Fitting data points to an ellipse with its center

I have a question about fitting ellipses to data with the ellipse center at the origin. I have explored two methods that fit ellipses but generate an arbitrary center unless I manipulate the data with some imaginary mirror points.

Method#01

This portion of the script directly comes from this useful post. I'm copying the codes directly here for ease.

fit.ellipse <- function (x, y = NULL) {
  # from:
  # http://r.789695.n4.nabble.com/Fitting-a-half-ellipse-curve-tp2719037p2720560.html
  #
  # Least squares fitting of an ellipse to point data
  # using the algorithm described in: 
  #   Radim Halir & Jan Flusser. 1998. 
  #   Numerically stable direct least squares fitting of ellipses. 
  #   Proceedings of the 6th International Conference in Central Europe 
  #   on Computer Graphics and Visualization. WSCG '98, p. 125-132 
  #
  # Adapted from the original Matlab code by Michael Bedward (2010)
  # michael.bedward@gmail.com
  #
  # Subsequently improved by John Minter (2012)
  # 
  # Arguments: 
  # x, y - x and y coordinates of the data points.
  #        If a single arg is provided it is assumed to be a
  #        two column matrix.
  #
  # Returns a list with the following elements: 
  #
  # coef - coefficients of the ellipse as described by the general 
  #        quadratic:  ax^2 + bxy + cy^2 + dx + ey + f = 0 
  #
  # center - center x and y
  #
  # major - major semi-axis length
  #
  # minor - minor semi-axis length
  #
  EPS <- 1.0e-8 
  dat <- xy.coords(x, y) 

  D1 <- cbind(dat$x * dat$x, dat$x * dat$y, dat$y * dat$y) 
  D2 <- cbind(dat$x, dat$y, 1) 
  S1 <- t(D1) %*% D1 
  S2 <- t(D1) %*% D2 
  S3 <- t(D2) %*% D2 
  T <- -solve(S3) %*% t(S2) 
  M <- S1 + S2 %*% T 
  M <- rbind(M[3,] / 2, -M[2,], M[1,] / 2) 
  evec <- eigen(M)$vec 
  cond <- 4 * evec[1,] * evec[3,] - evec[2,]^2 
  a1 <- evec[, which(cond > 0)] 
  f <- c(a1, T %*% a1) 
  names(f) <- letters[1:6] 

  # calculate the center and lengths of the semi-axes 
  #
  # see http://www.ncbi.nlm.nih.gov/pmc/articles/PMC2288654/
  # J. R. Minter
  # for the center, linear algebra to the rescue
  # center is the solution to the pair of equations
  # 2ax +  by + d = 0
  # bx  + 2cy + e = 0
  # or
  # | 2a   b |   |x|   |-d|
  # |  b  2c | * |y| = |-e|
  # or
  # A x = b
  # or
  # x = Ainv b
  # or
  # x = solve(A) %*% b
  A <- matrix(c(2*f[1], f[2], f[2], 2*f[3]), nrow=2, ncol=2, byrow=T )
  b <- matrix(c(-f[4], -f[5]), nrow=2, ncol=1, byrow=T)
  soln <- solve(A) %*% b

  b2 <- f[2]^2 / 4

  center <- c(soln[1], soln[2]) 
  names(center) <- c("x", "y") 

  num  <- 2 * (f[1] * f[5]^2 / 4 + f[3] * f[4]^2 / 4 + f[6] * b2 - f[2]*f[4]*f[5]/4 - f[1]*f[3]*f[6]) 
  den1 <- (b2 - f[1]*f[3]) 
  den2 <- sqrt((f[1] - f[3])^2 + 4*b2) 
  den3 <- f[1] + f[3] 

  semi.axes <- sqrt(c( num / (den1 * (den2 - den3)),  num / (den1 * (-den2 - den3)) )) 

  # calculate the angle of rotation 
  term <- (f[1] - f[3]) / f[2] 
  angle <- atan(1 / term) / 2 

  list(coef=f, center = center, major = max(semi.axes), minor = min(semi.axes), angle = unname(angle)) 
}

