plotting in R with absolute values

Question

I wonder if there is a straightforward way to plot the following two equations in R

x and y are variables and the rest are known parameters.

when X is a vector of dimension n then

Answer 1

This is a math problem rather than programming. A bit of calculation would help the programming task a lot easier.

First, assume c_1 and c_2 equal zero for simplicity. We can easily recover the original scale by shifting axes. Then, the matrix calculation can be written as follows.

Now let z = ax + by and w = cx + dy. Then, the first equation with absolute value metric would be written as:

From this equation, assuming that gamma is positive, you can visualize z and w as below.

So, you can find a set of (z, w) combinations that satisfy the requirement and convert back to (x, y).

The second equation with the maximum metric can be written as follows:

This implies that (z, w) can be visualized as below.

Again, you can generate such (z, w) pairs and convert back to (x, y).

Here is an R code for the first equation. You can try the second on your own.

library(ggplot2)

# A is (a,b; c,d) matrix
A <- matrix(c(1, 2, -1, 0), 
            nrow=2, ncol=2, byrow=TRUE)
gamma <- 1
c1 <- 0.2
c2 <- 0.1

###############################
z <- seq(-gamma, gamma, length=100)
w <- abs(gamma - abs(z))

z <- c(z, z)
w <- c(w, -w)

qplot(z, w) + coord_fixed()

# computing back (x,y) from (z,w)
z_mat <- rbind(z, w)
x_mat <- solve(A, z_mat)
x <- x_mat[1,] + c1
y <- x_mat[2,] + c2

qplot(x, y) + coord_fixed()
################################