Abstract: This paper presents a basis for the space of hyperbolic polynomials Γ_m=span{1, sht, cht, sh2t, ch2t, …, shmt, chmt} on the interval [0,α] from an extended Tchebyshev system, which is analogous to the Bernstein basis for the space of polynomial used as a kind of well-known tool for free-form curves and surfaces in Computer Aided Geometry Design. Then from this basis, we construct quasi Bézier curves and discuss some of their properties. At last, we give an example and extend the range of the parameter variable t to arbitrary close interval [r, s] (r<s).

Key words: Bernstein basis, Bézier curve, Hyperbolic polynomials, Extended Tchebyshev system, B-base

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