Example
An example is useful to fix ideas. We define five points in the plane
by the following data.frame. In the figure, the arrows
indicate the ordering.
pts <- data.frame(x = c(0.286, 0.730, 0.861, 0.623, 0.100),
y = c(0.164, 0.206, 0.514, 0.666, 0.492))
with(pts, {
par(mar = c(4, 4, 1, 1), las = 1)
plot(x, y, asp = 1, col = 4, panel.first = grid(), pch = 1, cex = 2, bty = "n")
arrows(x[-5], y[-5], x[-1], y[-1], angle = 15, length = 0.125, col = 2)
})
Clearly stats::spline cannot be used directly to produce
an interpolating curve in this case as neither \(x-\) or \(y-\)coordinates are monotonically
ordered.
The simple solution we offer here is to use use arc length along the line segments joining the points as a parameter and fit interpolating splines to both \(x-\) and \(y-\) coordinates of the given points as a function of arc length.
More explicitly, we take the cumulative Euclidean distance lengths of the arrow segments in the diagram above as the parameter and fit interpolating splines to the \(x-\) and \(y-\) coordinates of the lengths separately, and use the splines as the coordinates of the interpolating curve. The method is shown in the code below.
s <- with(pts, cumsum(c(0, sqrt(diff(x)^2 + diff(y)^2))))
icurve <- with(pts, data.frame(x = spline(s, x, n = 500)$y,
y = spline(s, y, n = 500)$y))
with(pts, {
par(mar = c(4, 4, 1, 1), las = 1)
with(icurve, plot(x, y, asp = 1, panel.first = grid(), type = "l", bty = "n", col = 2))
points(x, y)
})
This is essentially the operation of the function
frenchCurve::open_curve. The function produces an
S3 object with class "curve" for which several
methods are available, including its own plot and
lines methods for traditional graphics.
Scale dependence
One further tweak is provided by the two main functions of the
package. The Euclidean distances used in the computation are critically
dependent on the relative scales of the \(x-\) and \(y-\)coordinates. It is up to the user to
use the functions with the coordinates scaled in such a way as to make
the Euclidean distance the appropriate metric. To help with this, both
functions frenchCurve::open_curve and
frenchCurve::closed_curve provide an argument
asp to specify a scale adjustment. Specifically, the two
coordinates x and y * asp are used in the
distance computations for arc length.
The asp argument is a single numerical value with a
default of 1. However it may be supplied as a character
string and asp = "range" specifies that the value
asp = diff(range(x))/diff(range(y)) should be used. The
effect is shown in the following extension to the running example
below.
icurve <- open_curve(pts)
jcurve <- open_curve(pts, asp = "range")
plot(icurve, bty = "n", col = 2, asp = 1)
grid()
lines(jcurve, col = 4)
legend("topright", legend = c("asp = 1", 'asp = "range"'),
lty = "solid", col = c(2,4), pch=20, bty = "n", cex = 0.75)
Notice particularly that the asp argument to
open_curve and the asp argument to
graphics::plot are different, but have a similar
purpose.
Closed curves
If the curve is required to link back from the last point to the first in a smooth continuous way, the algorithm we propose is simply to repeat the points three times and choose the middle section of the result. This may be overkill, but the computation is relatively cheap and the result usually appears satisfactory for most aesthetic purposes.
The results for the running example are shown in the figure below:
iccurve <- closed_curve(pts)
jccurve <- closed_curve(pts, asp = "range")
plot(iccurve, bty = "n", col = 2, asp = 1)
grid()
lines(jccurve, col = 4)
legend("topright", legend = c("asp = 1", 'asp = "range"'),
lty = "solid", col = c(2,4), pch=20, bty = "n", cex = 0.75)
