B-Spline Pythagorean Hodograph Curves in Clifford Algebras

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216208%3A11320%2F23%3A10474949" target="_blank" >RIV/00216208:11320/23:10474949 - isvavai.cz</a>

Nalezeny alternativní kódy

RIV/49777513:23520/23:43967429

Výsledek na webu

<a href="https://verso.is.cuni.cz/pub/verso.fpl?fname=obd_publikace_handle&handle=sdMltY-_xv" target="_blank" >https://verso.is.cuni.cz/pub/verso.fpl?fname=obd_publikace_handle&handle=sdMltY-_xv</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1007/s00006-022-01255-7" target="_blank" >10.1007/s00006-022-01255-7</a>

Jazyk výsledku

angličtina

Název v původním jazyce

B-Spline Pythagorean Hodograph Curves in Clifford Algebras

Popis výsledku v původním jazyce

In several recent publications B-spline functions appeared with control points from abstract algebras, e.g. complex numbers, quaternions or Clifford algebras. In the context of constructions of Pythagorean hodograph curves, computations with these B-splines occur, mixing the components of the control points. In this paper we detect certain unifying patterns common to all these computations. We show that two essential components can be separated. The first one is the usual B-spline function squaring and integration, producing a new knot sequence and a new array of real coefficients for the control point computation. The second one is a special commutative multiplication which can be defined even in non-commutative algebras. We use this general Clifford algebra based approach to reconstruct some known results for the signatures (2, 0), (3, 0) and (2, 1) and add a new construction for the signature (3, 1). This last case is essential for the description of canal surfaces. It is shown that Clifford algebra is an especially suitable tool for the general description of B-spline curves with Pythagorean hodograph property. The presented unifying definition of PH B-splines is general and is not limited to any particular knot sequences or control points. In a certain sense, this paper can be considered as a continuation of the 2002 article by Choi et al. with regard to the B-splines.

Název v anglickém jazyce

B-Spline Pythagorean Hodograph Curves in Clifford Algebras

Popis výsledku anglicky

In several recent publications B-spline functions appeared with control points from abstract algebras, e.g. complex numbers, quaternions or Clifford algebras. In the context of constructions of Pythagorean hodograph curves, computations with these B-splines occur, mixing the components of the control points. In this paper we detect certain unifying patterns common to all these computations. We show that two essential components can be separated. The first one is the usual B-spline function squaring and integration, producing a new knot sequence and a new array of real coefficients for the control point computation. The second one is a special commutative multiplication which can be defined even in non-commutative algebras. We use this general Clifford algebra based approach to reconstruct some known results for the signatures (2, 0), (3, 0) and (2, 1) and add a new construction for the signature (3, 1). This last case is essential for the description of canal surfaces. It is shown that Clifford algebra is an especially suitable tool for the general description of B-spline curves with Pythagorean hodograph property. The presented unifying definition of PH B-splines is general and is not limited to any particular knot sequences or control points. In a certain sense, this paper can be considered as a continuation of the 2002 article by Choi et al. with regard to the B-splines.

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

—

OECD FORD obor

10102 - Applied mathematics

Projekt

<a href="/cs/project/GA20-11473S" target="_blank" >GA20-11473S: Symetrie a invariance v analýze, geometrickém modelování a teorii optimálního řízení</a><br>

Návaznosti

P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)

Rok uplatnění

2023

Kód důvěrnosti údajů

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Název periodika

Advances in Applied Clifford Algebras

ISSN

0188-7009

e-ISSN

1661-4909

Svazek periodika

33

Číslo periodika v rámci svazku

1

Stát vydavatele periodika

CH - Švýcarská konfederace

Počet stran výsledku

23

Strana od-do

9

Kód UT WoS článku

000906898000001

EID výsledku v databázi Scopus

2-s2.0-85145611027