Basis functions, alternative illustrations

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Let us set some global options for all code chunks in this document.

# Set seed for reproducibility
set.seed(1982) 
# Set global options for all code chunks
knitr::opts_chunk$set(
  # Disable messages printed by R code chunks
  message = FALSE,    
  # Disable warnings printed by R code chunks
  warning = FALSE,    
  # Show R code within code chunks in output
  echo = TRUE,        
  # Include both R code and its results in output
  include = TRUE,     
  # Evaluate R code chunks
  eval = TRUE,       
  # Enable caching of R code chunks for faster rendering
  cache = FALSE,      
  # Align figures in the center of the output
  fig.align = "center",
  # Enable retina display for high-resolution figures
  retina = 2,
  # Show errors in the output instead of stopping rendering
  error = TRUE,
  # Do not collapse code and output into a single block
  collapse = FALSE
)
# Start the figure counter
fig_count <- 0
# Define the captioner function
captioner <- function(caption) {
  fig_count <<- fig_count + 1
  paste0("Figure ", fig_count, ": ", caption)
}

Import libraries

library(MetricGraph)
library(dplyr)
library(plotly)

library(here)
library(rmarkdown)
library(grateful) # to cite

By hand

---

Press the Show button below to reveal the code.

---

V <- rbind(c(0, 0), c(1, 0), c(1, 1), c(0, 1), c(-1, 1), c(-1, 0), c(0, -1))
E <- rbind(c(1, 2), c(2, 3), c(3, 4), c(4, 5),
           c(5, 6), c(6, 1), c(4, 1),c(1, 7))
graph <- metric_graph$new(V = V, E = E)

