# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Curves with chord length parameterization. (English) Zbl 1205.65093
Summary: Motivated by the recent work [{\it G. Farin}, Comput. Aided Geom. Des. 23, No. 9, 722--724 (2006; Zbl 1171.65330)], we identify a family of curves that can be parameterized by chord length. The $\alpha$- and $(\alpha ,\beta )$-schemes are presented for characterizing planar and spatial curves respectively. Rational chord-length parameterizations are thoroughly investigated. In particular, the low-degree rational curves such as cubics and quartics are studied and applied to geometric Hermite interpolation. The results advise that this new class of curves, subsuming straight lines and circular arcs, have several obvious advantages over general polynomial and rational curves.

##### MSC:
 65D17 Computer aided design (modeling of curves and surfaces)
Full Text:
##### References:
 [1] Bangert, T.; Prautzsch, H.: Circles and spheres as rational splines, Neural parallel sci. Computing 5, 153-161 (1997) · Zbl 0871.68171 [2] De Boor, C.: A practical guide to splines, (1978) · Zbl 0406.41003 [3] Chou, J.: Higher order Bézier circles, Computer aided design 37, 303-309 (1995) · Zbl 0827.65015 · doi:10.1016/0010-4485(95)91140-G [4] Farin, G.: NURBS curves and surfaces, (2001) [5] Farin, G.: Rational quadratic circles are parameterized by chord length, Computer aided geometric design 23, 722-724 (2006) · Zbl 1171.65330 · doi:10.1016/j.cagd.2006.08.002 [6] Farouki, R. T.: The conformal map z - >z2 of hodograph plane, Computer aided geometric design 11, 363-390 (1994) · Zbl 0806.65005 · doi:10.1016/0167-8396(94)90204-6 [7] Farouki, R. T.; Sakkalis, T.: Real rational curves are not ”unit speed”, Computer aided geometric design 8, 151-157 (1991) · Zbl 0746.41019 · doi:10.1016/0167-8396(91)90040-I [8] Floater, M. S.; Surazhsky, T.: Parameterization for curve interpolation, Topics in multivariate approximation and interpolation, 101-115 (2005) · Zbl 1205.65036 [9] Lü, W.: Offset-rational parametric plane curves, Computer aided geometric design 12, 601-616 (1995) · Zbl 0875.68853 · doi:10.1016/0167-8396(94)00036-R [10] Sánchez-Reyes, J.: Higher-order Bézier circles, Computer aided design 29, No. 6, 469-472 (1997) [11] Zwikker, C.: The advanced geometry of plane curves and its applications, (1963) · Zbl 0151.26501