Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.a. A set of parametric equations for a given curve is always unique.

Accepted Answer

Understand the meaning of parametric equations: Parametric equations express the coordinates of points on a curve as functions of a parameter, usually denoted as $$t. $$For example, $$x = f(t)$$ and $$y = g(t)$$ describe a curve in the plane. Consider the uniqueness of parametric representations: Ask yourself if there can be more than one way to describe the same curve using different parametric equations. Think about how changing the parameter or reparametrizing the curve might affect the equations. Recall that a curve can be reparametrized: If you have one set of parametric equations $$x = f(t)$$ and $$y = g(t)$$, you can often create another set by substituting $$t$$ with a different parameter function $$s = h(t)$$, where $$h is a $$one-to-one function. This produces a different parametric form for the same curve. Conclude about uniqueness: Since multiple parameterizations can describe the same curve, the set of parametric equations for a given curve is generally not unique. Provide a counterexample: For instance, the circle $$x = \cos t$$, $$y = \sin t$$ can also be described by $$x = \cos(2t)$$, $$y = \sin(2t)$$, which traces the same circle but at a different speed and parameterization.

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
a. A set of parametric equations for a given curve is always unique.

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Uniqueness of Parametric Representations

Counterexamples in Mathematics