#vertical 
df01 = data.frame(x = c(-1,-1), y = c(0,0), z = c(0,1))
df02 = data.frame(x = c(-2/3,-2/3), y = c(0,0), z = c(0,1))
df03 = data.frame(x = c(-1/3,-1/3), y = c(0,0), z = c(0,1))
df04 = data.frame(x = c(0,0), y = c(0,0), z = c(0,1))
df05 = data.frame(x = c(0,0), y = c(-1/3,-1/3), z = c(0,1))
df06 = data.frame(x = c(0,0), y = c(-2/3,-2/3), z = c(0,1))
df07 = data.frame(x = c(0,0), y = c(-1,-1), z = c(0,1))
df08 = data.frame(x = c(2/3,2/3), y = c(-1,-1), z = c(0,1))
df09 = data.frame(x = c(1/3,1/3), y = c(-1,-1), z = c(0,1))
df10 = data.frame(x = c(1,1), y = c(-1,-1), z = c(0,1))
df11 = data.frame(x = c(2/3,2/3), y = c(0,0), z = c(0,1))
df12 = data.frame(x = c(1/3,1/3), y = c(0,0), z = c(0,1))
df13 = data.frame(x = c(1,1), y = c(0,0), z = c(0,1))
df14 = data.frame(x = c(2/3,2/3), y = c(1,1), z = c(0,1))
df15 = data.frame(x = c(1/3,1/3), y = c(1,1), z = c(0,1))
df16 = data.frame(x = c(1,1), y = c(1,1), z = c(0,1))
df17 = data.frame(x = c(1,1), y = c(-1/3,-1/3), z = c(0,1))
df18 = data.frame(x = c(1,1), y = c(-2/3,-2/3), z = c(0,1))
df19 = data.frame(x = c(1,1), y = c(1/3,1/3), z = c(0,1))
df20 = data.frame(x = c(1,1), y = c(2/3,2/3), z = c(0,1))
df21 = data.frame(x = c(0,0), y = c(1/3,1/3), z = c(0,1))
df22 = data.frame(x = c(0,0), y = c(2/3,2/3), z = c(0,1))
df23 = data.frame(x = c(0,0), y = c(1,1), z = c(0,1))
#inclined
d01 = data.frame(x = c(-1,-2/3), y = c(0,0), z = c(0,1))
d02 = data.frame(x = c(-2/3,-1/3), y = c(0,0), z = c(0,1))
d03 = data.frame(x = c(-1/3,0), y = c(0,0), z = c(0,1)) #central
d04 = data.frame(x = c(0,1/3), y = c(0,0), z = c(0,1))
d05 = data.frame(x = c(1/3,2/3), y = c(0,0), z = c(0,1))
d06 = data.frame(x = c(2/3,1), y = c(0,0), z = c(0,1))
d07 = data.frame(x = c(1,2/3), y = c(0,0), z = c(0,1))
d08 = data.frame(x = c(2/3,1/3), y = c(0,0), z = c(0,1))
d09 = data.frame(x = c(1/3,0), y = c(0,0), z = c(0,1)) #central
d10 = data.frame(x = c(0,-1/3), y = c(0,0), z = c(0,1))
d11 = data.frame(x = c(-1/3,-2/3), y = c(0,0), z = c(0,1))
d12 = data.frame(x = c(-2/3,-1), y = c(0,0), z = c(0,1))
d13 = data.frame(x = c(0,1/3), y = c(1,1), z = c(0,1))
d14 = data.frame(x = c(1/3,2/3), y = c(1,1), z = c(0,1))#basis
d15 = data.frame(x = c(2/3,1), y = c(1,1), z = c(0,1))
d16 = data.frame(x = c(1,2/3), y = c(1,1), z = c(0,1))#basis
d17 = data.frame(x = c(2/3,1/3), y = c(1,1), z = c(0,1))
d18 = data.frame(x = c(1/3,0), y = c(1,1), z = c(0,1))
d19 = data.frame(x = c(0,1/3), y = c(-1,-1), z = c(0,1))
d20 = data.frame(x = c(1/3,2/3), y = c(-1,-1), z = c(0,1))
d21 = data.frame(x = c(2/3,1), y = c(-1,-1), z = c(0,1))
d22 = data.frame(x = c(1,2/3), y = c(-1,-1), z = c(0,1))
d23 = data.frame(x = c(2/3,1/3), y = c(-1,-1), z = c(0,1))
d24 = data.frame(x = c(1/3,0), y = c(-1,-1), z = c(0,1))
d25 = data.frame(y = c(-1,-2/3), x = c(0,0), z = c(0,1))
d26 = data.frame(y = c(-2/3,-1/3), x = c(0,0), z = c(0,1))
d27 = data.frame(y = c(-1/3,0), x = c(0,0), z = c(0,1)) #central
d28 = data.frame(y = c(0,1/3), x = c(0,0), z = c(0,1))
d29 = data.frame(y = c(1/3,2/3), x = c(0,0), z = c(0,1))
d30 = data.frame(y = c(2/3,1), x = c(0,0), z = c(0,1))
d31 = data.frame(y = c(1,2/3), x = c(0,0), z = c(0,1))
d32 = data.frame(y = c(2/3,1/3), x = c(0,0), z = c(0,1))
d33 = data.frame(y = c(1/3,0), x = c(0,0), z = c(0,1)) #central
d34 = data.frame(y = c(0,-1/3), x = c(0,0), z = c(0,1))
d35 = data.frame(y = c(-1/3,-2/3), x = c(0,0), z = c(0,1))
d36 = data.frame(y = c(-2/3,-1), x = c(0,0), z = c(0,1))
d37 = data.frame(y = c(-1,-2/3), x = c(1,1), z = c(0,1))
d38 = data.frame(y = c(-2/3,-1/3), x = c(1,1), z = c(0,1))
d39 = data.frame(y = c(-1/3,0), x = c(1,1), z = c(0,1))
d40 = data.frame(y = c(0,1/3), x = c(1,1), z = c(0,1))
d41 = data.frame(y = c(1/3,2/3), x = c(1,1), z = c(0,1))
d42 = data.frame(y = c(2/3,1), x = c(1,1), z = c(0,1))
d43 = data.frame(y = c(1,2/3), x = c(1,1), z = c(0,1))
d44 = data.frame(y = c(2/3,1/3), x = c(1,1), z = c(0,1))
d45 = data.frame(y = c(1/3,0), x = c(1,1), z = c(0,1))
d46 = data.frame(y = c(0,-1/3), x = c(1,1), z = c(0,1))
d47 = data.frame(y = c(-1/3,-2/3), x = c(1,1), z = c(0,1))
d48 = data.frame(y = c(-2/3,-1), x = c(1,1), z = c(0,1))
# star set
d49 = data.frame(y = c(1/3,0), x = c(0,0), z = c(0,0))
d50 = data.frame(y = c(0,0), x = c(1/3,0), z = c(0,0))
d51 = data.frame(y = c(-1/3,0), x = c(0,0), z = c(0,0))
d52 = data.frame(y = c(0,0), x = c(-1/3,0), z = c(0,0))

