We study the problem of determining the complexity of the lower envelope of a collection of $n$ geometric objects. For collections of rays; unit length line segments; and collections of unit squares to which we apply at most two transformations from translation, rotation, and scaling, we prove a complexity of $\Theta(n)$.If all three transformations are applied to unit squares, then we show the complexity becomes $\Theta\big(n\alpha(n)\big)$, where $\alpha(n)$ is the slowly growing inverse of Ackermann's function.