How to draw an ellipse using its center and two points on its perimeter Suppose I have an ellipse centered at the origin, and two points on its perimeter which are not antipodal to one another (i.e. not negative to each other as vectors in $\mathbb R^2$). How can I draw the whole ellipse?
 A: The Wikipedia animation you point to show an ellipse being generated by two vectors $\mathbf a$ and $\mathbf b$ via the parametric equation
$$ t \mapsto \cos(t)\cdot \mathbf a + \sin(t)\cdot \mathbf b $$
with $t$ ranging from $0$ to $2\pi$, say.
Getting an equation for the ellipse is not as simple to describe. What you need there is to express an arbitrary point in the (non-rectangular) coordinate system that has $\mathbf a$ and $\mathbf b$ as basis vectors. Then the equation for (say) the closed ellipse is
$$ \{ v \mid p^2+q^2\le 1 \text{ where } v=p\mathbf a+q\mathbf b \}$$
A: Hint: Write vectors in polar coordinates $(r,\theta )$ and use the polar coordinate ellipse equation for both vector and find (a and e) for the ellipse 
with the origin at one focus, with the angular coordinate $\theta = 0 $ still measured from the major axis, 
the ellipse's equation is
    $r(\theta)=\frac{a (1-e^{2})}{1 \pm e \cos\theta} $  . 
The eccentricity of the ellipse (commonly denoted as either e or $\epsilon$) is
    $e=\varepsilon=\sqrt{\frac{a^2-b^2}{a^2}} =\sqrt{1-\left(\frac{b}{a}\right)^2} =f/a$ 
For more information :
http://en.wikipedia.org/wiki/Ellipse