graph$build_mesh(h = 1/40)
graph$plot(plotly = TRUE, vertex_size = 4, vertex_color = "blue",
           edge_color = "black", edge_width = 2) %>%  
  config(mathjax = 'cdn') %>% 
  layout(title = TeX("\\text{Basis functions on a metric graph }\\Gamma = (\\mathcal{V},\\mathcal{E})"), 
         showlegend = FALSE,
                    font = list(family = "Palatino"),
                    scene = list(
                      aspectratio = list(x = 1.8, y = 1.8, z = 0.4),
                      annotations = list(
                        list(
                          x = -1, y = 0, z = 0,
                          text = TeX("v_1"),
                          textangle = 0, ax = 0, ay = 25,
                          font = list(color = "black", size = 16),
                          arrowcolor = "black", arrowsize = 1, arrowwidth = 1,arrowhead = 1),
                        list(
                          x = 0, y = 0, z = 0,
                          text = TeX("v_2"),
                          textangle = 0, ax = 0, ay = 25,
                          font = list(color = "black", size = 16),
                          arrowcolor = "black", arrowsize = 1, arrowwidth = 1,arrowhead = 1),
                        list(
                          x = 0, y = -1, z = 0,
                          text = TeX("v_3"),
                          textangle = 0, ax = 0, ay = 25,
                          font = list(color = "black", size = 16),
                          arrowcolor = "black", arrowsize = 1, arrowwidth = 1,arrowhead = 1),
                        list(
                          x = 1, y = -1, z = 0,
                          text = TeX("v_4"),
                          textangle = 0, ax = 0, ay = 25,
                          font = list(color = "black", size = 16),
                          arrowcolor = "black", arrowsize = 1, arrowwidth = 1,arrowhead = 1),
                        list(
                          x = 1, y = 0, z = 0,
                          text = TeX("v_5"),
                          textangle = 0, ax = 0, ay = 25,
                          font = list(color = "black", size = 16),
                          arrowcolor = "black", arrowsize = 1, arrowwidth = 1,arrowhead = 1),
                        list(
                          x = 1, y = 1, z = 0,
                          text = TeX("v_6,x_0^{e_7}"),
                          textangle = 0, ax = 0, ay = 25,
                          font = list(color = "black", size = 16),
                          arrowcolor = "black", arrowsize = 1, arrowwidth = 1,arrowhead = 1),
                        list(
                          x = 0, y = 1, z = 0,
                          text = TeX("v_7,x_3^{e_7}"),
                          textangle = 0, ax = 0, ay = 25,
                          font = list(color = "black", size = 16),
                          arrowcolor = "black", arrowsize = 1, arrowwidth = 1,arrowhead = 1),
                        list(
                          x = 2/3, y = 1, z = 0,
                          text = TeX("x_1^{e_7}"),
                          textangle = 0, ax = 0, ay = 25,
                          font = list(color = "black", size = 16),
                          arrowcolor = "black", arrowsize = 1, arrowwidth = 1,arrowhead = 1),
                        list(
                          x = 1/3, y = 1, z = 0,
                          text = TeX("x_2^{e_7}"),
                          textangle = 0, ax = 0, ay = 25,
                          font = list(color = "black", size = 16),
                          arrowcolor = "black", arrowsize = 1, arrowwidth = 1,arrowhead = 1),
                        list(
                          x = 0.15, y = 0, z = 0,
                          text = TeX("\\mathcal{N}_{v_2}"),
                          textangle = 0, ax = 0, ay = 25,
                          font = list(color = "green", size = 16),
                          arrowcolor = "white", arrowsize = 1, arrowwidth = 1,arrowhead = 1),
                        list(
                          x = 2/3, y = 1, z = 1,
                          text = TeX("\\varphi_1^{e_7}"),
                          textangle = 0, ax = 0, ay = 75,
                          font = list(color = "blue", size = 16),
                          arrowcolor = "white", arrowsize = 1, arrowwidth = 1,arrowhead = 1),
                        list(
