Abstract
Matching curves to represent diverse fold morphologies is an active research field in structural geology that can have far-reaching implication in resource studies and in engineering geology. Further, some of the recent literatures show that 2D and 3D restoration and best fit of folds have been researched actively in petroleum geosciences. The significant advantage of the method presented here using cubic Bézier curve is that the profile of a fold could be represented in terms of four “controlling points”, which can be re-synthesized in 2D graphical plot using the spreadsheet programme such as Microsoft Excel, Apache Open Office Spreadsheet etc. by simply developing a tabular spreadsheet based on the equation of cubic Bézier curve. The method is simple and has been tested successfully to synthesize a few fold profiles by changing the values of coordinates of the controlling points and on photographs of two natural examples of folds. Bézier curves of different order have been used along with multi-paradigm programming/numerical computing software such as MATLAB, mathematical symbolic computation program: Wolfram Mathematica, and vector graphics designing. However, it is not easy for all learners or researchers to rapidly use or develop such programmes. On the other hand, spreadsheets programmes, both commercial and open-source, well known to the majority of the population having a general knowledge in computers.
This is a preview of subscription content, log in via an institution to check access.
Source of the background photo is Fig. 1.71 of Mukherjee (2015)
Similar content being viewed by others
References
Agoston MK (2005) Computer graphics and geometric modeling: implementation and algorithms. Springer, Verlag, London, p 907
Aller J, Bobillo-Ares NC, Bastida F, Lisle RJ, Menéndez CO (2010) Kinematic analysis of asymmetric folds in competent layers using mathematical modelling. J Struct Geol 32:1170–1184
Bastida F, Aller J, Bobillo RS (1999) Geometrical analysis of folded surfaces using simple functions. Tectonophys 21:729–742
Bézier P (1966) Definition numérique des courbeset surfaces; I. Automatisme 11:625–632
Bézier P (1967) Definition numérique des courbeset surfaces; II. Automatisme 12:17–21
Biswas B, Lovell BC (2008) Bézier and Splines in Image Processing and Machine Vision. Springer, London, pp 8–9
Biswas T, Mukherjee S. Submitted. Non-uniform B-spline curve analyses of sigmoid brittle shear Y- and ductile shear S-planes. Int J Earth Sci.
Chew DM (2003) An Excel Spreadsheet for finite strain analysis using the Rf/φ technique. Comp. Geosci. 29:795–799
Chun L, Zhang Y, Wang Y (2009) Analysis of complete fold shape based on quadratic Bézier curves. J Struct Geol 31:575–581
Davies BL, Robotham AJ, Yarwood A (1986) Computer-aided drawing and design. Chapman and Hall, London, p 328p
de Cemp EA (1999) Visualization of complex geological structures using 3-D Bézier construction tools. Comput Geosci 25:581–597
De Paor DG (1996) Bézier curves and geological design: structural geology and personal computers. Elsevier Science Ltd., Amsterdam, pp 389–417
Fletcher R (1979) The shape of single-layer folds at small but finite amplitude. Tectonophys 60(256):77–87
Ghassemi MR, Schmalholz SW, Ghassemi AR (2010) Kinematics of constant arc length folding for different fold shapes. J Struct Geol 32:755–765
Gogoi MP, Mukherjee S (2019) Synthesis of folds in 3D with Bézier surface. In: Billi A, Fagereng A (Eds) Problems and solutions in structural geology and tectonics. Developments in structural geology and tectonics book series. Volume 5. Series Editor: Mukherjee S. Elsevier. pp. 279–290. ISSN: 2542–9000. ISBN: 9780128140482
Gogoi MP, Mukherjee S, Goswami TK (2017) Analyses of fold profiles by changing weight parameters of NURB Curves. J Earth Sys Sci 126:98
