= Solution
Take the lower coordinate in each direction from the smaller lower bound, and the upper coordinate from the larger upper bound. This gives an <axis-aligned bounding box>, rather than the generally nonrectangular <set union> of the two regions. Use <defensive copying> because the paper's `Point` has mutable fields.
``
final class Rect {
private final Point lowerLeft, upperRight;
public Rect(Point ll, Point ur) {
if (ll == null || ur == null)
throw new IllegalArgumentException("null corner");
double xl = ll.getX(), yl = ll.getY();
double xh = ur.getX(), yh = ur.getY();
if (!Double.isFinite(xl) || !Double.isFinite(yl)
|| !Double.isFinite(xh) || !Double.isFinite(yh)
|| xl > xh || yl > yh)
throw new IllegalArgumentException("invalid rectangle");
lowerLeft = new Point(xl, yl);
upperRight = new Point(xh, yh);
}
public Rect Union(Rect r) {
return new Rect(
new Point(Math.min(lowerLeft.getX(), r.lowerLeft.getX()),
Math.min(lowerLeft.getY(), r.lowerLeft.getY())),
new Point(Math.max(upperRight.getX(), r.upperRight.getX()),
Math.max(upperRight.getY(), r.upperRight.getY())));
}
public Point lowerLeft() { return lowerLeft.clone(); }
public Point upperRight() { return upperRight.clone(); }
}
``
Any containing <rectangle> must reach at least as far as each input <rectangle> in all four coordinate directions. The constructed <rectangle> reaches exactly those four limits, which proves both containment and minimality:
$$
\boxed{[\min(x_l,x'_l),\max(x_h,x'_h)]\times[\min(y_l,y'_l),\max(y_h,y'_h)].}
$$
The corner accessors return copies, so changing a constructor argument or returned `Point` cannot alter an existing `Rect`.
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