Representation Forms of Pseudo Null Curves in the 3-D Minkowski Space

doi:10.12068/j.issn.1005-3026.2020.07.023

Abstract

Abstract: In the 3-D Minkowski space， the spacelike curves can be divided into the spacelike curves of the first kind， the second kind and the pseudo null curves according to the nature of their principal normal vectors. The representation forms of pseudo null curves were surveyed. First， two structure functions were defined by the concept of pseudo null curves. Then the Frenet frames and the curvature functions were expressed by the defined structure functions. At the same time， the relationship between the two structure functions were found. Finally， the representation forms of pseudo null curves with constant curvatures and their structure functions were given， furthermore， the corresponding examples and their graphs were given.

Key words: Minkowski space, pseudo null curve, structure function, curvature function, representation form

CLC Number:

O186

QIAN Jin-hua， TIAN Xue-qian. Representation Forms of Pseudo Null Curves in the 3-D Minkowski Space[J]. Journal of Northeastern University Natural Science, 2020, 41(7): 1061-1064.

References

[1]Kim Y H，Yoon D W.On non-developable ruled surface in Lorentz-Minkowski 3-spaces［J］.Taiwanese Journal of Mathematics，2007，11(1):197-214.
[2]Ferrandez A，Gimenez A，Lucas P.Null generalized helices in Lorentz-Minkowski spaces［J］.Journal of Physics A:Mathematical and General，2002，35(39):8243-8251.
[3]Izumiya S，Takiyama A.A time-like surface in Minkowski 3-space which contains light-like lines［J］.Journal of Geometry，1999，64(1):95-101.
[4]Ali A T，Turgut M.Position vector of a time-like slant helix in Minkowski 3-space［J］.Journal of Mathematical Analysis and Applications，2010，365(2):559-569.
[5]Ilarslan K.Spacelike normal curves in Minkowski 3-space［J］.Turkish Journal of Mathematics，2005，29(1):53-63.
[6]Ilarslan K，Nesovic E.Spacelike and timelike normal curves in Minkowski space-time［J］.Publications de I′Institut Mathematique，2009，85(99):111-118.
[7]Inoguchi J，Lee S.Null curves in Minkowski 3-space［J］.International Electronic Journal of Geometry，2008，1(2):40-83.
[8]Qian J H，Kim Y H.Null helix and k-type null slant helices in Minkowski 4-space［J］.Revista de la Union Matematica Argentina，2016，57(1):71-83.
[9]Ilarslan K，Nesovic E.Some characterizations of pseudo and partially null osculating curves in Minkowski space［J］.International Electronic Journal of Geometry，2011，4(2):1-12.
[10]Liu H L，Meng Q X.Representation formulas of curves in a two-and three-dimensional lightlike cone［J］.Results in Mathematics，2011，59(3/4):437-451.
[11]Qian J H，Kim Y H.Directional associated curves of a null curve in Minkowski 3-space［J］.Bulletin of the Korean Mathematical Society，2015，52(1):183-200.
[12]Nesovic E，Ozturk U，Ozturk E B K.On k-type pseudo null Darboux helices in Minkowski 3-space［J］.Journal of Mathematical Analysis and Applications，2016，439(2):690-700.
[13]Grbovic M，Ilarslan K，Nesovic E.On null and pseudo null Mannheim curves in Minkowski 3-space［J］.Journal of Geometry，2014，105(1):177-183.
[14]Ilarslan K，Nesovic E.Some characterizations of null，pseudo null and partially null rectifying curves in Minkowski space-time［J］.Taiwanese Journal of Mathematics，2008，12(5):1035-1044.
[15]Nesovic E，Ozturk U，Ozturk E B K，et al.On ruled surfaces with pseudo null base curve in Minkowski 3-space［J］.International Electronic Journal of Geometry，2016，9(2):9-20.
[15]关守平，房少纯.一种新型的区间-粒子群优化算法［J］.东北大学学报(自然科学版)，2012，33(10):1381-1384.(Guan Shou-ping，Fang Shao-chun.A new interval particle swarm optimization algorithm［J］.Journal of Northeastern University(Natural Science)，2012，33(10):1381-1384.)(上接第1060页)
[6]Ding H Y，Sun Y G，Tian D T.Computer-assisted cervical smears screening technology review［J］.International Journal of Biomedical Engineering，2000(2):85-90.
[7]Zhao Y H，Hong Y L，Sun C J，et al.Segmentation for cervical cancer cell based on k-means clustering algorithm［J］.Clinical Medical Engineering，2014，21(9):1089-1090.
[8]Ning Z Y，Luo W，Li Y，et al.Based on the feature of the nucleus of cervical cancer cell image identification and classification［J］.Automation and Instrumentation，2016(10):197-199.
[9]Wied G，Bartels P，Bahr G，et al.Taxonomic intra-cellular analytic system(TIICAS) for cell identification［J］.Acta Cytologica，1968，12(3):180-204.
[10]齐守良，张波，李萌，等.基于图论的分裂型人格障碍多加权脑结构网络研究［J］.东北大学学报(自然科学版)，2019，40(8):1093-1098.(Qi Shou-liang，Zhang Bo，Li Meng，et al.Graph theory based study on multi-weighted structural brain network of schizotypal personality disorder［J］.Journal of Northeastern University(Natural Science)，2019，40(8):1093-1098.)
[11]Szegedy C，Liu W，Jia Y，et al.Going deeper with convolutions［C］//2015 IEEE Conference on Computer Vision and Pattern Recognition.Boston，2015:1-9.
[12]Xiao C W，Li B，Zhu J Y，et al.Generating adversarial examples with adversarial networks［C］//Proceedings of the Twenty-Seventh International Joint Conference on Artificial Intelligence.Stockholm，2018:3905-3911.
[13]Song Y，Zhang L，Chen S，et al.A deep learning based framework for accurate segmentation of cervical cytoplasm and nuclei［C］//The 36th Annual International Conference of the IEEE Engineering in Medicine & Biology Society.Chicago，2014:2903-2906.
[14]Purwanti E，Bustomi M A，Aldian R D.Applied computing based artificial neural network for classification of cervical cancer［C］//The 6th Conference of Indonesian Students Association in Korea.Daejeon，2013:4-7.
[15]Buja L.Netter’s illustrated human pathology［M］.Beijing:People’s Medical Publishing House，2008.
[16]Macqueen J.Some methods for classification and analysis of multivariate observations［C］//Proceedings of Berkeley Symposium on Mathematical Statistics & Probability.Berkeley，1965:281-297.
[17]Alex K，Ilya S，Hg E.Imagenet classification with deep convolutional neural networks［C］//Proceedings of Neural Information Processing Systems.Lake Tahoe，2012:1097-1105.
[18]Goodfellow I J，Pouget-Abadie J，Mirza M，et al.Generative adversarial networks［J］.Advances in Neural Information Processing System，2014，3:2672-2680.
[19]Murphy K.Machine learning:a probabilistic perspective［M］.Cambridge:MIT Press，2012.
[20]Szegedy C，Vanhoucke V，Ioffe S，et al.Rethinking the inception architecture for computer vision［C］//2016 IEEE Conference on Computer Vision and Pattern Recognition(CVPR).Las Vegas，2016:2818-2826.
[15]关守平，房少纯.一种新型的区间-粒子群优化算法［J］.东北大学学报(自然科学版)，2012，33(10):1381-1384.(Guan Shou-ping，Fang Shao-chun.A new interval particle swarm optimization algorithm［J］.Journal of Northeastern University(Natural Science)，2012，33(10):1381-1384.)