Abstract: In this paper, we present two new unified mathematics models of conics and polynomialcurves, called \it algebraic hyperbolic trigonometric (\it AHT)\it B\'ezier curves and \it non-uniform algebraic hyperbolictrigonometric (NUAHT) B-spline curves of order n, which aregenerated over the space \rm span\\sin t,\cos t,\sinh t,\cosht,1,t,\ldots,t^n-5\, n\ge 5. The two kinds of curvesshare most of the properties as those of the B\'ezier curves andB-spline curves in polynomial space. In particular, they canrepresent exactly some remarkable transcendental curves such as thehelix, the cycloid and the catenary. The subdivision formulae ofthese new kinds of curves are also given. The generations of thetensor product surfaces are straightforward. Using the newmathematics models, we present the control mesh representations oftwo classes of minimal surfaces.