Quick start

I made a gist² containing the functions needed to produce a simple animation of a 2×2 matrix transformation. If you are reading this post with the sole goal of creating an animation like the one below³, you can copy and run this code chunk to render a 2×2 matrix transformation gif (the input to argument m can be any 2×2 matrix of interest).

if (!requireNamespace("devtools")) install.packages("devtools")
devtools::source_gist("https://gist.github.com/brshallo/6a125f9c96dac5445cebb97cc62bfc9c")

animate_matrix_transformation(m = matrix(c(0.5, 0.5, 0.5, -0.25), nrow = 2))

Over the next several sections I’ll walk through the thinking behind this code (culminating in the Visualizations section, where this animation will be shown again). Sections in the Appendix contain variations on this animation that add-on additional simple transformations and layers.

Helper functions

construct_grid(): given vectors of x and y intercepts, return a dataframe with columns x, y, xend, yend (meant for input into geom_segment()).

library(tidyverse)
construct_grid <- function(xintercepts = -5:5, yintercepts = -5:5){
  bind_rows(
    crossing(x = xintercepts,
             y = min(yintercepts),
             yend = max(yintercepts)) %>%
      mutate(xend = x),
    crossing(y = yintercepts,
             x = min(xintercepts),
             xend = max(xintercepts)) %>%
      mutate(yend = y)
  ) %>% 
    select(x, y, xend, yend)
}

Use with geom_segment() to make simple grids:

construct_grid() %>% 
  ggplot(aes(x = x, y = y, xend = xend, yend = yend))+
  geom_segment()+
  coord_fixed()+
  ggforce::theme_no_axes()+
  theme(panel.border = element_blank())

transform_df_coords(): Given dataframe, column names of coordinates⁴, and a transformation matrix, return dataframe with transformed coordinates.

transform_df_coords <- function(df, ..., m = diag(length(df))){
  
  df_names <- names(df)
  
  df_coords <- df %>% 
    select(...)
  
  df_coords_names <- names(df_coords)
  
  df_matrix <- df_coords %>% 
    as.matrix() %>% 
    t()
  
  df_coords_new <- (m %*% df_matrix) %>% 
    t() %>% 
    as_tibble() %>% 
    set_names(df_coords_names)
  
  df_other <- df %>% 
    select(-one_of(df_coords_names))
  
  bind_cols(df_coords_new, df_other) %>% 
    select(df_names)
}

transform_df_coords() is just matrix multiplication, but facilitates applying matrix transformations on a dataframe where each row (in specified columns) represents a vector / coordinate point⁵.

Example in \(\mathbb{R}^2\):

transform_df_coords(tibble(x = 1:4, y = 1:4), x, y, m = matrix(1:4, nrow = 2)) %>% 
  knitr::kable()

Again, this is the same as:

\[ \left(\begin{array}{cc} 1 & 3\\ 2 & 4 \end{array}\right) \left(\begin{array}{cc} 1 & 2 & 3 & 4 \\ 1 & 2 & 3 & 4 \end{array}\right) = \left(\begin{array}{cc} 4 & 8 & 12 & 16 \\ 6 & 12 & 18 & 24 \end{array}\right)\]

(Just with a ‘tidy’ dataframe as output.)

Also works with more dimensions, see example in \(\mathbb{R}^3\):

transform_df_coords(tibble(x = 1:5, y = 1:5, z = 1:5), x, y, z, m = matrix(1:9, nrow = 3)) %>% 
  knitr::kable()

However for our visualizations, we only care about examples in 2 dimensions (when we are applying a 2×2 matrix transformation).

Construct objects for graph

For a simple animation I will build dataframes that contain the coordinates for the following objects⁶:

• a starting grid and a transformed grid
• a starting basis vector and a transformed basis vector

To play nicely with gganimate the start and transformed objects need to have additional properties⁷:

• a field that groups like objects across the animation (e.g. id column)
• a field that designates transitions between start and transformed states (e.g. time column)

For my example I will be applying the following matrix transformation to our basis vectors⁸. \[ \left(\begin{array}{cc} 0.5 & 0.5\\ 0.5 & -0.25 \end{array}\right)\]

Define transformation matrix:

