zbMATH — the first resource for mathematics

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.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
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)
Nonexistence of rational rotation-minimizing frames on cubic curves. (English) Zbl 1172.65327
Summary: We prove there is no rational rotation-minimizing frame (RMF) along any non-planar regular cubic polynomial curve. Although several schemes have been proposed to generate rational frames that approximate RMF’s, exact rational RMF’s have been only observed on certain Pythagorean-hodograph curves of degree seven. Using the Euler-Rodrigues frames naturally defined on Pythagorean-hodograph curves, we characterize the condition for the given curve to allow a rational RMF and rigorously prove its nonexistence in the case of cubic curves.

65D17Computer aided design (modeling of curves and surfaces)
53A04Curves in Euclidean space
Full Text: DOI
[1] Altmann, S. L.: Rotations, quaternions, and double groups, (1986) · Zbl 0683.20037
[2] Bishop, R. L.: There is more than one way to frame a curve, American mathematical monthly 82, 246-251 (1975) · Zbl 0298.53001 · doi:10.2307/2319846
[3] Choi, H. I.; Han, C. Y.: Euler -- rodrigues frames on spatial Pythagorean-hododgraph curves, Computer aided geometric design 19, 603-620 (2002) · Zbl 1043.53005 · doi:10.1016/S0167-8396(02)00165-6
[4] Choi, H. I.; Lee, D. S.; Moon, H. P.: Clifford algebra, spin representation, and rational parameterization of curves and surfaces, Advances in computational mathematics 17, 5-48 (2002) · Zbl 0998.65024 · doi:10.1023/A:1015294029079
[5] Farouki, R. T.: Exact rotation-minimizing frames for spatial Pythagorean-hodograph curves, Graphical models 64, 382-395 (2003) · Zbl 1055.68124 · doi:10.1016/S1524-0703(03)00002-X
[6] Farouki, R. T.; Han, C. Y.: Rational approximation schemes for rotation-minimizing frames on Pythagorean-hodograph curves, Computer aided geometric design 20, 435-454 (2003) · Zbl 1069.65551 · doi:10.1016/S0167-8396(03)00095-5
[7] Farouki, R. T.; Han, C. Y.; Manni, C.; Sestini, A.: Characterization and construction of helical polynomial space curves, Journal of computational and applied mathematics 162, 365-392 (2004) · Zbl 1059.65016 · doi:10.1016/j.cam.2003.08.030
[8] Farouki, R. T.; Al-Kandari, M.; Sakkalis, T.: Structural invariance of spatial Pythagorean-hodographs, Computer aided geometric design 19, 365-407 (2002)
[9] Farouki, R. T.; Sakkalis, T.: Pythagorean hodographs, IBM J. Res. develop. 34, 736-752 (1990)
[10] Jüttler, B.; Mäurer, C.: Cubic Pythagorean hodograph spline curves and applications to sweep surface modeling, Computer-aided design 31, 73-83 (1999) · Zbl 1054.68748 · doi:10.1016/S0010-4485(98)00081-5
[11] Jüttler, B.; Wagner, M. G.: Rational motion-based surface generation, Computer-aided design 31, 203-213 (1999) · Zbl 1053.68744 · doi:10.1016/S0010-4485(99)00016-0
[12] Klok, F.: Two moving coordinate frames for sweeping along a 3D trajectory, Computer aided geometric design 3, 217-229 (1986) · Zbl 0631.65145 · doi:10.1016/0167-8396(86)90039-7
[13] Lyons, D. W.: An elementary introduction to the Hopf fibration, Mathematics magazine 76, 87-98 (2003)
[14] Mäurer, C.; Jüttler, B.: Rational approximation of rotation minimizing frames using Pythagorean-hodograph cubics, Journal for geometry and graphics 3, 141-159 (1999) · Zbl 0976.53003 · emis:journals/JGG/3.2/2.html
[15] Wang, W.; Joe, B.: Robust computation of the rotation minimizing frame for sweep surface modeling, Computer-aided design 29, 379-391 (1997)