Gueziec A (1996) Spline curves and surfaces from data modeling. In: Marcus LF, Corti M, Loy A, Naylor GJP, Slice DE (eds) Advances in morphometrics. Plenum Press, New York, pp 253–262
Hjlle O, Petersen SA, Bruaset AMA (2013) Numerical framework for modeling folds in structural geology. Math Geol 45:255–276
Janke SJ (2015) Mathematical structures for computer graphics. John Wiley & Sons, Inc., pp 223–230. ISBN: 978-1-118-71219-1
Karataş M (2013) A multi foci closed curve: cassini oval, its properties and applications. Doğuş Üniversitesi Dergisi 14(2):231–248
Lisle RJ, Fernandez Martinez JL, Bobillo-Ares N, Menendez O, Aller J, Bastida F (2006) Fold profiler: a matlab based program for fold shape classification. Comp Geosci 32:102–108
Liu C, Zhang Y, Wang Y (2009) Analysis of complete fold shape based on quadratic Bézier curves. J Struct Geol 31:575–581
Liu C, Zhang Y, Shi B (2009) Geometric and kinematic modeling of detachment folds with growth strata based on Bézier curves. J Struct Geol 31:260–269
Marsh D (2005) Applied geometry for computer graphics and CAD, 2nd edn. Springer-Verlag, London, p 350
Masood A, Ejaz S (2010) An efficient algorithm for robust curve fitting using cubic Bézier curves. In: Huang D-S, Zhang X, Reyes Garcia CS, Zhang L (eds) Advanced intelligent computing theories and applications: with aspects of artificial intelligence. Springer-Verlag, Berlin, pp 255–262
Mukherjee S (2014) Review of flanking structures in meso- and micro-scales. Geol Mag 151:957–974
Mukherjee S (2015) Atlas of structural geology. Elsevier, Amsterdam
Mukherjee S, Koyi HA (2010) Higher himalayan shear zone, sutlej section- structural geology and extrusion mechanism by various combinations of simple shear, pure shear and channel flow in shifting modes. Int J Earth Sci 99:1267–1303
Mukherjee S, Punekar J, Mahadani T, Mukherjee R (2015) A review on intrafolial folds and their morphologies from the detachments of the western Indian Higher Himalaya. In: Mukherjee S, Mulchrone KF (eds) Ductile shear zones: from micro- to macro-scales. Wiley Blackwell, Hoboken, pp 182–205
Parent R (2012) Computer animation: algorithms and techniques, 3rd edn. Elsevier, Amsterdam, p 460
Riskus A (2006) Approximation of a cubic Bézier curve by circular arcs and vice versa. Information Technology and Control. p 35
Rouby D, Xiao H, Suppe J (2000) 3-D restoration of complexly folded and faulted surfaces using multiple unfolding mechanisms. AAPG Bull 84:805–829
Sederberg TW (2016) Computer aided geometric design course notes; September 28
Shao L, Zhou H (1996) Curve fitting with Bézier cubics. Graph Models Image Process 58:223–232
Stabler CL (1968) Simplified Fourier analysis of fold shapes. Tectonophys 6:343–350
Venkataraman P (2009) Applied optimization with MATLAB programming, 2nd edn. Wiley, Hoboken, p 462
Vince J (2010) Mathematics for computer graphics, 3rd edn. Springer-Verlag, London, p 310
Zhong D, Li M, Wang G, Wei H (2004) NURBS-based 3D graphical modeling and visualization of geological structures. In: Third International Conference on Image and Graphics (ICIG'04) (pp 414–417). IEEE
Acknowledgements
Self-funded research. Thanks to the authors’ Head of the Departments for providing research facilities. We thank Chief Editor: Wolf-Christian Dullo, Managing Editor: Monika Dullo, Reviewers: Anonymous and Subhobroto Mazumder and the Springer proofreading team for handling the work, sending review, and preparing the proof.
Author information
Authors and Affiliations
Corresponding author
Electronic supplementary material
Below is the link to the electronic supplementary material.
Rights and permissions
About this article
Cite this article
Gogoi, M.P., Mukherjee, S. & Goswami, T.K. Analyses of fold profiles using cubic Bézier curve. Int J Earth Sci (Geol Rundsch) 110, 183–191 (2021). https://doi.org/10.1007/s00531-020-01945-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00531-020-01945-2