146CHAPTER 2. CURVESor, in terms of coordinates,braceleftBiggx=1+a√32cosθ−b2sinθy=2−a2cosθ+b√32sinθ.The general relation between a plane curve, given as the locus of anequation, and its possible parametrizations will be clarified by means ofthe Implicit Function Theorem in Chapter3.Analyzing a Curve from a ParametrizationThe examples in the preceding section all went from a static expression of acurve as the locus of an equation to a dynamic description as the image ofa vector-valued function. The converse process can be difficult, but given afunction−→p:R→R2, we can try to “trace out” the path as the point moves.As an example, consider the function−→p:R→R2defined byx(t) =t3y(t) =t2with domain (−∞,∞). We note thaty(t)≥0, with equality only fort= 0,
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