How to determine whether a points lies in an ellipse

Question

I had posted a similar question earlier. I was trying to determine whether a point lies within an ellipse. Basically I generate some bivariate normal data and create an ellipse. Heres the code I use

 library(MASS)
 set.seed(1234)
 x1<-NULL
 x2<-NULL
 k<-1
 Sigma2 <- matrix(c(.72,.57,.57,.46),2,2)
 Sigma2
 rho <- Sigma2[1,2]/sqrt(Sigma2[1,1]*Sigma2[2,2])

 eta<-replicate(300,mvrnorm(k, mu=c(-2.503,-1.632), Sigma2)) 

 p1<-exp(eta)/(1+exp(eta))
 n<-60
 x1<-replicate(300,rbinom(k,n,p1[,1]))
 x2<-replicate(300,rbinom(k,n,p1[,2]))

 rate1<-x1/60
 rate2<-x2/60

 library(car)
 dataEllipse(rate1,rate2,levels=c(0.05, 0.95)) 

I need to find out whether the pair (p1[,1],p1[,2]) lies within the area of the ellipse above.

Answer 1

dataEllipse returns the ellipses as polygons, so you could use the point.in.polygon function from the sp library to check whether the points are inside the ellipse:

ell = dataEllipse(rate1, rate2, levels=c(0.05, 0.95)) 
point.in.polygon(rate1, rate2, ell$`0.95`[,1], ell$`0.95`[,2])

---

When I run the following code...

library(MASS)
set.seed(1234)
x1<-NULL
x2<-NULL
k<-1
Sigma2 <- matrix(c(.72,.57,.57,.46),2,2)
Sigma2
rho <- Sigma2[1,2]/sqrt(Sigma2[1,1]*Sigma2[2,2])
eta<-replicate(300,mvrnorm(k, mu=c(-2.503,-1.632), Sigma2))
p1<-exp(eta)/(1+exp(eta))
n<-60
x1<-replicate(300,rbinom(k,n,p1[,1]))
x2<-replicate(300,rbinom(k,n,p1[,2]))
rate1<-x1/60
rate2<-x2/60
library(car)
ell = dataEllipse(rate1, rate2, levels=c(0.05, 0.95))
library(sp)
point.in.polygon(rate1, rate2, ell$`0.95`[,1], ell$`0.95`[,2])

... I get the following output

  [1] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1
 [56] 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
[111] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
[166] 1 1 1 1 1 1 1 0 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1
[221] 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
[276] 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

Answer 2

Just find C and subtract radius