x, y for point of intersection

Hi everyone,

I am trying to get a point of intersection between a
polyline and a straight line 

.. and get the x and y

coordinates of this point.
For exemplification consider this:

set.seed(123)
k1 <-rnorm(100, mean=1.77, sd=3.33)
k1 <- sort(k1)
q1 <- rnorm(100, mean=2.37, sd=0.74)
q1 <- sort(q1, decreasing = TRUE)
plot(k1, q1, xlim <- c((min(k1)-5), (max(k1)+5)), type="l")

ya <- 2
xa = -5
yb=4
xb=12
lines(c(xa, xb), c(ya, yb), col = 2)

# I want to get the x and y coordinates of the
# intersection of the 2 lines.

m <- (ya-yb)/(xa-xb)
b <- ya-m*xa
ln <- loess(q1~k1)
lines(ln)

It is clear that the x, y will satisfy both linear equations,
y = m*x + b and the ln polyline - .. but while I can visualize
the equation of the straight line  - I have problems with the
polyline. I will appreciate any ideas to solve this problem.
I thought it a trivial solution but it seems I cannot see it.
Thanks,
Monica

ya <- 2
  xa = -5
  yb =  4
  xb = 12

These define a line

  y = ya + (x - xa)*(yb - ya)/(xb - xa)

so write this as

  y = A + B*x

Then points above the line satisfy

  y > A + B*X

and points below the line satisfy

  Y < A + B*X

  A <- ya - xa*(yb - ya)/(xb - xa)
  B <- (yb - ya)/(xb - xa)

So now extract the points (x,y) fron the loess fit:

  x.ln <- ln$x
  y.ln <- ln$y

and now find the points on 'ln' which are above, and
the points on ln which are below, which will locate the
segment which crosses the (X,Y) line:

  ix.upper <- which(y.ln >  A + B*ln$y)
  ix.lower <- which(y.ln <=3D A + B*ln$y)

  ix.upper
  # 1  2  3  4  5  6  7  8  9 10 11 12 13 14 15=20

So now you have the line segment from

  (x.ln[15],y.ln[15])
to
  (x.ln[16],y.ln[16])

and now all you need to do is to find the intersection
of the line from (ln.x[15],ln.y[15]) to (ln.x[16],ln.y[16])
with the line from (xa,ya) to (xb,yb).

(There could be complications if the y-values of ln do not
continually decrease in value; but happily they do decrease
in your example).

Hoping this helps!
Ted.



--------------------------------------------------------------------
E-Mail: (Ted Harding) <ted.harding at wlandres.net>
Fax-to-email: +44 (0)870 094 0861
Date: 22-Nov-11                                       Time: 22:49:54
------------------------------ XFMail ------------------------------