使用r在曲线下方或上方的点

Question

I try to figure out, if my point is below or above a simple curve and struggling with my math at the moment, I guess...

I prepared a working example, so but the math first.

I have some points and I want to check if they are above or below a curve... The curve has the function y=1/(x-.5) So I thought I will set the function to 0 and get 0=1/(x-.5)-y

Afterwards I will get negative values if the point is on one side of the curve, and positive values on the other side... Hope this is right so far...

I realised a problem, if the x values is smaller then .5, then the part below 1/ gets negative and all my values are also negative...

I added a special point (5) which gives the expected positive value, but how about the other ones, how should I test those? Thanks for help!

points <- data.frame(
  x=c(-3.6030515,-0.2791478,10.2045860,-0.7457344,1,0.4037591,0.1555678,
      6.1525442,1.9831603),
  y=c(0.95715140,0.18139107,2.87456154,0.17190597,0.5,0.09778570,0.02708183,
      2.69455955,1.09943870)
)
curves <- data.frame(x=c(seq(.1,10,.1)))
curves$y <- 1/(curves$x-.5)

plot(points$x,points$y)
lines(curves$x,curves$y)
lines(-curves$x,curves$y)

1/(points$x-.5)-points$y >= 0

Thanks in advance!

Answer 1

Unless I've misunderstood the question, you should be able to just evaluate the function at your points' x values, and compare the outcome (ie the y value according to the function) to your points' y values.

f <- function(x) 1 / (x-0.5)
f(points$x) < points$y

# [1]  TRUE  TRUE  TRUE  TRUE FALSE  TRUE  TRUE  TRUE  TRUE

With the way I've structured the inequality, TRUE indicates that the curve is below the corresponding y value in the points vector. In other words, all but the fifth point are above the curve.

Answer 2

To count the number below the curve :

## count the number below the curve 
sum(points$y<1/(points$x-0.5) )

To show it graphically :

## plot it using plot and curve
plot(points$x,points$y,col=ifelse(points$y<1/(points$x-0.5) ,'blue','red'),pch=20)
curve(1/(x-.5),-4,10,add=TRUE,col='green',lwd=2)

discontinuity part :

To show the discontinuity part graphically you should use curve :

curve(1/(x-.5),0,1,col='green',lwd=2)
abline(v=0.5,lwd=3)

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