Example 10 Let $x, y \in \mathbf{R}^{+}$, find the minimum value of the function
$f(x, y)=\sqrt{x^{2}-3 x+3}+\sqrt{y^{2}-3 y+3}+\sqrt{x^{2}-\sqrt{3} x y+y^{2}}$.
Solution: Due to the special structural form of the problem, it inspires special associations. As shown in Figure $5-4(1)$, consider the trihedral angle $O-ABC$, making its side edges $OA, OB, OC$ have pairwise angles of $30^{\circ}$. Take a point $P$ on $OA$ such that $OP=\sqrt{3}$. Draw a plane through point $P$ intersecting $OB$ and $OC$ at $Q$ and $R$ respectively, and let $OQ=x, OR=y$. Then the perimeter of $\triangle PQR$ is $\sqrt{x^{2}-3x+3} + \sqrt{y^{2}-3y+3} + \sqrt{x^{2}-\sqrt{3}xy+y^{2}}$.

Thus, the problem is converted to finding the minimum perimeter of the triangular section passing through point $P$.
Figure 5 - 4
Unfold this trihedral angle along the edge $OA$, as shown in Figure $5-4(2)$. Then the perimeter of $\triangle PQR$ equals the length of the broken line $PQR P'$. Clearly, $PQ + QR + RP' \geqslant PP'$, and by $\angle AOA' = 90^{\circ}$ and $OP = \sqrt{3}$, we know $PP' = \sqrt{6}$. Therefore, the minimum value of the given algebraic expression is $\sqrt{6}$.