Shape Preserving Interpolation to Convex Data by Rational Functions with Quadratic Numerator and Linear Denominator

C2rational quartic interpolation spline with local shape preserving property

“…For all w i = µ i = 0, S(x) is precisely the C 1 convexity-preserving rational quadratic/linear interpolant given in [13]. For all w i = 1 and µ i = 0, S(x) is the C 2 convexity-preserving rational quartic interpolation spline developed in [12].…”

“…This form is the C 1 convexity-preserving interpolant given in [13]. This means that a large value of λ i will give approximately a convexity-preserving rational quadratic/linear interpolant in the subinterval [x i , x i+1 ].…”

“…By introducing a new family of local shape parameters w i into the rational quartic interpolation spline given in [15], we give an explicit representation of a general C 2 rational quartic interpolation spline with three families of local shape parameters, which possesses local shape-preserving property and includes several interpolation splines developed in [12,13,14,15] as special cases. The new introduced local shape parameters w i have foreseeable shape control effect on generating interpolation curves, which can provide users more degrees of freedom to fulfill their different design intent.…”

Polynomial cubic splines with tension properties

“…Spaces of the form (8) are fundamental in the following section and have been introduced several years ago (see, e.g., the pioneering papers Delbourgo and Gregory, 1985;Delbourgo, 1989) to solve shape-preserving interpolation problems. μ and ν act as tension parameters; indeed, if ψ satisfies (7) and ρ is the straight line trough the points (0,…”

Two shape preserving lagrangeC 2-interpolants

“…Delbourgo and Gregory [3][4][5] solve the problem by using rational splines and without introducing further knots of interpolation. They obtain O(h 4) error when the function to be interpolated is c~4.…”