                          x = 0, y = 0, z = 1,
                          text = TeX("\\phi_{v_2}"),
                          textangle = 0, ax = 0, ay = 75,
                          font = list(color = "red", size = 16),
                          arrowcolor = "white", arrowsize = 1, arrowwidth = 1,arrowhead = 1)
                        ))) %>% 
  add_trace(x = -2/3, y = 0, z = 0, mode = "markers", type = "scatter3d", 
            marker = list(size = 4, color = "black", symbol = 104)) %>%
  add_trace(x = -1/3, y = 0, z = 0, mode = "markers", type = "scatter3d", 
            marker = list(size = 4, color = "black", symbol = 104)) %>%
  add_trace(x = 0, y = -1/3, z = 0, mode = "markers", type = "scatter3d", 
            marker = list(size = 4, color = "black", symbol = 104)) %>%
  add_trace(x = 0, y = -2/3, z = 0, mode = "markers", type = "scatter3d", 
            marker = list(size = 4, color = "black", symbol = 104)) %>%
  add_trace(x = 1/3, y = -1, z = 0, mode = "markers", type = "scatter3d", 
            marker = list(size = 4, color = "black", symbol = 104)) %>%
  add_trace(x = 2/3, y = -1, z = 0, mode = "markers", type = "scatter3d", 
            marker = list(size = 4, color = "black", symbol = 104)) %>%
  add_trace(x = 0, y = 1/3, z = 0, mode = "markers", type = "scatter3d", 
            marker = list(size = 4, color = "black", symbol = 104)) %>%
  add_trace(x = 0, y = 2/3, z = 0, mode = "markers", type = "scatter3d", 
            marker = list(size = 4, color = "black", symbol = 104)) %>% 
  add_trace(x = 1/3, y = 0, z = 0, mode = "markers", type = "scatter3d", 
            marker = list(size = 4, color = "black", symbol = 104)) %>%
  add_trace(x = 2/3, y = 0, z = 0, mode = "markers", type = "scatter3d", 
            marker = list(size = 4, color = "black", symbol = 104)) %>% 
  add_trace(x = 1/3, y = 1, z = 0, mode = "markers", type = "scatter3d", 
            marker = list(size = 4, color = "black", symbol = 104)) %>%
  add_trace(x = 2/3, y = 1, z = 0, mode = "markers", type = "scatter3d", 
            marker = list(size = 4, color = "black", symbol = 104)) %>% 
  add_trace(x = 1, y = 1/3, z = 0, mode = "markers", type = "scatter3d", 
            marker = list(size = 4, color = "black", symbol = 104)) %>%
  add_trace(x = 1, y = 2/3, z = 0, mode = "markers", type = "scatter3d", 
            marker = list(size = 4, color = "black", symbol = 104)) %>% 
  add_trace(x = 1, y = -1/3, z = 0, mode = "markers", type = "scatter3d", 
            marker = list(size = 4, color = "black", symbol = 104)) %>%
  add_trace(x = 1, y = -2/3, z = 0, mode = "markers", type = "scatter3d", 
            marker = list(size = 4, color = "black", symbol = 104)) %>% #so far the botton
  add_trace(x = -2/3, y = 0, z = 1, mode = "markers", type = "scatter3d", 
            marker = list(size = 1, color = "black", symbol = 104)) %>%
  add_trace(x = -1/3, y = 0, z = 1, mode = "markers", type = "scatter3d", 
            marker = list(size = 1, color = "black", symbol = 104)) %>%
  add_trace(x = 0, y = -1/3, z = 1, mode = "markers", type = "scatter3d", 
            marker = list(size = 1, color = "black", symbol = 104)) %>%
  add_trace(x = 0, y = -2/3, z = 1, mode = "markers", type = "scatter3d", 
            marker = list(size = 1, color = "black", symbol = 104)) %>%
  add_trace(x = 1/3, y = -1, z = 1, mode = "markers", type = "scatter3d", 
            marker = list(size = 1, color = "black", symbol = 104)) %>%
  add_trace(x = 2/3, y = -1, z = 1, mode = "markers", type = "scatter3d", 
            marker = list(size = 1, color = "black", symbol = 104)) %>%
  add_trace(x = 0, y = 1/3, z = 1, mode = "markers", type = "scatter3d", 
            marker = list(size = 1, color = "black", symbol = 104)) %>%
  add_trace(x = 0, y = 2/3, z = 1, mode = "markers", type = "scatter3d", 
            marker = list(size = 1, color = "black", symbol = 104)) %>% 
  add_trace(x = 1/3, y = 0, z = 1, mode = "markers", type = "scatter3d", 
            marker = list(size = 1, color = "black", symbol = 104)) %>%
  add_trace(x = 2/3, y = 0, z = 1, mode = "markers", type = "scatter3d", 
            marker = list(size = 1, color = "black", symbol = 104)) %>% 
  add_trace(x = 1/3, y = 1, z = 1, mode = "markers", type = "scatter3d", 
            marker = list(size = 1, color = "black", symbol = 104)) %>%
  add_trace(x = 2/3, y = 1, z = 1, mode = "markers", type = "scatter3d", 
