length(theta)))

p1 <- plot_ly() |> 
  config(mathjax = 'cdn') |>
  add_trace(x = 0, y = 0, z = 0, mode = "markers", type = "scatter3d", 
            marker = list(size = gsw, color = "black", symbol = 104),
            showlegend = FALSE) |>
  add_trace(x = 1, y = 0, z = 0, mode = "markers", type = "scatter3d", 
            marker = list(size = gsw, color = "black", symbol = 104),
            showlegend = FALSE) |>
  add_trace(x = 0.25, y = 0, z = 0, mode = "markers", type = "scatter3d", 
            marker = list(size = gsw, color = "red", symbol = 104),
            showlegend = FALSE) |> 
  add_trace(x = 1+1/pi, y = 1/pi, z = 0, mode = "markers", type = "scatter3d", 
            marker = list(size = gsw, color = "red", symbol = 104),
            showlegend = FALSE) |> 
  add_trace(data = SS1, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
            line = list(width = gsw, color = "black", symbol = 104),
            showlegend = FALSE) |>
  add_trace(data = SS2, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
            line = list(width = gsw, color = "black", symbol = 104),
            showlegend = FALSE) |>
  add_trace(data = df1, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
            line = list(width = gsw, color = "red", symbol = 104),
            showlegend = FALSE) |>
  add_trace(data = df2, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
            line = list(width = gsw, color = "red", symbol = 104),
            showlegend = FALSE) |>
  add_trace(x = x, 
            y = y, 
            z = f, 
            type = "scatter3d",
            mode = "lines",  
            line = list(color = "rgb(0,0,200)", width = gsw),
            showlegend = FALSE) |>
  add_trace(x = rep(x, each = 3), 
            y = rep(y, each = 3), 
            z = unlist(lapply(f, function(zj) c(0, zj, NA))),
            type = "scatter3d", 
            mode = "lines",
            line = list(color = "lightgray", width = 0.5),
            showlegend = FALSE) |>
  layout(font = list(family = "Palatino"),
         scene = c(tadpole.layout(x_range, y_range, z_range),notes1),
         margin = list(l = 0, r = 0, b = 0, t = 0),
         showlegend = FALSE,
         font = list(family = "Palatino"))
save(p1, file = here::here("data_files/tadpole_graph_function.Rdata"))

2.2 Mesh

Below we plot the mesh of the tadpole graph.

df3 <- data.frame(x = graph$mesh$V[, 1], 
                  y = graph$mesh$V[, 2], 
                  z = rep(0, length(graph$mesh$V[, 1])))
df4 <- df3[6:nrow(df3), ]


