Laguerre's differential geometry and kinematics

Zdeněk Jankovský

Mathematica Bohemica (1995)

  • Volume: 120, Issue: 1, page 29-40
  • ISSN: 0862-7959

Abstract

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In this paper the plane Laguerre’s geometry in the augmented plane of dual numbers is presented. Basic integral and differential invariants of -curves in the plane are deduced, i.e. the -curve arc, -curvature, -minimal curves, -circle. Furthermore the contact of -curves, -osculating circle, -evolute of a curve and some special -motions are studied from the point of view of -Differential geometry.

How to cite

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Jankovský, Zdeněk. "Laguerresche Differentialgeometrie und Kinematik." Mathematica Bohemica 120.1 (1995): 29-40. <http://eudml.org/doc/247796>.

@article{Jankovský1995,
author = {Jankovský, Zdeněk},
journal = {Mathematica Bohemica},
keywords = {Laguerre geometry in the isotropic plane; differential geometric properties of curves; Laguerre geometries; Laguerre geometry in the isotropic plane; differential geometric properties of curves},
language = {ger},
number = {1},
pages = {29-40},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Laguerresche Differentialgeometrie und Kinematik},
url = {http://eudml.org/doc/247796},
volume = {120},
year = {1995},
}

TY - JOUR
AU - Jankovský, Zdeněk
TI - Laguerresche Differentialgeometrie und Kinematik
JO - Mathematica Bohemica
PY - 1995
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 120
IS - 1
SP - 29
EP - 40
LA - ger
KW - Laguerre geometry in the isotropic plane; differential geometric properties of curves; Laguerre geometries; Laguerre geometry in the isotropic plane; differential geometric properties of curves
UR - http://eudml.org/doc/247796
ER -

References

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  1. W. Benz, Vorlesungen über Geometrie der Algebren, Springer-Verlag, Berlin-Heidelberg-New York, 1973. (1973) Zbl0258.50024MR0353137
  2. W. Blaschke G. Thomsen, Differentialgeometrie III. Differentialgeometrie der Kreise und Kugeln, Verlag J. Springer, Berlin, 1929. (1929) 
  3. F. M. Dimentberg, Theorie der Schrauben und ihre Anwendungen, Nauka, Moskva, 1978. (Russisch.) (1978) 
  4. L. Granát, Duale Zahlen und ihre Anwendungen in der Liniengeometrie und in räumlicher Kinematik, Acta polytechnica - Práce ČVUT v Praze IV (1967), no. 1, 5-19. (1967) 
  5. H. Frank, 10.1007/BF01918306, J. Geometry 7/2 (1976), 53-84. (1976) Zbl0297.50020MR0390935DOI10.1007/BF01918306
  6. I. M. Jaglom, Complex Numbers in Geometry, Leicester, 1967. (Englisch.) (1967) MR0220134
  7. Z. Jankovský, Zur Laguerreschen Ebenegeometrie, Čas. pěst. mat. 109 (1984), 236-249. (1984) MR0755587
  8. Z. Jankovský, Die Phasenbilder der Geschwindigkeitenfelder der Laguerreschen Bewegungen, Apl. mat. 33 (1988), no. 5, 374-388. (1988) MR0961315
  9. Z. Jankovský, Zu den Automorphismen der Laguerreschen Bewegungen, Beiträge zur Algebra und Geometrie 29 (1989), 35-49. (1989) MR1036089
  10. Z. Jankovský, Zur Approximation der Bahnkurven der M-Bewegung, Apl. mat. 24 (1979), no. 5, 389-395. (1979) MR0547042
  11. J. Maeda, 10.4099/jjm1924.18.0_385, Jap. J. Math. 18 (1941-43), 385-583. (1941) MR0018031DOI10.4099/jjm1924.18.0_385
  12. H. Mäurer, 10.1007/BF01112655, Math. Zeitschr. 98 (1967), 355-386. (1967) MR0216361DOI10.1007/BF01112655
  13. H. Mäurer, 10.1007/BF01917428, J. Geometry 8, 1/2 (1976), 79-93. (1976) MR0500514DOI10.1007/BF01917428
  14. H. Mäurer, 10.1007/BF01320205, Monatsht. f. Math. 86 (1978), no. 2, 131-142. (1978) MR0516836DOI10.1007/BF01320205
  15. H. Pottmann, Zur globalen Kinematik einer fünfgliedrigen Untergruppe der ebenen Laguerre-Gruppe, Čas. pěst. mat., 112 (1987), no. 4, 401-409. (1987) Zbl0633.53013MR0921331
  16. U. Schober, Zur Differentialgeometrie der Kurven in der Laguerre-Ebene, Diss. TU Karlsruhe. 1966. (1966) 
  17. E. Study, Zur Theorie der Riccatischen und der Schwarzschen Differentialgleichungen, Archiv Math. Phys. 16 (1910), 113-144. (1910) 
  18. G. I. Tlupov, Die Interpretationen der Ebenen mit der projektiven Metrik auf den Ebenen der komplexen, dualen und anormal komplexen Zahlen, Sb. nauč. rabot po izbr. voprosam mat. i mech. Vyp. 36, Kabardino-Balkarskij gos. Univ., Nalčik. 1972, pp. 157-168. (Russisch.) (1972) 
  19. T. Takasu, Differentialgeometrie in den Kugelräumen (2 Bände). II. Laguerresche Differentialgeometrie, Tokyo, 1939. (1939) 
  20. A. T. Yang F. Freudenstein, 10.1115/1.3629601, ASME J. Appl. Mech. 31 (1964), 300-308. (1964) MR0163452DOI10.1115/1.3629601
  21. A. T. Yang, 10.1115/1.3591499, ASME, J. Eng. Ind. 91 (1969), 152-157. (1969) DOI10.1115/1.3591499

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