# Fast(?) cubic beziers at variable resolution

• I wanted to calculate some points along cubic bezier curves
• Current R grid bezier implementation is not a real bezier! According to docs for grid::bezierGrob(): “The curve is currently drawn using an approximation based on X-splines.”
• Code below is the fastest thing I could come up with (I’m only 80% sure it’s correct)
• It vectorized implemntation of the explicit form of the curve as explained on Wikipedia
• I need something that can quickly change between low-complexity and high-complexity approximations to the curve by changing the number of line segments generated
• Speed-ups welcomed

## Code

#~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
# Pre-calculate sets of control-point weights for each for a range of
# different desired points in the output
#~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
bf3s <- lapply(1:100, function(N) {
t  <- seq.int(0, 1, length.out = N)

matrix(c(
1 * t^0 * (1 - t)^3,
3 * t^1 * (1 - t)^2,
3 * t^2 * (1 - t)^1,
1 * t^3 * (1 - t)^0
), ncol = 4
)
})

#~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#' Convert cubic  beziers to data.frame of coordinates along curve
#'
#' @param x,y coords of control points
#' @param N Number of output points
#'
#' @return list of coords
#~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
bezier3 <- function(x, y, N = 5) {
stopifnot(length(x)== 4, length(y) == 4)
list(
x = rowSums(bf3s[[N]] * matrix(x, nrow = N, ncol = 4, byrow = TRUE)),
y = rowSums(bf3s[[N]] * matrix(y, nrow = N, ncol = 4, byrow = TRUE))
)
}

## Demo

library(grid)

# Random control points
set.seed(1)
xcp <- runif(4)
ycp <- runif(4)

# Generate low and high complexity approximations
pt_lo <- bezier3(xcp, ycp,  4)
pt_hi <- bezier3(xcp, ycp, 40)

# Draw my approximations
grid.lines(pt_lo$x, pt_lo$y, gp = gpar(col = 'red' ))
grid.lines(pt_hi$x, pt_hi$y, gp = gpar(col = 'blue'))

## Comparison to grid::grid.bezier()

# Overlay the grid version of this bezier
grid.lines(pt_lo$x, pt_lo$y, gp = gpar(col = 'red' ))
grid.lines(pt_hi$x, pt_hi$y, gp = gpar(col = 'blue'))
grid.bezier(xcp, ycp)

## Benchmark

• About 30,000 beziers/second
expression N min median itr/sec
bezier3(xcp, ycp, N) 5 22.4µs 26.9µs 35103.10
bezier3(xcp, ycp, N) 10 22.3µs 27.5µs 31933.63
bezier3(xcp, ycp, N) 20 22.7µs 27.8µs 33358.24
bezier3(xcp, ycp, N) 50 24µs 30.5µs 30847.32
bezier3(xcp, ycp, N) 100 27µs 33µs 27591.29