Finding where two linear fits intersect in R

Question

I have two linear fits that I've gotten from lm calls in my R script. For instance...

fit1 <- lm(y1 ~ x1)
fit2 <- lm(y2 ~ x2)

I'd like to find the (x,y) point at which these two lines (fit1 and fit2) intersect, if they intersect at all.

Answer 1

One way to avoid the geometry is to re-parameterize the equations as:

y1 = m1 * (x1 - x0) + y0
y2 = m2 * (x2 - x0) + y0

in terms of their intersection point (x0, y0) and then perform the fit of both at once using nls so that the returned values of x0 and y0 give the result:

# test data
set.seed(123)
x1 <- 1:10
y1 <- -5 + x1 + rnorm(10)
x2 <- 1:10
y2 <- 5 - x1 + rnorm(10)
g <- rep(1:2, each = 10) # first 10 are from x1,y1 and second 10 are from x2,y2

xx <- c(x1, x2)
yy <- c(y1, y2)
nls(yy ~ ifelse(g == 1, m1 * (xx - x0) + y0, m2 * (xx - x0) + y0),
    start = c(m1 = -1, m2 = 1, y0 = 0, x0 = 0))

EDIT: Note that the lines xx<-... and yy<-... are new and the nls line has been specified in terms of those and corrected.

Answer 2

Here's some high school geometry then ;-)

# First two models
df1 <- data.frame(x=1:50, y=1:50/2+rnorm(50)+10)
m1 <- lm(y~x, df1)

df2 <- data.frame(x=1:25, y=25:1*2+rnorm(25)-10)
m2 <- lm(y~x, df2)

# Plot them to show the intersection visually    
plot(df1)
points(df2)

# Now calculate it!    
a <- coef(m1)-coef(m2)
c(x=-a[[1]]/a[[2]], y=coef(m1)[[2]]*x + coef(m1)[[1]])

Or, to simplify the solve-based solution by @Dwin:

cm <- rbind(coef(m1),coef(m2)) # Coefficient matrix
c(-solve(cbind(cm[,2],-1)) %*% cm[,1])
# [1] 12.68034 16.57181