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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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    "The definition above" is incorrect; you're only specifying one coordinate.2012-06-06
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    @QiaochuYuan seems to make sense, since I want to accomplish with the segment between $m$ and $m + a$, $m + b$, where $m,a,b$ are vectors2012-06-06
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    I think you want to write that as $$x = a \cos t$$ $$y = b \sin t$$ Then note that $${\left( {\frac{x}{a}} \right)^2} + {\left( {\frac{y}{b}} \right)^2} = 1$$2012-06-06
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    @PeterTamaroff Okay, but $a$ and $b$ are vectors, how are these written for plotting purposes? For instance gnuplot?2012-06-06
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    No idea. I'm just letting you know that is usually how the ellipse with center $(0,0)$ and radii $a$ and $b$ is usually parametrized.2012-06-06
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    @PeterTamaroff Ah the center (0,0) I have given a center with a vector $m$, what does that mean for the equation? Is it then for instance $x = (m + a) \cdot \cos t$ ?2012-06-06
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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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    Almost! But how do I change the center from $(0,0)$ to $(m_1,m_2)$ with your equation?2012-06-06
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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