Irrlicht Engine

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// getClosestPointToEllipsoid
// gives the closest point to an axis-aligned ellipsoid
// assumes that the ellipsoid is at the origin
// treats the ellispoid-problem as a sphere-problem
// \param Point: the point you want to project onto the ellipsoid
// \param Radius: the ellipsoid radius
// \return the closest point ont the ellipsoid
// issues:
// - fails on the inner side of the ellipsoid (the more deeper "Point" is)
// - issues towards the highest peaks
// - not punctual
core::vector3df CGame::getClosestPointToEllipsoid(const core::vector3df& Point, const core::vector3df& Radius)
{
    // prepare "mn" (= "a"), "mx" (= "b"), and later "c"
    float mx = core::max_(Radius.X, Radius.Y, Radius.Z);
    float mn = core::min_(Radius.X, Radius.Y, Radius.Z);
    core::vector3df mpr = core::vector3df(mn) / Radius;
    core::vector3df rpm = Radius / core::vector3df(mn);

    // now check if the "Point" is inside, outside, or on the ellipsoid
    // the original direction to "Point"
    core::vector3df dir = core::vector3df(Point).normalize();
    // the intersection point on the ellipsoid
    core::vector3df isect = core::vector3df((dir * mpr).normalize() * Radius);

    // if it's on the ellipsoid, return
    float plen = Point.getLengthSQ(), ilen = isect.getLengthSQ();
    if (core::equals(plen, ilen)) return isect;

    // transform "Point" to sphere-space
    core::vector3df tp = Point * mpr;

    // calculate "q" (for the inside, it's really bad)
    core::vector3df q = (plen > ilen) ? (Point - tp) : (Point - (tp * mpr));
    // clamp "q" to the nearest focus point of the ellipse
    float c = sqrt((mx * mx) - (mn * mn));
    if (q.getLength() > c) q.setLength(c);
    // transform "q" to sphere-space
    core::vector3df tq = q * mpr;

    // calculate intersection of the sphere with ray "tp-tq"
    core::line3df ray(tp, tq);
    double dist = 1;
    ray.getIntersectionWithSphere(core::vector3df(), mn, dist);
    core::vector3df n = (tq - tp).normalize();
    // the intersection point
    core::vector3df i = tp + (n * dist);

    // transform the intersection point back to ellipsoid-space
    core::vector3df ti = i * rpm;

    return ti;
}