# Parameters for an ellipse given measures of ellipticity

I am trying to visualize some data in the form:

{
x: 455.53 //the center x coordinate
y: 122.44 //the center y coordinate
e1: .24101 //value from -1 to 1, represents stretching along x when positive, along y when negative
e2: -.44211 //value from -1 to 1, represents stretching along the 45 degree line when positive and 135 when negative
}


This is intended to describe an ellipse, and I guess I can picture how it would work in my head. What I have no idea how to do, is transform this data into a form I can use to draw an ellipse. What I really need is the scale factor of the major axis, the minor axis and the angle of rotation, since its essentially describing a deformed circle of arbitrary size. Any other forms of ellipse are also welcome. Thanks!

The ellipiticity of the circle at a position $\left(x,y\right)$ tangential to a point $\left(x',y'\right)$ is
$$e_{tangential} = -(e_1\cos(2\phi) +e_2\sin(2\phi)))$$
$$\phi = \arctan\left(\frac{y-y'}{x-x'}\right)$$