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Precalculus Mathematics Homework Help
Parametrization of implicit curve
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[QUOTE="Ray Vickson, post: 5240428, member: 330118"] Before attempting to parametrize the curve (if possible), try to understand it first. For example, if you take ##C = 1## you have two curves [tex] y = \sqrt{ x^3 - 3 x + 1} \;\;\; \text{(Curve C1)} [/tex] and [tex] y = - \sqrt{ x^3 - 3 x + 1} \;\;\; \text{(Curve C2)} [/tex] If ##x_1, x_2, x_3## are the real roots of ##f(x) = x^3 - 3 x + 1##, we have ##x_1 = -1.879385242##, ##x_2 = 0.3472963553## and ##x_3 = 1.532088886##. We have ##f(x) > 0## on ##(x_1,x_2)## and on ##(x_3 \infty)##, so the values of ##y## are real on those two intervals. Thus, there are really two pairs of (C1,C2) curves, one pair for ##x_1 \leq x \leq x_2## and another pair for ##x \geq x_3##. All the curves satisfy ##y = 0## at ##x = x_1, x_2, x_3##. Somehow, any parametrization you cook up would need to recognize all those issues. [/QUOTE]
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Parametrization of implicit curve
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