notes2 <- list(annotations = list(
                   list(
                     x = 0, y = 0, z = 0,
                     text = TeX("v_1"),
                     textangle = 0, ax = 0, ay = -35,
                     font = list(color = "black", size = gfsize),
                     arrowcolor = "gray", arrowsize = 1, arrowwidth = 0.5, arrowhead = 1),
                   list(
                     x = 1, y = 0, z = 0,
                     text = TeX("v_2"),
                     textangle = 0, ax = 0, ay = -35,
                     font = list(color = "black", size = gfsize),
                     arrowcolor = "gray", arrowsize = 1, arrowwidth = 0.5, arrowhead = 1),
                   list(
                     x = df4[1,1], y = df4[1,2], z = 0,
                     text = TeX("x_1^{e_2}"),
                     textangle = 0, ax = 0, ay = -35,
                     font = list(color = "black", size = gfsize),
                     arrowcolor = "gray", arrowsize = 1, arrowwidth = 0.5, arrowhead = 1),
                   list(
                     x = df4[2,1], y = df4[2,2], z = 0,
                     text = TeX("x_2^{e_2}"),
                     textangle = 0, ax = 0, ay = -35,
                     font = list(color = "black", size = gfsize),
                     arrowcolor = "gray", arrowsize = 1, arrowwidth = 0.5, arrowhead = 1),
                   list(
                     x = df4[3,1], y = df4[3,2], z = 0,
                     text = TeX("x_3^{e_2}"),
                     textangle = 0, ax = 0, ay = -35,
                     font = list(color = "black", size = gfsize),
                     arrowcolor = "gray", arrowsize = 1, arrowwidth = 0.5, arrowhead = 1),
                   list(
                     x = df4[4,1], y = df4[4,2], z = 0,
                     text = TeX("x_4^{e_2}"),
                     textangle = 0, ax = 0, ay = -35,
                     font = list(color = "black", size = gfsize),
                     arrowcolor = "gray", arrowsize = 1, arrowwidth = 0.5, arrowhead = 1),
                   list(
                     x = df4[5,1], y = df4[5,2], z = 0,
                     text = TeX("x_5^{e_2}"),
                     textangle = 0, ax = 0, ay = -35,
                     font = list(color = "black", size = gfsize),
                     arrowcolor = "gray", arrowsize = 1, arrowwidth = 0.5, arrowhead = 1),
                   list(
                     x = df4[6,1], y = df4[6,2], z = 0,
                     text = TeX("x_6^{e_2}"),
                     textangle = 0, ax = 0, ay = -35,
                     font = list(color = "black", size = gfsize),
                     arrowcolor = "gray", arrowsize = 1, arrowwidth = 0.5, arrowhead = 1),
                   list(
                     x = df4[7,1], y = df4[7,2], z = 0,
                     text = TeX("x_7^{e_2}"),
                     textangle = 0, ax = 0, ay = -35,
                     font = list(color = "black", size = gfsize),
                     arrowcolor = "gray", arrowsize = 1, arrowwidth = 0.5, arrowhead = 1),
                   list(
                     x = 0.25, y = 0, z = 0,
                     text = TeX("x_1^{e_1}"),
                     textangle = 0, ax = 0, ay = -35,
                     font = list(color = "black", size = gfsize),
                     arrowcolor = "gray", arrowsize = 1, arrowwidth = 0.5, arrowhead = 1),
                   list(
                     x = 0.5, y = 0, z = 0,
                     text = TeX("x_2^{e_1}"),
                     textangle = 0, ax = 0, ay = -35,
                     font = list(color = "black", size = gfsize),
                     arrowcolor = "gray", arrowsize = 1, arrowwidth = 0.5, arrowhead = 1),
                   list(
                     x = 0.75, y = 0, z = 0,
                     text = TeX("x_3^{e_1}"),
                     textangle = 0, ax = 0, ay = -35,
                     font = list(color = "black", size = gfsize),
                     arrowcolor = "gray", arrowsize = 1, arrowwidth = 0.5, arrowhead = 1)))

p2 <- plot_ly() |> 
  config(mathjax = 'cdn') |>
  add_trace(x = x, 
            y = y, 
            z = x*0, 
            type = "scatter3d",
            mode = "lines",  
            line = list(color = "black", width = gsw),
            showlegend = FALSE) |>
  add_trace(data = df3, x = ~x, y = ~y, z = ~z, mode = "markers", type = "scatter3d", 
            marker = list(size = gsw, color = "gray", symbol = 104)) |>
  add_trace(x = 0, y = 0, z = 0, mode = "markers", type = "scatter3d", 
            marker = list(size = gsw, color = "black", symbol = 104),
            showlegend = FALSE) |>
  add_trace(x = 1, y = 0, z = 0, mode = "markers", type = "scatter3d", 
            marker = list(size = gsw, color = "black", symbol = 104),
            showlegend = FALSE) |>
  layout(font = list(family = "Palatino"),
         scene = c(tadpole.layout(x_range, y_range, z_range), notes2),
         margin = list(l = 0, r = 0, b = 0, t = 0),
         showlegend = FALSE,
         font = list(family = "Palatino"))
save(p2, file = here::here("data_files/tadpole_graph_mesh.Rdata"))

2.3 Basis functions on a metric graph

Let each edge \(e\in\mathcal{E}\) be subdivided into \(n_{e}\geq 2\) regular segments of length \(h_{e}\), and be delimited by the nodes \(0 = x_0^{e},x_1^{e},\dots,x_{n_{e}-1}^{e}, x_{n_{e}}^{e} = \ell_{e}\). For each \(j = 1,\dots,n_{e}-1\), we consider the following standard hat basis functions \[\begin{equation*} \varphi_j^{e}(x)=\begin{cases} 1-\dfrac{|x_j^{e}-x|}{h_{e}},&\text{ if }x_{j-1}^{e}\leq x\leq x_{j+1}^{e},\\ 0,&\text{ otherwise}. \end{cases} \end{equation*}\] For each \(e\in\mathcal{E}\), the set of hat functions \(\left\{\varphi_1^{e},\dots,\varphi_{n_{e}-1}^{e}\right\}\) is a basis for the space \[\begin{equation*} V_{h_{e}} = \left\{w\in H_0^1(e)\;\Big|\;\forall j = 0,1,\dots,n_{e}-1:w|_{[x_j^{e}, x_{j+1}^{e}]}\in\mathbb{P}^1\right\}, \end{equation*}\] where \(\mathbb{P}^1\) is the space of linear functions on \([0,\ell_{e}]\). For each vertex \(v\in\mathcal{V}\), we define \[\begin{equation*} \mathcal{N}_v = \left\{\bigcup_{e\in\left\{e\in\mathcal{E}_v: v = x_0^e\right\}}[v,x_1^e]\right\}\bigcup\left\{\bigcup_{e\in\left\{e\in\mathcal{E}_v: v = x^e_{n_e}\right\}}[x^e_{n_e-1},v]\right\}, \end{equation*}\] which is a star-shaped set with center at \(v\) and rays made of the segments contiguous to \(v\). On \(\mathcal{N}_v\), we define the hat functions as \[\begin{equation*} \phi_v(x)=\begin{cases} 1-\dfrac{|x_v^{e}-x|}{h_{e}},&\text{ if }x\in\mathcal{N}_v\cap e \text{ and }e\in\mathcal{E}_v,\\ 0,&\text{ otherwise}, \end{cases} \end{equation*}\] where \(x_v^e\) is either \(x_0^e\) or \(x_{n_e}^e\) depending on the edge direction and its parameterization. See (Arioli and Benzi 2018) for more. Figure 3 below provides an illustration of the system of basis functions \(\{\varphi_j^e, \phi_v\}\) together with the set \(\mathcal{N}_v\).

