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I'm writing a visualization tool for magnetic fields in DirectX. I am currently building a model for 3D curves which will be used to describe field lines. The curves will be built as staight pipes joining several points. The lighting and shading will then take care of smoothing things.

Now back to maths. I have two direction vectors describing the direction of two pipes. They join at the origin. Now the part that joins the two pipes needs to be an ellipse at an angle that bisects the two vectors. Knowing the radius of the pipes (equal radius) and the two vectors, is it possible to come up with the equation of the ellipse in 3D? thank you and sorry if this is not so clear.