Finds the shortest distance between two lines

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Description

Finds the shortest distance between two lines.

Consider the two lines

x(s) = a_x + b_x*s and y(t) = a_y + b_y*t

in an K-space where the offset and direction vectors are a_x and b_x (in R^K) that define the line x(s) (s is a scalar). Similar for the line y(t). This function finds the point (s,t) for which |x(s)-x(t)| is minimal.

Usage

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## Default S3 method:
distanceBetweenLines(ax, bx, ay, by, ...)

Arguments

ax,bx

Offset and direction vector of length K for line z_x.

ay,by

Offset and direction vector of length K for line z_y.

...

Not used.

Value

Returns the a list containing

ax,bx

The given line x(s).

ay,by

The given line y(t).

s,t

The values of s and t such that |x(s)-y(t)| is minimal.

xs,yt

The values of x(s) and y(t) at the optimal point (s,t).

distance

The distance between the lines, i.e. |x(s)-y(t)| at the optimal point (s,t).

Author(s)

Henrik Bengtsson

References

[1] M. Bard and D. Himel, The Minimum Distance Between Two Lines in n-Space, September 2001, Advisor Dennis Merino.
[2] Dan Sunday, Distance between 3D Lines and Segments, Jan 2016, http://geomalgorithms.com/a07-_distance.html.

Examples

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for (zzz in 0) {

# This example requires plot3d() in R.basic [http://www.braju.com/R/]
if (!require(pkgName <- "R.basic", character.only=TRUE)) break

layout(matrix(1:4, nrow=2, ncol=2, byrow=TRUE))

############################################################
# Lines in two-dimensions
############################################################
x <- list(a=c(1,0), b=c(1,2))
y <- list(a=c(0,2), b=c(1,1))
fit <- distanceBetweenLines(ax=x$a, bx=x$b, ay=y$a, by=y$b)

xlim <- ylim <- c(-1,8)
plot(NA, xlab="", ylab="", xlim=ylim, ylim=ylim)

# Highlight the offset coordinates for both lines
points(t(x$a), pch="+", col="red")
text(t(x$a), label=expression(a[x]), adj=c(-1,0.5))
points(t(y$a), pch="+", col="blue")
text(t(y$a), label=expression(a[y]), adj=c(-1,0.5))

v <- c(-1,1)*10;
xv <- list(x=x$a[1]+x$b[1]*v, y=x$a[2]+x$b[2]*v)
yv <- list(x=y$a[1]+y$b[1]*v, y=y$a[2]+y$b[2]*v)

lines(xv, col="red")
lines(yv, col="blue")

points(t(fit$xs), cex=2.0, col="red")
text(t(fit$xs), label=expression(x(s)), adj=c(+2,0.5))
points(t(fit$yt), cex=1.5, col="blue")
text(t(fit$yt), label=expression(y(t)), adj=c(-1,0.5))
print(fit)


############################################################
# Lines in three-dimensions
############################################################
x <- list(a=c(0,0,0), b=c(1,1,1))  # The 'diagonal'
y <- list(a=c(2,1,2), b=c(2,1,3))  # A 'fitted' line
fit <- distanceBetweenLines(ax=x$a, bx=x$b, ay=y$a, by=y$b)

xlim <- ylim <- zlim <- c(-1,3)
dummy <- t(c(1,1,1))*100;

# Coordinates for the lines in 3d
v <- seq(-10,10, by=1);
xv <- list(x=x$a[1]+x$b[1]*v, y=x$a[2]+x$b[2]*v, z=x$a[3]+x$b[3]*v)
yv <- list(x=y$a[1]+y$b[1]*v, y=y$a[2]+y$b[2]*v, z=y$a[3]+y$b[3]*v)

for (theta in seq(30,140,length=3)) {
  plot3d(dummy, theta=theta, phi=30, xlab="", ylab="", zlab="",
                             xlim=ylim, ylim=ylim, zlim=zlim)

  # Highlight the offset coordinates for both lines
  points3d(t(x$a), pch="+", col="red")
  text3d(t(x$a), label=expression(a[x]), adj=c(-1,0.5))
  points3d(t(y$a), pch="+", col="blue")
  text3d(t(y$a), label=expression(a[y]), adj=c(-1,0.5))

  # Draw the lines
  lines3d(xv, col="red")
  lines3d(yv, col="blue")

  # Draw the two points that are closest to each other
  points3d(t(fit$xs), cex=2.0, col="red")
  text3d(t(fit$xs), label=expression(x(s)), adj=c(+2,0.5))
  points3d(t(fit$yt), cex=1.5, col="blue")
  text3d(t(fit$yt), label=expression(y(t)), adj=c(-1,0.5))

  # Draw the distance between the two points
  lines3d(rbind(fit$xs,fit$yt), col="purple", lwd=2)
}

print(fit)

} # for (zzz in 0)
rm(zzz)

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