            marker = list(size = 1, color = "black", symbol = 104)) %>% 
  add_trace(x = 1, y = 1/3, z = 1, mode = "markers", type = "scatter3d", 
            marker = list(size = 1, color = "black", symbol = 104)) %>%
  add_trace(x = 1, y = 2/3, z = 1, mode = "markers", type = "scatter3d", 
            marker = list(size = 1, color = "black", symbol = 104)) %>% 
  add_trace(x = 1, y = -1/3, z = 1, mode = "markers", type = "scatter3d", 
            marker = list(size = 1, color = "black", symbol = 104)) %>%
  add_trace(x = 1, y = -2/3, z = 1, mode = "markers", type = "scatter3d", 
            marker = list(size = 1, color = "gray", symbol = 104)) %>% # now the vextex above
  add_trace(x = -1, y = 0, z = 1, mode = "markers", type = "scatter3d", 
            marker = list(size = 1, color = "gray", symbol = 104)) %>% 
  add_trace(x = 0, y = 0, z = 1, mode = "markers", type = "scatter3d", 
            marker = list(size = 1, color = "gray", symbol = 104)) %>%
  add_trace(x = 0, y = 1, z = 1, mode = "markers", type = "scatter3d", 
            marker = list(size = 1, color = "gray", symbol = 104)) %>% 
  add_trace(x = 0, y = -1, z = 1, mode = "markers", type = "scatter3d", 
            marker = list(size = 1, color = "gray", symbol = 104)) %>%
  add_trace(x = 1, y = 0, z = 1, mode = "markers", type = "scatter3d", 
            marker = list(size = 1, color = "gray", symbol = 104)) %>% 
  add_trace(x = 1, y = 1, z = 1, mode = "markers", type = "scatter3d", 
            marker = list(size = 1, color = "gray", symbol = 104)) %>%
  add_trace(x = 1, y = -1, z = 1, mode = "markers", type = "scatter3d", 
            marker = list(size = 1, color = "gray", symbol = 104)) %>% 
  add_trace(data = df01, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
            line = list(width = 1, color = "gray", symbol = 104, dash = 'dot')) %>% 
  add_trace(data = df02, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
            line = list(width = 1, color = "gray", symbol = 104, dash = 'dot')) %>% 
  add_trace(data = df03, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
            line = list(width = 1, color = "gray", symbol = 104, dash = 'dot')) %>% 
  add_trace(data = df04, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
            line = list(width = 1, color = "gray", symbol = 104, dash = 'dot')) %>% 
  add_trace(data = df05, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
            line = list(width = 1, color = "gray", symbol = 104, dash = 'dot')) %>% 
  add_trace(data = df06, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
            line = list(width = 1, color = "gray", symbol = 104, dash = 'dot')) %>% 
  add_trace(data = df07, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
            line = list(width = 1, color = "gray", symbol = 104, dash = 'dot')) %>% 
  add_trace(data = df08, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
            line = list(width = 1, color = "gray", symbol = 104, dash = 'dot')) %>% 
  add_trace(data = df09, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
            line = list(width = 1, color = "gray", symbol = 104, dash = 'dot')) %>% 
  add_trace(data = df10, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
            line = list(width = 1, color = "gray", symbol = 104, dash = 'dot')) %>% 
  add_trace(data = df11, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
            line = list(width = 1, color = "gray", symbol = 104, dash = 'dot')) %>% 
  add_trace(data = df12, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
            line = list(width = 1, color = "gray", symbol = 104, dash = 'dot')) %>% 
  add_trace(data = df13, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
            line = list(width = 1, color = "gray", symbol = 104, dash = 'dot')) %>% 
  add_trace(data = df14, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
            line = list(width = 1, color = "gray", symbol = 104, dash = 'dot')) %>% 
  add_trace(data = df15, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
            line = list(width = 1, color = "gray", symbol = 104, dash = 'dot')) %>% 
  add_trace(data = df16, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
            line = list(width = 1, color = "gray", symbol = 104, dash = 'dot')) %>% 
  add_trace(data = df17, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
