Abstract
In this chapter we introduce the first interesting class of planar curves, namely the conic sections. This leads to a first discussion of singular and non singular points. Closely tied to these concepts is the notion of the tangent at a point on a curve. We then move on to a discussion of curves of higher degrees, and introduce the concepts of tangent lines, the tangent cone and the multiplicity of a point on a curve which may have singularities. A number of important examples of higher order curves are discussed. Elliptic curves are briefly discussed, this class of curves (which are certainly not ellipses) played an important role for the fruitful interplay between geometry and function theory, so central in the pathbreaking work of Niels Henrik Abel.
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- 1.
We also say conic curve or just a conic.
- 2.
This concept will be explained in more detail in Sect. 2.9.
References
Holme, A.: Geometry. Our Cultural Heritage. Springer, Berlin (2001)
Holme, A.: Geometry. Our Cultural Heritage, 2nd edn. Springer, Berlin (2010)
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© 2012 Springer-Verlag Berlin Heidelberg
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Holme, A. (2012). Curves in \(\mathbb{A}^{2}_{k}\mbox{ and in }\mathbb{P}^{2}_{k}\) . In: A Royal Road to Algebraic Geometry. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-19225-8_2
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DOI: https://doi.org/10.1007/978-3-642-19225-8_2
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-19224-1
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