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The graph of a function \(y = f(x)\) represents a curve in the plane. This concept, however, is too tight to represent more intricate curves, like loops, self-intersections or even curves of fractal dimension. The aim of this chapter is to introduce the concept of parametrised curves and to study, in particular, the case of differentiable curves. For the visualisation of the trajectory of a curve, the notions of velocity vector, moving frame and curvature are important. The chapter contains a collection of geometrically interesting examples of curves and several of their construction principles. Further, the computation of the arc length of differentiable curves is discussed, and an example of a continuous, bounded curve of infinite length is given. The chapter ends with a short outlook on spatial curves. For the vector algebra used in this chapter, we refer to Appendix A.
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Concerning the vector notation we remark that x( t), y( t) actually represent the coordinates of a point in \(\mathbb R^2\). It is, however, common practise and useful to write this point as a position vector, thus the column notation.
W. Neil, 1637–1670.
Archimedes of Syracuse, 287–212 B.C.
J.A. Lissajous, 1822–1880.
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- Chapter 14