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I have two vectors $a$ and $b$ and want to perform a parametric plot that generates an ellipse out of it. Normally you would do that for:

$t \mapsto \cos(t) \cdot a + \sin(t) \cdot b$

But when I look up parametric plot, I only find examples like:

$\sin(t), \cos(t)$

or

$t \cdot \sin(t), 5 \cdot \cos(t)$

These plots are defined through commas, one value for $x$ and one for $y$. But how do I know map the definition above to this comma notation?

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    Just make the shift, $x \mapsto x-h$, yes.2012-06-06

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a and b are vectors (over $\mathbb{R}$), so one can write them as $a=(a_1,a_2)$ and $b = (b_1,b_2)$. Thus your parameterization is

$t \rightarrow (a_1 \cos(t) + b_1 \sin(t), a_2 \cos(t)+b_2 \sin(t)) $ for $t \in [0,2\pi]$.

That should do the trick. Have a play with the numbers $a_1,..,b_2$ and see what the results are when you plot them!

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    $(m_1 + a_1 \cos(t) + b_1 \sin(t), m_2 + a_2 \cos(t) + b_2 \sin(t))$. Note that you'll always get an ellipse unless $a$ and $b$ are linearly dependent.2012-06-06