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Unformatted text preview: 138 CHAPTER 2. CURVES point. A second and equally important source of curves is dynamic in nature: a curve can be generated as the path of a moving point . This is the fundamental viewpoint in Newtons Principia (as well as the work of Newtons older contemporary Christian Huygens (1629-1695)), but mechanical constructions of curves also go back to antiquity, for example in Archimedes spiral (p. 150 ). We have seen in the case of lines in the plane how these two approaches interact: for example, the intersection of two lines is easier to find as the simultaneous solution of their equations, but a parametrized version more naturally encodes intrinsic geometric properties like the direction of a line. We have also seen that when one goes from lines in the plane to lines in space, the static formulation becomes unwieldy, requiring two equations, whileespecially with the language of vectorsthe dynamic formulation extends quite naturally. For this reason, we will adopt the dynamicextends quite naturally....
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- Fall '08