Let's take a example distribution of polar points for illustration purpose

X<-structure(list(x_polar = c(0, 229.777200000011, 246.746099999989, 
-10.8621999999741, -60.8808999999892, 75.8904999999795, -83.938199999975, 
-62.9770000000135, 49.1650999999838, 52.3093000000226, 49.6891000000178, 
-66.4248999999836, 34.3671999999788, 242.386400000018, 343.60619999998
), y_polar = c(0, 214.868299999973, 161.063599999994, -68.8972000000067, 
-77.0230000000447, 93.2863000000361, -16.2356000000145, 27.7828000000445, 
-17.8077000000048, 2.10540000000037, 25.6866000000155, -84.6034999999683, 
-31.1800000000512, 192.010800000047, 222.003700000001)), .Names = c("x_polar", 
"y_polar"), row.names = c(NA, -15L), class = "data.frame")


efit <- fit.ellipse(X)
e <- get.ellipse(efit)
#plot
par(bg=NA)
plot(X, pch=3, col='gray', lwd=2, axes=F, xlab="", ylab="", type='n',
     ylim=c(min(X$y_polar)-150, max(X$y_polar)), xlim=c(min(X$x_polar)-150, max(X$x_polar))) #blank plot
points(X$x_polar, X$y_polar, pch=3, col='gray', lwd=2, axes=F, xlab="", ylab="") #observations
lines(e, col="red", lwd=3, lty=2) #plotting the ellipse
points(0,0,col=2, lwd=2, cex=2) #center/origin

To bring the origin of the ellipse at the center we could modify as follows (surely not the best way of doing it)

#generate mirror coordinates
X$x_polar_mirror<- -X$x_polar
X$y_polar_mirror<- -X$y_polar

mydata<-as.matrix(data.frame(c(X$x_polar, X$x_polar_mirror), c(X$y_polar, X$y_polar_mirror)))
#fit the data
efit <- fit.ellipse(mydata)
e <- get.ellipse(efit)
par(bg=NA)
plot(mydata, pch=3, col='gray', lwd=2, axes=F, xlab="", ylab="", type='n',
     ylim=c(min(X$y_polar)-150, max(X$y_polar)), xlim=c(min(X$x_polar)-150, max(X$x_polar)))
points(X$x_polar, X$y_polar, pch=3, col='gray', lwd=2, axes=F, xlab="", ylab="")
lines(e, col="red", lwd=3, lty=2) 
points(0,0,col=2, lwd=2, cex=2)  #center

Well ... it sort of does the job but none would be happy with all those imaginary points considered in the calculation.

Method#02

This is another indirect way of fitting the data but again the ellipse center is not at the origin. Any workaround?

require(car)
dataEllipse(X$x_polar, X$y_polar, levels=c(0.15, 0.7), 
            xlim=c(-150, 400), ylim=c(-200,300))

My questions: (a) is there a robust alternative way of fitting these points with the ellipse center at the origin (0,0)? (b) is there a measure of the goodness of ellipse fit? Thank you in advance.

I'm not really happy with aproach I've concieved, there should be a closed form solution, but still:

# Ellipse equasion with center in (0, 0) with semiaxis pars[1] and pars[2] rotated by pars[3].
# t and pars[3] in radians

ellipsePoints <- function(t, pars) {
  data.frame(x = cos(pars[3]) * pars[1] * cos(t) - sin(pars[3]) * pars[2] * sin(t),
             y = sin(pars[3]) * pars[1] * cos(t) + cos(pars[3]) * pars[2] * sin(t))
}


# Way to fit an ellipse through  minimising distance to data points.
# If weighted then points which are most remote from center will have bigger impact.

ellipseBrute <- function(x, y, pars, weighted = FALSE) {

  d <- sqrt(x**2 + y**2)
  t <- asin(y/d)
  w <- (d/sum(d))**weighted

  t[x == 0 & y == 0] <- 0

  ep <- ellipsePoints(t, pars)

  sum(w*(sqrt(ep$x**2 + ep$y**2) - d)**2)
}

# Fit through optim.

opt_res <- optim(c(diff(range(X$x_polar)),
                   diff(range(X$y_polar)),
                   2*pi)/2,
                 ellipseBrute,
                 x = X$x_polar, y = X$y_polar,
                 weighted = TRUE
                 )
# Check resulting ellipse throuh plot

df <- ellipsePoints(seq(0, 2*pi, length.out = 1e3), opt_res$par) 

plot(y ~ x, df, col = 'blue', t = 'l',
     xlim = range(c(X$x_polar, df$x)),
     ylim = range(c(X$y_polar, df$y)))
points(0, 0, pch = 3, col = 'blue')

points(y_polar ~ x_polar, X)