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I was wondering how to find a parametrization of $(x-t)^2+(y-t^2)^2-t^2=0$, and how to use a software like Mathematica to draw a picture of this equation based on the parametrization.

Thanks in advance.

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    While this is $n$ot strictly speaking off-topic here in Math.SE, for future reference, the website Arjang pointed you to is http://mathematica.stackexchange.com and is likely much better for such questions.2012-04-02

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You can use ContourPlot3D. Here is an example parametrization:

ContourPlot3D[(x - t)^2 + (y - t^2)^2 == t^2, {x, -2, 2}, {y, -2, 2}, {t, -1, 1}] 

Result of running ContourPlot3D