            line = list(width = 1, color = "gray", symbol = 104, dash = 'dot')) %>% 
  add_trace(data = df18, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
            line = list(width = 1, color = "gray", symbol = 104, dash = 'dot')) %>% 
  add_trace(data = df19, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
            line = list(width = 1, color = "gray", symbol = 104, dash = 'dot')) %>% 
  add_trace(data = df20, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
            line = list(width = 1, color = "gray", symbol = 104, dash = 'dot')) %>% 
  add_trace(data = df21, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
            line = list(width = 1, color = "gray", symbol = 104, dash = 'dot')) %>% 
  add_trace(data = df22, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
            line = list(width = 1, color = "gray", symbol = 104, dash = 'dot')) %>% 
  add_trace(data = df23, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
            line = list(width = 1, color = "gray", symbol = 104, dash = 'dot')) %>% #so far vertical
  add_trace(data = d01, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
            line = list(width = 1, color = "gray", symbol = 104)) %>% 
  add_trace(data = d02, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
            line = list(width = 1, color = "gray", symbol = 104)) %>% 
  add_trace(data = d03, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
            line = list(width = 2, color = "red", symbol = 104)) %>% 
  add_trace(data = d04, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
            line = list(width = 1, color = "gray", symbol = 104)) %>% 
  add_trace(data = d05, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
            line = list(width = 1, color = "gray", symbol = 104)) %>% 
  add_trace(data = d06, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
            line = list(width = 1, color = "gray", symbol = 104)) %>% 
  add_trace(data = d07, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
            line = list(width = 1, color = "gray", symbol = 104)) %>% 
  add_trace(data = d08, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
            line = list(width = 1, color = "gray", symbol = 104)) %>% 
  add_trace(data = d09, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
            line = list(width = 2, color = "red", symbol = 104)) %>% 
  add_trace(data = d10, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
            line = list(width = 1, color = "gray", symbol = 104)) %>% 
  add_trace(data = d11, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
            line = list(width = 1, color = "gray", symbol = 104)) %>% 
  add_trace(data = d12, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
            line = list(width = 1, color = "gray", symbol = 104)) %>% 
  add_trace(data = d13, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
            line = list(width = 1, color = "gray", symbol = 104)) %>% 
  add_trace(data = d14, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
            line = list(width = 2, color = "blue", symbol = 104)) %>% 
  add_trace(data = d15, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
            line = list(width = 1, color = "gray", symbol = 104)) %>% 
  add_trace(data = d16, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
            line = list(width = 2, color = "blue", symbol = 104)) %>% 
  add_trace(data = d17, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
            line = list(width = 1, color = "gray", symbol = 104)) %>% 
  add_trace(data = d18, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
            line = list(width = 1, color = "gray", symbol = 104)) %>% 
  add_trace(data = d19, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
            line = list(width = 1, color = "gray", symbol = 104)) %>% 
  add_trace(data = d20, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
            line = list(width = 1, color = "gray", symbol = 104)) %>% 
  add_trace(data = d21, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
            line = list(width = 1, color = "gray", symbol = 104)) %>% 
  add_trace(data = d22, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
            line = list(width = 1, color = "gray", symbol = 104)) %>% 
  add_trace(data = d23, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
