Abstract
The chapter is devoted to the study of plane curves from the point of view of differential geometry. We focus our attention to classical curves and also on conics, and provide examples and other applications in architectural elements of the mathematical theory.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
We thank Kim Williams for completing this information.
- 2.
A more general procedure to obtain the parametrization by lines of a given irreducible conic and other classes of curves can be found in Section 4.6 of Sendra et al. (2007).
References
Brander, D., A. Bærentzen, K. Clausen, Fisker, A.-S. Gravesen, M. N. Lund, T. B. Nørbjerg, K. Steenstrup, and A. Søndergaard. (2016). Designing for hot-blade cutting. In Proc. AAG 2016, 305–326.
Burden, R. L., and J. D. Faires. (2000). Numerical Analysis. Brooks/Cole.
Caliò, F., and E. Marchetti. (2015). Generation of Architectural Forms Through Linear Algebra. In Architecture and Mathematics from Antiquity to the Future, ed. K. Williams and M. Ostwald. Birkhäuser, Cham.
Čučaković, A., and M. Paunović. (2015). Cylindrical mirror anamorphosis and urban-architectural ambience. Nexus Network Journal 17:605–622.
do Carmo, M.-P. (1976). Differential geometry of curves and surfaces. Englewood Cliffs, NJ: Prentice-Hall, Inc.
Dunham, D. (2003). Hyperbolic spirals and spiral patterns. Proceedings of Bridges 2003: Mathematics, Music, Art, Architecture, Education, Culture, 521–528.
Echevarría, L., C. Garnica, and J. P. Gutiérrez. (2014). La costilla laminar del Instituto de Ciencias de la Construcción Eduardo Torroja (IETcc-CSIC). Levantamiento mediante láser-escáner y evaluación estructural. Informes de la Construcción 66(536), e038. https://doi.org/10.3989/ic.14.116.
Gailiunas, P. (2014). Recursive rosettes. Proceedings of Bridges 2014: Mathematics, Music, Art, Architecture, Education, Culture, 127–134.
Gerbino, A. (2014). Geometrical objects. Architecture and the mathematical sciences 1400-1800. Springer International Publishing.
Gibson, C. (2001). Elementary geometry of differentiable curves: An undergraduate introduction. Cambridge: Cambridge University Press.
Hanh, A. J. (2012). Mathematical excursions to the World’s Great Buildings. Princeton University Press.
Hart, G., and E. Heathfield. (2018). Catenary arch constructions. Proceedings of Bridges 2018: Mathematics, Art, Music, Architecture, Education, Culture, 325–332.
Kimberling, C. (2004). The shape and history of the ellipse in washington, d.c. Proceedings of Bridges 2004: Mathematics, Music, Art, Architecture, Education, Culture, 1–12.
Lang, S. (1986). Introduction to linear algebra. Undergraduate Texts in Mathematics. New York: Springer-Verlag.
Langbein, W., and L. B. Leopold. (1970). River Meanders and the Theory of Minimum Variance. In Rivers and River Terraces. Geographical Readings, ed. G.H. Dury. London: Palgrave Macmillan.
Lastra, A., J. R. Sendra, and J. Sendra. (2018). Hybrid trigonometric varieties. Preprint. https://arxiv.org/pdf/1711.07728.pdf.
Marchetti, E., and L. R. Costa. (2015). What geometries in milan cathedral? architecture and mathematics from antiquity to the future, vol. i, 509–534. Cham: Birkhäuser/Springer.
Picon, A. (2010). Culture numérique et architecture : une introduction. Editions Birkhäuser.
Pottmann, H., A. Asperl, M. Hofer, and A. Kilian. (2007). Architectural geometry. Bentley Institute Press.
Sendra, J., F. Winkler, and S. Perez-Diaz. (2007). Rational algebraic curves: A computer algebra approach. Series: algorithms and computation in mathematics, Vol. 22. Springer Verlag.
Sharpe, T. (2010). The role of aesthetics, visual and physical integration in building mounted wind turbines - An alternative approach, paths to sustainable energy. Jatin Nathwani and Artie Ng. IntechOpen.
Shen, Y., E. Zhang, Y. Feng, S. Liu, and J. Wang. (2021). Parameterizing the curvilinear roofs of traditional chinese architecture. Nexus Network Journal, 475–492.
Tapp, K. (2016). Differential geometry of curves and surfaces. Undergraduate Texts in Mathematics. Springer.
Umehara, M., Y. Kotaro, and W. Rossman. (2017). Differential geometry of curves and surfaces. World Scientific.
Wikipedia (2019b). List of hyperbolid structures, wikipedia.com, accessed 30 april 2019. https://en.wikipedia.org/wiki/List_of_hyperboloid_structures.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2021 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this chapter
Cite this chapter
Lastra, A. (2021). Parametrizations and Plane Curves. In: Parametric Geometry of Curves and Surfaces. Mathematics and the Built Environment, vol 5. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-81317-8_1
Download citation
DOI: https://doi.org/10.1007/978-3-030-81317-8_1
Published:
Publisher Name: Birkhäuser, Cham
Print ISBN: 978-3-030-81316-1
Online ISBN: 978-3-030-81317-8
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)