Find xy coordinates of a point knowing its distance to other 2 points

Question

I'm writing a code in R to calculate the xy coordinates of point, using the law of cosines. I have two reference points (1 and 2) which the xy coordinates are known. I want to find the coordinates of the other point (3). I know the distances 3-1, 3-2 and 1-2, but I don't know the angles between them.

Thanks in advance for any help!

I've tried some trigonometric equations I've found on web and Rohlf&Archie 1978 paper, but they don't work.

Answer 1

You haven't told us your set-up exactly, but it sounds as though you have two known x, y, co-ordinates:

x1 <- 1
x2 <- 5

y1 <- 3
y2 <- 6

And known distances between these two points plus a third point:

d12 <- 5
d13 <- 8
d23 <- 5

We can draw them like this:

plot(c(x1, x2), c(y1, y2), xlim = c(0, 12), ylim = c(0, 12))
text(c(x1, x2), c(y1, y2) + 0.5, labels = c('1', '2'))

Now, it's obviously easy to calculate the angle between the horizontal line and the segment joining points 1 and two - it's just the arctangent of the slope:

abline(h = y1, lty = 2)
theta <- atan2(y2 - y1, x2 - x1)

segments(x1, y1, x1 + d12 * cos(theta), y1 + d12 * sin(theta), lty = 2)

Now, although we don't know where point 3 is, we can use the law of cosines to calculate the angle 3-1-2 like this:

angle_312 <- acos((d12^2 + d13^2 - d23^2)/(2 * d12 * d13))

To get this in terms of angle from the horizontal we can do:

angle_13 <- angle_312 - theta

This allows us to work out the co-ordinates of point 3:

x3 <- x1 + d13 * cos(angle_13)
y3 <- y1 + d13 * sin(angle_13)

We can draw this point on our plot as follows:

points(x3, y3)
text(x3, y3 + 0.5, '3')

And we can show that it is correct by drawing circles of the correct radius around points one and two. Point 3 should be at the meeting point of the two circles:

polygon(x1 + dist_1_3 * cos(seq(0, 2 * pi, length = 100)), 
        y1 + dist_1_3 * sin(seq(0, 2 * pi, length = 100)), lty = 2)
polygon(x2 + dist_2_3 * cos(seq(0, 2 * pi, length = 100)), 
        y2 + dist_2_3 * sin(seq(0, 2 * pi, length = 100)), lty = 2)

Note that there is a second solution at the other point where the circles meet: in this case we would get that by changing angle_13 <- angle_312 - theta to angle_13 <- angle_312 + theta:

angle_13 <- angle_312 + theta

x3 <- x1 + d13 * cos(angle_13)
y3 <- y1 + d13 * sin(angle_13)

points(x3, y3)
text(x3, y3 + 0.5, '3')