# Hyperbolic Spline Interpolation (Hyperbolic Spline Interpolation)

Jump to navigation
Jump to search

## Description

The problem of restoring complex curves and surfaces from discrete data so that their shape is preserved is called isogeometric interpolation. A very popular tool for solving this problem are hyperbolic splines in tension, which were introduced in 1966 by Schweikert. These splines have smoothness sufficient for many applications; combined with algorithms for the automatic selection of the tension parameters, they adapt well to the given data. Unfortunately, the evaluation of hyperbolic splines is a very difficult problem because of roundoff errors (for small values of the tension parameters) and overflows (for large values of these parameters).�

## Parameters

$n$: number of points

## Table of Algorithms

Name | Year | Time | Space | Approximation Factor | Model | Reference |
---|---|---|---|---|---|---|

B.I. Kvasov | 2008 | $O(n^{3} \log^{2}K)$ | $O(n)$? | Exact | Deterministic | Time |

V. A. Lyul’ka and A. V. Romanenko | 1994 | $O(n^{5})$ | Exact | Deterministic | Time | |

V. A. Lyul’ka and I. E. Mikhailov | 2003 | $O(n^{4})$ | Exact | Deterministic | Time | |

V. I. Paasonen | 1968 | $O(n^{5} \log K)$ | Exact | Deterministic | ||

P. Costantini, B. I. Kvasov, and C. Manni | 1999 | $O(n^{5} \log K)$ | $O(n)$? | Exact | Deterministic | Time |

B. I. Kvasov | 2000 | $O(n^{4})$ | $O(n)$?? | Exact | Deterministic | Time |