How to approximate an ellipse from the exterior?

In order to solve this problem about the overlap of two ellipses, we approximate the ellipses from their interior. Indeed an ellipse is convex and we construct a path by joining successive points on the perimeter. This gives a polygon contained in the ellipse.

So the area we find at the end is necessarily less than the true area: we get a lower bound of the area.

How to construct a polygon containing the ellipse (and approximating it of course), in order to get an upper bound? It does not seem easy to construct an envelope by taking small pieces of tangents.

Constructing the tangents isn't too hard, using the ellipse formula given in this answer. It gives the ellipse in parametric form; to get the slopes of the tangents, use (dy/dtheta)/(dx/dtheta), i.e.

slopes <- (-a * sin(theta) * sin(angle) + b * cos(theta) * cos(angle))/
            (-a * sin(theta) * cos(angle) - b * cos(theta) * sin(angle))

Then the intercepts come from intercepts <- y - slopes*x, where x and y are from points on the ellipse.

Finally, you would intersect successive pairs of those tangent lines to get the vertices of the outer polygon. Here's a complete solution:

innerouter <- function(x0, y0, a, b, angle, n = 360) {
    angle <- angle/360 * 2 * pi
    theta <- c(seq(0, 2 * pi, length.out = n), 0)
    
    slopes <- (-a * sin(theta) * sin(angle) + b * cos(theta) * cos(angle))/
        (-a * sin(theta) * cos(angle) - b * cos(theta) * sin(angle))
    crds <- cbind(a * cos(theta) * cos(angle) - b * sin(theta) * sin(angle) + x0,
                                a * cos(theta) * sin(angle) + b * sin(theta) * cos(angle) + y0)
    intercepts <- crds[,2] - slopes*crds[,1]
    i <- 1:(n-1)
    x <- (intercepts[i] - intercepts[i+1])/(slopes[i+1] - slopes[i])
    y <- slopes[i]*x + intercepts[i]
    outer <- cbind(c(x, x[1]),  c(y, y[1]))
    inner <- crds
    list(inner = inner, outer = outer) 
}

both <- innerouter(0,0,5, 10, 45, n=10)
plot(both$outer, col = "green", type = "l")
lines(both$inner, col = "red")

^{Created on 2023-05-10 with reprex v2.0.2}