Question

I have a 3D ellipsoid function:

ellipsoid <- function(center=c(0, 0, 0), radius=1, shape=diag(3),
  segments=51) {
    angles <- (0:segments)*2*pi/segments
    ecoord2 <- function(p) {
      c(cos(p[1])*sin(p[2]), sin(p[1])*sin(p[2]), cos(p[2])) }
    unit.sphere <- t(apply(expand.grid(angles, angles), 1, ecoord2))
    t(center + radius * t(unit.sphere %*% chol(shape))) 
  }

that makes an ellipsoid with a given center and radius. Then I can draw it using:

q <- quads3d(ellips[,1], ellips[,2], ellips[,3], front="lines",
  back="lines", alpha=.5, 
                  lit=FALSE, col=surface.col[1])

But, how can I determine if a point (x,y,z) falls inside this ellipsoid? Specifically, how do I figure out the semiaxes of the ellipsoid?

for instance,

fitsInEllipsoid <- function(ellipsoid, x, y, z) {
#returns true if (x,y,z) fits inside the ellipsoid
}

Answer 1

A point (x,y,z) is inside if

where a and b are the equatorial radii (along the x and y axes) and c is the polar radius (along the z-axis), i.e. the square root of the diagonal of the shape parameter.
The center of the ellipsoid is denoted by (x0, y0, z0) (variable center in your function).