Hallo,
ja, Udo Du hast Recht, alle Splines sind gleich gut, und
das Konstruieren eines InterpolationFunction[] mit stetigen
Ableitungen ist ein Kinderspiel.
Ich liebe z. B. Catmull-Rom Splines die auch stetige Ableitungen
haben und man braucht daf"ur auch keine Gleichungssysteme
l"osen. MathGL3d benutzt die z. B. um die Volumendaten & deren
Gradienten zu interpolieren
Leider will Herr Heitmann nun mal klassische kubische
Splines. Das kann man ihm nicht ausreden.
Weil man es Ihm nicht ausreden kann h"angt ein Notebook mit
den klassischen kubischen Splines dran, leider war ich zu faul
noch schicke Optionen f"ur die Ableitungen an den Enden zu machen
...
"Ubrigens, wenn das Bild auf der home-page von M. Heitmann echt ist,
dann macht er wohl schon lange *keine* "(Haus)Aufgaben" mehr.
Gruss
Jens
Udo und Susanne Krause wrote:
>
> Guten Tag miteinander,
>
> unbekannt ist, ob Martin unterdessen seine (Haus)Aufgabe gelöst hat; wenn nicht,
> kann er einen Blick in das beiliegende Notebook werfen. Man muss verstehen, dass
> die 1. und 2. (..., n., ...) Ableitung bei einer Liste (endlicher Länge) von
> Messpunkten reine Fiktion sind. Man kann daher, wie von Hrn. Kuska vorgeschlagen,
> die Ableitungen aus einem SplineFit Objekt beziehen. Man kann ebensogut durch je 3
> aufeinanderfolgende (Mess)Punkte einen Kreis legen (dieser ist eindeutig
> bestimmt), von dieser Kreislinie die erste und zweite Ableitung im mittleren Punkt
> nehmen und das Ganze bei Interpolation[] abliefern. Das ist im Notebook
> heimannSpline.nb implementiert. Die Ergebnisse sind interessant: die kubischen
> Splines neigen im Sinne praller Kurven zur Erzeugung lokaler Extrema ausserhalb
> der Messpunkte. Das macht die hier implementierte "Kreismethode" weniger (vor der
> allfälligen Verwendung müssten Sie alle Formeln kontrollieren, Martin (this IS a
> disclaimer)). Die "Kreismethode" ist sinnvoll, wenn an inhaltlich interessanten
> Stellen mehr Messpunkte erhoben werden, dann kommt sie den Intentionen von C. F.
> Gauss nahe, der erstmals solche Konstruktionen in der Geometrie verwandte.
> Natürlich liefert sie ein InterpolatingFunction Objekt, sie liefert
> vorteilhafterweise keine kubischen Splines - und sie ist billig.
>
> Mit den besten Grüssen
> Udo.
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