notes3 <- list(annotations = list(
                   list(
                     x = 0, y = 0, z = 1,
                     text = TeX("\\phi_{v_1}"),
                     textangle = 0, ax = 0, ay = -15,
                     font = list(color = "red", size = gfsize),
                     arrowcolor = "rgba(0,0,0,0)"),
                   list(
                     x = 0, y = 0, z = 0,
                     text = TeX("\\mathcal{N}_{v_1}"),
                     textangle = 0, ax = 25, ay = 5,
                     font = list(color = "green", size = gfsize),
                     arrowcolor = "rgba(0,0,0,0)"),
                   list(
                     x = 1, y = 0, z = 1,
                     text = TeX("\\phi_{v_2}"),
                     textangle = 0, ax = 0, ay = -15,
                     font = list(color = "red", size = gfsize),
                     arrowcolor = "rgba(0,0,0,0)"),
                   list(
                     x = 0, y = 0, z = 0,
                     text = TeX("v_1"),
                     textangle = 0, ax = 0, ay = 35,
                     font = list(color = "black", size = gfsize),
                     arrowcolor = "gray", arrowsize = 1, arrowwidth = 0.5, arrowhead = 1),
                   list(
                     x = 1, y = 0, z = 0,
                     text = TeX("v_2"),
                     textangle = 0, ax = 0, ay = 35,
                     font = list(color = "black", size = gfsize),
                     arrowcolor = "gray", arrowsize = 1, arrowwidth = 0.5, arrowhead = 1),
                   list(
                     x = 1, y = 0, z = 0,
                     text = TeX("\\mathcal{N}_{v_2}"),
                     textangle = 0, ax = 40, ay = -10,
                     font = list(color = "green", size = gfsize),
                     arrowcolor = "rgba(0,0,0,0)"),
                   list(
                     x = df4[5,1], y = df4[5,2], z = 0,
                     text = TeX("x_5^{e_2}"),
                     textangle = 0, ax = 0, ay = 35,
                     font = list(color = "black", size = gfsize),
                     arrowcolor = "gray", arrowsize = 1, arrowwidth = 0.5, arrowhead = 1),
                   list(
                     x = df4[5,1], y = df4[5,2], z = 1,
                     text = TeX("\\varphi_5^{e_2}"),
                     textangle = 0, ax = 0, ay = -15,
                     font = list(color = "rgb(0,0,200)", size = gfsize),
                     arrowcolor = "rgba(0,0,0,0)")))

