Tangency to two ellipses how?

I have arbitrary ellipse with center at <x,y>, major and minor axis <a,b> parallel to x and y axis, respectively, and a point on it P <c+x,d+y> at arbitrary angle T where c=a*cos(T) and d=b*sin(T).  If the major axis is a, the angle of tangency is atan2(d,a/cos(T)-c), and if the major axis is b, the angle is atan2(b/cos(90-T)-d,c).

The standard form of the tangent is y=tan(T)*x+(d+y)-tan(T)*(c+x).

I have two ellipses with arbitrary loci and axes.  How do I with similar simplicity determine the four lines of tangency those ellipses share?

why would they share any lines of tangency?

Any two ellipses where one ellipse is not wholly contained by the other share at least two tangents.  If they intersect, or if they touch at a single point, they share two tangents.  If one is completely outside of the other, they share four tangents.

>>3
I'd to see a picture of that.  Can you post some

>>4
I f'shnikered. :P

>>4
Quite simple to draw.

Draw a capital "X" with two vertical lines on either side (left,right) of the cross.  Now draw the two ellipses (one above the cross, one below) so that the ellipses are both tangent to all four lines.

Right, then once you have that drawn and you can see the general form of the lines, draw two new ellipses on a fresh sheet of paper and try to connect the lines in the same way.

Anyways, you get two "inner" tangents and two "outer" tangent lines.  And it seems to me, by my rough sketches, that the inner tangents intersect on the line segment connecting the loci of the ellipses, while the two outer tangents intersect along that same line, but at a point behind the smaller ellipse.  If the two ellipses are the same size, the the outer tangents are parallel...

Is that right?

erm i mean centres or radii or wahtever the hell they are called for ellipses

>>3
they also share lots of curvy tangents tho right.

>>9

OP here...ick, I don't even want to think about constructing an ellipse that's tangental to two other ellipses...

Ewwwwww.

It turns out that if you graph the slope vs y-intercept of all the angles of tangency to a single ellipse you get a hyperbola.  So how does I find intersection points for two hyperbolas?  :o

>>10
I guess you isolate say y for each and set the half-equations equal to each other to solve for x, etc