Four point parabolic interpolation

We show that four points in the plane may be interpolated by one or two parabolas or possibly by no parabola, depending on the configuration of points. We provide methods for distinguishing the cases and constructing the parabolas.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/29347/1/0000415.pd

Constructing parametric quadratic curves

Abstract Constructing a parametric spline curve to pass through a set of data points requires assigning a knot to each data point. In this paper we discuss the construction of parametric quadratic splines and present a method to assign knots to a set of planar data points. The assigned knots are invariant under a ne transformations of the data points, and can be used to construct a parametric quadratic spline which reproduces parametric quadratic polynomials. Results of comparisons of the new method with several known methods are included

Geometric four-point parabolic interpolation

V delu diplomskega seminarja bomo obravnavali interpolacijo štirih točk v ravnini s parametrično podano parabolično krivuljo. Dokazali bomo izrek, ki povezuje število interpolacijskih krivulj skozi dane točke z obliko lika, katerega oglišča so te točke, in opisali praktično konstukcijo interpolacijske krivulje na primerih. Istega problema se bomo lotili še s pomočjo kubičnih Lagrangeevih baznih polinomov, ki jim bomo s pravilno izbiro prostih parametrov znižali stopnjo in tako dobili parabolično krivuljo. Obravnavali bomo Hermitov problem, torej problem interpolacije dveh točk in tangentnih vektorjev v teh točkah s parabolično krivuljo, nazadnje pa bomo numerično izračunali red konvergence pri aproksimaciji parametrično podanih krivulj s paraboličnimi krivuljami.In this thesis we present the solution to four-point parabolic interpolation problem. The theorem that shows how the number of interpolation curves is related to the shape of the quadrilateral that has the given points as its vertices is proven and the construction of the interpolant in some practical examples is described. The same problem is solved again with a different approach, that is with cubic Lagrange polynomials. We find such parameters that lower the interpolant’s degree to obtain a parabolic curve. Furthermore, the Hermite’s problem is discussed, where we find a parabolic interpolant for two points and two tangent vectors. Lastly, we numerically calculate the convergence rate for approximation of parametrically given curves with parabolic curves