p3 <- plot_ly(
  #width = 1500, height = 2000
         ) |> 
  config(mathjax = 'cdn') |>
  add_trace(x = rep(x, times = graph$nV), 
            y = rep(y, times = graph$nV), 
            z = as.vector(A[, 1:graph$nV]), 
            type = "scatter3d", 
            mode = "lines",
            line = list(color = "red", width = gsw), 
            showlegend = FALSE) |>
  add_trace(x = rep(x, times = ncol(A) - graph$nV), 
            y = rep(y, times = ncol(A) - graph$nV), 
            z = as.vector(A[, (graph$nV+1):ncol(A)]), 
            type = "scatter3d",
            mode = "lines", 
            line = list(color = "darkgray", width = gsw), 
            showlegend = FALSE) |>
  add_trace(x = rep(x, each = 3), 
            y = rep(y, each = 3), 
            z = unlist(lapply(apply(A, 1, max, na.rm = TRUE), function(zj) c(0, zj, NA))),
            type = "scatter3d", 
            mode = "lines",
            line = list(color = "lightgray", width = 0.5),
            showlegend = FALSE) |>
  add_trace(x = x, 
            y = y, 
            z = as.vector(A[, 10]), 
            type = "scatter3d",
            mode = "lines",  
            line = list(color = "rgb(0,0,200)", width = gsw),
            showlegend = FALSE) |>
  add_trace(x = x, 
            y = y, 
            z = x*0, 
            type = "scatter3d",
            mode = "lines",  
            line = list(color = "black", width = gsw),
            showlegend = FALSE) |>
  add_trace(x = rep(x, times = graph$nV), 
            y = rep(y, times = graph$nV), 
            z = c(replace(rep(NA, nrow(A)), 1:11, 0), 
                  replace(rep(NA, nrow(A)), c(31:51, 111:121), 0)), 
            type = "scatter3d", 
            mode = "lines",
            line = list(color = "green", width = gsw), 
            showlegend = FALSE) |>
  add_trace(x = df4[5,1], y = df4[5,2], z = 0, mode = "markers", type = "scatter3d", 
            marker = list(size = gsw, color = "rgb(0,0,200)", symbol = 104),
            showlegend = FALSE) |>
  add_trace(x = 0, y = 0, z = 0, mode = "markers", type = "scatter3d", 
            marker = list(size = gsw, color = "black", symbol = 104),
            showlegend = FALSE) |>
  add_trace(x = 1, y = 0, z = 0, mode = "markers", type = "scatter3d", 
            marker = list(size = gsw, color = "black", symbol = 104),
            showlegend = FALSE) |>
  layout(font = list(family = "Palatino"),
         scene = c(tadpole.layout(x_range, y_range, z_range), notes3),
         margin = list(l = 0, r = 0, b = 0, t = 0),
  paper_bgcolor = "white",  # the overall canvas background
  plot_bgcolor  = "white",   # the area inside axes
         showlegend = FALSE,
         font = list(family = "Palatino"))
save(p3, file = here::here("data_files/tadpole_graph_basis_functions.Rdata"))

# combine_plotly_grid_pdf(list(p1,p2,p3), output_pdf = here::here("data_files/plotlypic/tadpole_graph_basis_functions.pdf"), ncol = 3)
# combine_plotly_pdf_single(p3, here::here("data_files/plotlypic/p3.pdf"))

3 Plots part 1

load(here::here("data_files/interval_graph.Rdata"))
p11

Figure 1: Interval graph \(\Gamma_I = (\mathcal{V}, \mathcal{E})\) with \(\mathcal{V} = \{v_1, v_2\}\) and \(\mathcal{E} = \{e_1\}\).

load(here::here("data_files/circle_graph.Rdata"))
p22

Figure 2: Circle graph \(\Gamma_C = (\mathcal{V}, \mathcal{E})\) with \(\mathcal{V} = \{v_1\}\) and \(\mathcal{E} = \{e_1\}\).

load(here::here("data_files/tadpole_graph.Rdata"))
p33

Figure 3: Tadpole graph \(\Gamma_T = (\mathcal{V}, \mathcal{E})\) with \(\mathcal{V} = \{v_1, v_2\}\) and \(\mathcal{E} = \{e_1,e_2\}\).

4 Plots part 2

load(here::here("data_files/tadpole_graph_function.Rdata"))
p1

Figure 4: A function \(f=\gamma\big|_\Gamma\) on the tadpole graph, where \(\gamma(x,y) = e^{-x^2-y^2}\), together with a path that illustrates the geodesic distance between points \(s_1\) and \(s_2\).

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

Figure 5: Mesh nodes on the tadpole graph.

load(here::here("data_files/tadpole_graph_basis_functions.Rdata"))
p3

Figure 6: Basis functions on the tadpole graph.

5 Plots part 3

load(here::here("data_files/interval_graph.Rdata"))
p11

Figure 7: Interval graph \(\Gamma_I = (\mathcal{V}, \mathcal{E})\) with \(\mathcal{V} = \{v_1, v_2\}\) and \(\mathcal{E} = \{e_1\}\).

load(here::here("data_files/interval_graph_flip_edge.Rdata"))
p2int

Figure 8: Interval graph \(\Gamma_I = (\mathcal{V}, \mathcal{E})\) with \(\mathcal{V} = \{v_1, v_2\}\) and \(\mathcal{E} = \{e_1\}\), where \(e_1\) is flipped.

6 References

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