Radial extension of a square boundary homeomorphism

A homeomorphism $g:\partial Q\to \partial Q$ of the boundary of a closed square $Q$ extends to a homeomorphism of $Q$ by fixing its centre $c$ and setting $H(c+r(p-c))=c+r(g(p)-c)$ for $p\in \partial Q$ and $0<r\le 1$. Each noncentral point lies on a unique ray from $c$ meeting the boundary once, so the formula is well defined and its inverse is obtained from $g^{-1}$. Both maps are continuous off the centre by the radial parametrization, and at the centre because their distances from $c$ are bounded by $r$ times the radius of the square. It follows that complements of any two closed boundary arcs are homeomorphic: first map the arcs to one another by a boundary homeomorphism, then extend radially.