            line = list(width = 1, color = "gray", symbol = 104)) %>% 
  add_trace(data = d24, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
            line = list(width = 1, color = "gray", symbol = 104)) %>% 
  add_trace(data = d25, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
            line = list(width = 1, color = "gray", symbol = 104)) %>% 
  add_trace(data = d26, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
            line = list(width = 1, color = "gray", symbol = 104)) %>% 
  add_trace(data = d27, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
            line = list(width = 2, color = "red", symbol = 104)) %>% 
  add_trace(data = d28, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
            line = list(width = 1, color = "gray", symbol = 104)) %>% 
  add_trace(data = d29, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
            line = list(width = 1, color = "gray", symbol = 104)) %>% 
  add_trace(data = d30, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
            line = list(width = 1, color = "gray", symbol = 104)) %>% 
  add_trace(data = d31, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
            line = list(width = 1, color = "gray", symbol = 104)) %>% 
  add_trace(data = d32, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
            line = list(width = 1, color = "gray", symbol = 104)) %>% 
  add_trace(data = d33, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
            line = list(width = 2, color = "red", symbol = 104)) %>% 
  add_trace(data = d34, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
            line = list(width = 1, color = "gray", symbol = 104)) %>% 
  add_trace(data = d35, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
            line = list(width = 1, color = "gray", symbol = 104)) %>% 
  add_trace(data = d36, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
            line = list(width = 1, color = "gray", symbol = 104)) %>% 
  add_trace(data = d37, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
            line = list(width = 1, color = "gray", symbol = 104)) %>% 
  add_trace(data = d38, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
            line = list(width = 1, color = "gray", symbol = 104)) %>% 
  add_trace(data = d39, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
            line = list(width = 1, color = "gray", symbol = 104)) %>% 
  add_trace(data = d40, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
            line = list(width = 1, color = "gray", symbol = 104)) %>% 
  add_trace(data = d41, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
            line = list(width = 1, color = "gray", symbol = 104)) %>% 
  add_trace(data = d42, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
            line = list(width = 1, color = "gray", symbol = 104)) %>% 
  add_trace(data = d43, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
            line = list(width = 1, color = "gray", symbol = 104)) %>% 
  add_trace(data = d44, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
            line = list(width = 1, color = "gray", symbol = 104)) %>% 
  add_trace(data = d45, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
            line = list(width = 1, color = "gray", symbol = 104)) %>% 
  add_trace(data = d46, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
            line = list(width = 1, color = "gray", symbol = 104)) %>% 
  add_trace(data = d47, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
            line = list(width = 1, color = "gray", symbol = 104)) %>% 
  add_trace(data = d48, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
            line = list(width = 1, color = "gray", symbol = 104)) %>% #star set
  add_trace(data = d49, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
            line = list(width = 3, color = "green", symbol = 104)) %>% 
  add_trace(data = d50, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
            line = list(width = 3, color = "green", symbol = 104)) %>% 
  add_trace(data = d51, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
            line = list(width = 3, color = "green", symbol = 104)) %>% 
  add_trace(data = d52, x = ~x, y = ~y, z = ~z, mode = "lines", type = "scatter3d",
            line = list(width = 3, color = "green", symbol = 104))

