Abstract: In the space Γ_n=span1,sin t,cos t,sinh t,cosh t,t,t²,…,t^n-4, a kind of algebraic trigonometric hyperbolic basis called ATH Bézier basis is constructed by an integral approach. By the symmetry of the ATH Bézier basis, we construct an orthogonal basis called quasi-Lengendre basis, then we present the conversion formula between the ATH Bézier basis and the orthogonal basis. In addition, the optimal lower degree approximation of the ATH Bézier curves is investigated.