The group \(\text{SL}(2,\mathbb{Z})\) acts on the circle consisting of rays from the origin in \(\mathbb{R}^2\). Every nontrivial parabolic element (transvection) has 2 fixed points. The restriction of the action to one of the invariant arcs extended by the identity on the other arc is called a half transvection. It is shown that the group \(G\) generated by half transvections is isomorphic to the Higman-Thompson group \(T\) which was known to be a finitely presented infinite simple group. Using the isomorphism, new finite presentations of \(T\) are obtained.