R查找连接4个坐标的两条线的交点

Question

I apologize in advance if my code looks very amateurish. I'm trying to assign quadrants to 4 measurement stations approximately located on the edges of a town.

I have the coordinates of these 4 stations:

a  <- c(13.2975,52.6556)
b  <- c(14.0083,52.5583)
c  <- c(13.3722,52.3997)
d  <- c(12.7417,52.6917)

Now my idea was to create lines connecting the north-south and east-west stations:

line.1 <- matrix(c(d[1],b[1],d[2],b[2]),ncol=2)
line.2 <- matrix(c(a[1],c[1],a[2],c[2]),ncol=2)

Plotting all the stations the connecting lines looks allright, however not very helpful for analyzing it on a computer.

So I calculated the eucledian vectors for the two lines:

vec.1 <- as.vector(c((b[1]-d[1]),(b[2]-d[2])))

vec.2 <- as.vector(c((c[1]-a[1]),(c[2]-a[2])))

which allowed me to calculate the angle between the two lines in degrees:

alpha <- acos((vec.1%*%vec.2) / (sqrt(vec.1[1]^2+vec.1[2]^2)*  
          sqrt(vec.2[1]^2+vec.2[2]^2)))) * 180/pi

The angle I get for alpha is 67.7146°. This looks fairly good. From this angle I can easily calculate the other 3 angles of the intersection, however I need values relative to the grid so I can assign values from 0°-360° for the wind directions.

Now my next planned step was to find the point where the two lines intersect, add a horizontal and vertical abline through that point and then calculate the angle relative to the grid. However I can't find a proper example that does that and I don't think I have a nice linear equation system I could solve.

Is my code way off? Or maybe anyone knows of a package which could help me? It feels like my whole approach is a bit wrong.

Answer 1

Okay I managed to calculate the intersection point, using line equations. Here is how.

The basic equation for two points is like this:

y - y_1 = (y_2-y_1/x_2-x_1) * (x-x_1)

If you make one for each of the two lines, you can just substitute the fractions.

k.1 <- ((c[2]-a[2])/(c[1]-a[1]))
k.2 <- ((b[2]-d[2])/(b[1]-d[1]))  

Reshaping the two functions you get a final form for y :

y <- (((-k.1/k.2)*d[2]+k.1*d[1]-k.1*c[1]+d[2])/(1-k.1/k.2))

This one you can now use to calculate the x -value:

x <- ((y-d[2])+d[1]*k.2)/k.2

In my case I get

y = 52.62319

x = 13.3922

I'm starting to really enjoy this program!

Answer 2

Wikipedia has a good article on finding the intersection between two line segments with an explicit formula. However, you don't need to know the point of intersection to calculate the angle to the grid (or axes of coordinate system.) Just compute the angles from your vec.1 and vec.2 to the basis vectors:

e1 <- c(1, 0)
e2 <- c(0, 1)

as you have done.