The Degree Elevation Of Algebraic Hyperbolic B-spline Curves

The NURBS has become a de facto standard for the representation of curves and surfaces inCAGD. However, the higher expressive power of a rational model comes at the expense ofintroducing several drawbacks. The algebraic hyperbolic(AH) B-splineis new model introducedrecently which can represent exactly some remarkable curves such as the hyperbola and thecatenary. We consider the degree elevation problem of algebraic hyperbolic B-spline curves inthis paper.The degree elevation algorithm is the foundational algorithm of CAGD, first we present theexact description of the degree elevation problem of AH B-spline curves. It's well known that thedegree elevation of Bézier curves is comer cutting. Comer cutting is attractive since it providessimple geometric constructions of curves and can be implemented easily and efficiently. In thispaper we construct a degree elevation algorithm which can be interpreted as comer cutting, wealso analyze the convergence of control polygons.In order to raise the degree of the algebraic hyperbolic B-spline curves, we present a newclass of basis functions, to be called bi-order algebraic hyperbolic B-spline basis functions, thisclass of basis functions don't have unique order, but have the bi-order property. We also analyzethe properties of bi-order AH B-spline basis functions. The transforming formulas between AHB-spline and bi-order AH B-spline lead to the comer cutting for degree elevation of AH B-splinecurves.It's well known that the repeated degree elevation of Bézier curves and B-spline curvesleads to a sequence of control polygons that converges to the curves. In this paper we firstestimate the integral of the AH B-spline basis function, then we prove that the control polygonsconverge to the AH B-spline curves when we repeat degree elevation of AH B-spline curves.