# same as above examples using `matrix()` but I find inputting into tribble more
# intuitive for 2x2 matrix
transformation_matrix <- tribble(~ x, ~ y,
                                 0.5, 0.5,
                                 0.5, -0.25) %>% 
  as.matrix()

Construct grids:

grid_start <- construct_grid() %>% 
  mutate(id = row_number())

grid_trans <- grid_start %>% 
  # need to `transform_df_coords()` twice as each segment is made up of 2 points
  transform_df_coords(x, y, m = transformation_matrix) %>% 
  transform_df_coords(xend, yend, m = transformation_matrix)

grid_all <- bind_rows(
  mutate(grid_start, time = 1),
  mutate(grid_trans, time = 2)
)

Construct basis vectors:

basis_start <- tibble(
  x = c(0, 0),
  y = c(0, 0),
  xend = c(1, 0),
  yend = c(0, 1),
  # `vec` is unnecessary, will just use to differentiate colors
  vec = c("i", "j")
) %>% 
  mutate(id = nrow(grid_start) + row_number())

basis_trans <- basis_start %>% 
  transform_df_coords(x, y, m = transformation_matrix) %>% 
  transform_df_coords(xend, yend, m = transformation_matrix)

basis_all <- bind_rows(
  mutate(basis_start, time = 1),
  mutate(basis_trans, time = 2)
)

Build visualization

Define breaks in grid:

# If you just want to use the starting grid for the breaks, could do
x_breaks <- unique(grid_start$x)
y_breaks <- unique(grid_start$y)

Define visualization:

p <- ggplot(aes(x = x, y = y, group = id), data = grid_all)+
  geom_segment(aes(xend = xend, yend = yend))+
  geom_segment(aes(xend = xend, yend = yend, colour = vec), data = basis_all, arrow = arrow(length = unit(0.02, "npc")), size = 1.2)+
  scale_x_continuous(breaks = x_breaks, minor_breaks = NULL)+
  scale_y_continuous(breaks = y_breaks, minor_breaks = NULL)+
  coord_fixed()+
  theme_minimal()+
  theme(axis.text = element_blank(),
        axis.title = element_blank(),
        legend.position = "none")

p

p + gganimate::transition_states(time, wrap = FALSE)

Appendix

The code used to construct images within the appendix is very similar to code already shown¹¹.

rotate_trans <- tribble(~ x, ~ y,
                        cos(pi / 2), -sin(pi / 2),
                        sin(pi / 2), cos(pi / 2)) %>%
  as.matrix()

sheer_trans <- tribble(~ x, ~ y,
                      1, 0,
                      0.5, 1) %>%
  as.matrix() 

grid_start <- construct_grid() %>% 
  mutate(id = row_number())

grid_trans <- grid_start %>% 
  # need to `transform_df_coords()` twice as each segment is made up of 2 points
  transform_df_coords(x, y, m = rotate_trans) %>% 
  transform_df_coords(xend, yend, m = rotate_trans)

grid_trans2 <- grid_trans %>% 
  # need to `transform_df_coords()` twice as each segment is made up of 2 points
  transform_df_coords(x, y, m = sheer_trans) %>% 
  transform_df_coords(xend, yend, m = sheer_trans)

grid_all <- bind_rows(
  mutate(grid_start, time = 1),
  mutate(grid_trans, time = 2),
  mutate(grid_trans2, time = 3)
) 

basis_start <- tibble(
  x = c(0, 0),
  y = c(0, 0),
  xend = c(1, 0),
  yend = c(0, 1),
  # `vec` is unnecessary, will just use to differentiate colors
  vec = c("i", "j")
) %>% 
  mutate(id = nrow(grid_start) + row_number())

basis_trans <- basis_start %>% 
  # need to `transform_df_coords()` twice as each segment is made up of 2 points
  transform_df_coords(x, y, m = rotate_trans) %>% 
  transform_df_coords(xend, yend, m = rotate_trans)

basis_trans2 <- basis_trans %>% 
  # need to `transform_df_coords()` twice as each segment is made up of 2 points
  transform_df_coords(x, y, m = sheer_trans) %>% 
  transform_df_coords(xend, yend, m = sheer_trans)

basis_all <- bind_rows(
  mutate(basis_start, time = 1),
  mutate(basis_trans, time = 2),
  mutate(basis_trans2, time = 3)
) 

