The Interpolation Problem The general interpolation problem We
The Interpolation Problem • The general interpolation problem Ø We have a set of abscissas and a function Ø Target: to find a polynomial P(x) such that Ø Observation: the function by the set of values , 04/03/2021 , . is not “critical”, one can substitute it. )c) Gershon Elber, Technion 1
Specific Interpolation Problems • We have a set of abscissas and a function • Target: to find a polynomial P(x) such that , • , . This is the classical Hermite interpolation. • Target: to find P(x) such that in each abscissa there is a certain degree of contact between f(x) and P(x). This is the general Hermite interpolation. • Observation: once again the function f(x) is not “critical”. 04/03/2021 )c) Gershon Elber, Technion 2
B-spline Functions Interpolation Problems • We have a set of knots • and a function • Requirements: Ø , s+1 constraints Ø , s-1 constraints Ø Two additional requirements: and 04/03/2021 )c) Gershon Elber, Technion 3
Complete Cubic Interpolation with B -splines • B-spline interpolants are polynomials by parts • There are discontinuities at knots • Requirements: Ø Degree 3 Ø Continuity C(2) • The abscissas are knots • First and immediate diagnosis: the internal knots multiplicity 1 04/03/2021 )c) Gershon Elber, Technion have 4
Complete Cubic Interpolation with B -splines • The B-spline curve: • We have to interpolate s+1 values to interpolate • We have s+3 degrees of freedom Pi 04/03/2021 )c) Gershon Elber, Technion 5
Complete Cubic Interpolation with B -splines • In order to interpolate conditions • The end points are: • In general: and we choose open end and • Two additional constraints: 04/03/2021 )c) Gershon Elber, Technion 6
Complete Cubic Interpolation with B -splines • At only and • Moreover, • Therefore, • Similarly, 04/03/2021 )c) Gershon Elber, Technion 7
Complete Cubic Interpolation with B -splines • Matriceal system of equations 04/03/2021 )c) Gershon Elber, Technion 8
Complete Cubic Interpolation with B -splines 04/03/2021 )c) Gershon Elber, Technion 9
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