Figure 1: Illustration of the system of basis functions \(\{\varphi_j^e, \phi_v\}\) (solid gray lines) on the graph \(\Gamma= (\mathcal{V},\mathcal{E})\), where \(\mathcal{E}= \left\{e_1,\dots,e_8\right\}\) and \(\mathcal{V}= \left\{v_1,\dots,v_7\right\}\). Note that for all \(e_i\in\mathcal{E}\), \(n_{e_i} = 3\) and \(h_{e_i} = 1/3\). In particular, for edge \(e_7\), observe that the three regular segments are delimited by the nodes \(v_6 = x_0^{e_7}, x_1^{e_7}, x_2^{e_7}, x_3^{e_7} = v_7\). Corresponding to node \(x_1^{e_7}\), we have plotted the basis function \(\varphi_1^{e_7}\) in blue. Notice also that the set \(\mathcal{N}_{v_2}\) is depicted in green and its corresponding basis function \(\phi_{v_2}\) is shown in red.

Using projection matrix

graph_to_get_loc <- graph$clone()
graph_to_get_loc$build_mesh(h = 1/150)

graph_to_get_loc$plot(mesh = TRUE) + 
  ggtitle("Fine mesh") + 
  theme_minimal() + 
  theme(text = element_text(family = "Palatino"))

Figure 2: Fine mesh

loc <- graph_to_get_loc$get_mesh_locations()

graph$build_mesh(h = 1/3)

graph$plot(mesh = TRUE) + 
  ggtitle("Coarse mesh") + 
  theme_minimal() + 
  theme(text = element_text(family = "Palatino"))

Figure 3: Coarse mesh

A <- graph$fem_basis(loc)

DD <- data.frame(A = as.matrix(A)) %>% 
  rowwise() %>%
  mutate(max_value = max(c_across(everything()))) %>%
  cbind(edge_number = loc[, 1], distance_on_edge = loc[, 2], A.24 = 0)

graph$add_observations(data = DD, edge_number = "edge_number",
  distance_on_edge = "distance_on_edge", data_coords = "PtE", normalized = TRUE, clear_obs = TRUE)

---

Press the Show button below to reveal the code.

---

graph$plot(data = "max_value", vertex_size = 0, plotly = TRUE, edge_width = 0.1) %>%  
  config(mathjax = 'cdn') %>% 
  layout(showlegend = FALSE,
         font = list(family = "Palatino"),
         scene = list(aspectratio = list(x = 1.8, y = 1.8, z = 0.4)))

Figure 4: Basis functions on a metric graph \(\Gamma = (\mathcal{V},\mathcal{E})\)

graph$plot(data = "A.1", vertex_size = 0, plotly = TRUE, edge_width = 0.1) %>%  
  config(mathjax = 'cdn') %>% 
  layout(showlegend = FALSE,
         font = list(family = "Palatino"),
         scene = list(aspectratio = list(x = 1.8, y = 1.8, z = 0.4)))

Figure 5: One basis function on a metric graph \(\Gamma = (\mathcal{V},\mathcal{E})\)

graph_to_get_loc$plot_function(X = DD$max_value, vertex_size = 0, plotly = TRUE) %>%  
  config(mathjax = 'cdn') %>% 
  layout(showlegend = FALSE,
         font = list(family = "Palatino"),
         scene = list(aspectratio = list(x = 1.8, y = 1.8, z = 0.4)))

Figure 6: Basis functions on a metric graph \(\Gamma = (\mathcal{V},\mathcal{E})\)

graph_to_get_loc$plot_function(X = DD$A.1, vertex_size = 0, plotly = TRUE) %>%  
  config(mathjax = 'cdn') %>% 
  layout(showlegend = FALSE,
         font = list(family = "Palatino"),
         scene = list(aspectratio = list(x = 1.8, y = 1.8, z = 0.4)))

Figure 7: One basis function on a metric graph \(\Gamma = (\mathcal{V},\mathcal{E})\)

References

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

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