p_mult <- ggplot(aes(x = x, y = y, group = id), data = grid_all)+
  geom_segment(aes(xend = xend, yend = yend))+
  geom_segment(aes(xend = xend, yend = yend, colour = vec), data = basis_all, arrow = arrow(length = unit(0.02, "npc")), size = 1.2)+
  scale_x_continuous(breaks = x_breaks, minor_breaks = NULL)+
  scale_y_continuous(breaks = y_breaks, minor_breaks = NULL)+
  coord_fixed()+
  theme_minimal()+
  theme(axis.text = element_blank(),
        axis.title = element_blank(),
        legend.position = "none") 

p_mult 

p_mult + 
  gganimate::transition_states(time, wrap = FALSE) 

circle_df <- tibble(x0 = 0, y0 = 0, r = 5)

p_rotation <- ggplot(aes(), data = filter(grid_all, time <= 2))+
  geom_segment(aes(x = x, y = y, group = id, xend = xend, yend = yend))+
  geom_segment(aes(x = x, y = y, group = id, xend = xend, yend = yend, colour = vec), arrow = arrow(length = unit(0.02, "npc")), size = 1.2, data = filter(basis_all, time <= 2 ))+
  scale_x_continuous(breaks = x_breaks, minor_breaks = NULL)+
  scale_y_continuous(breaks = y_breaks, minor_breaks = NULL)+
  coord_fixed()+
  ggforce::geom_circle(aes(x0 = 0, y0 = 0, r = 5), data = circle_df)+
  theme_minimal()+
  theme(axis.text = element_blank(),
        axis.title = element_blank(),
        legend.position = "none")

p_rotation + gganimate::transition_states(time, wrap = FALSE) 

animate_matrix_transformation(m = matrix(c(-1, 0, 0, 1), nrow = 2))

points_start <- crossing(x = c(-3.5:3.5), y = c(-3.5:3.5)) %>% 
  mutate(id = nrow(grid_start) + nrow(basis_start) + row_number())
 
points_trans <- points_start %>% 
  transform_df_coords(x, y, m = rotate_trans)

points_all <- bind_rows(
  mutate(points_start, time = 1),
  mutate(points_trans, time = 2))

p_points <- p +
  geom_point(data = points_all, colour = "royalblue3")

p_points + gganimate::transition_states(time, wrap = FALSE)

# maybe just my eyes... maybe need to increase framerate... or something

p_points <- p_rotation +
  geom_point(aes(x, y), data = points_all, colour = "royalblue3")

p_points + gganimate::transition_states(time, wrap = FALSE) 

x_breaks <-
  seq(
    from = 
      floor(min(c(grid_all$x, grid_all$xend))), 
    to = 
      ceiling(max(c(grid_all$x, grid_all$xend))), 
    by = 1)

y_breaks <-
  seq(
    from = 
      floor(min(c(grid_all$y, grid_all$yend))), 
    to = 
      ceiling(max(c(grid_all$y, grid_all$yend))), 
    by = 1)

1. Which are shown throughout the series and most notably in chapters 3 and 13.↩

2. See section [Problems and potential improvements] for notes on a couple potential updates I’ll make… not positive I’ll keep the gist code updated.↩

3. And may not care about understanding how to do multiple transformations, adding additional layers, etc.↩

4. / vectors↩

5. I’m guessing there is a better / more elegant function already out there for ‘tidy matrix multiplication’ or something… but couldn’t immediately think of anything.↩

6. You could add additional objects to the image – just need to ensure you create start and transformed versions of each object.↩

7. Creating these is not needed if you just wanted to create static images for the below examples.↩

8. No real reason for choosing this transformation, just thought it looked cool.↩

9. If wrap = TRUE (default) the reverse looping of the image is inaccurate as the transformation back to the original basis actually represents a transformation by the inverse of the transformation matrix. Though leaving it in would look cooler.↩

10. Could functionalize more… or make a shiny app, or do more with, see [Problems and potential improvements] for notes…↩

11. Can largely skim over↩

12. Though this gets into decomposing the rotation, etc. components of the matrix transformation of interest for each frame.↩

13. Seems issue has to do with group needing to apply both to the polygon at a given time as well as points on the polygon across time.↩

14. Would require latex title which I don’t know if is supported by gganimate↩