Question

I am working on a visualization for some data and I’ve run into a snag. I need to draw some ellipses based on data that looks like this:

{
    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 
}

Long story short, I have no idea how to do this… it is just for a one time visualization so efficiency isn’t a concern. If someone can suggest how to manipulate the e1/e2 to get the foci or major/minor axis and angle of rotation, that’d be super fancy. Thanks!

Answer 1

This form of specifying ellipticity is common in gravitational lensing. These ellipticity numbers are the real and imaginary parts of a complex ellipicity value; see the section Weak Lensing Observables and the expression for ε there.

I can’t do proper math notation here because of a policy decision; see this meta question. https://meta.stackexchange.com/questions/4152/adding-support-for-math-notation. Accordingly, I’ll simply point out that the magnitude of the vector is a transform of the major-minor axis ratio, and that the angle is half of the inverse tangent of the ratio of the two components.