Parallel ellisoid projection onto a plane, numerical method

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I am having the following problem. I need to find the contour points of a ellipsoidal rotated body on the xy-plane (see image for more detail). Now, I have found the analytic solution to this problem (using this method).

However, due to the use of symbolic values I use in Matlab this process is slow (and I need to repeat it this process thousands of times).

I have all information regarding the ellipsoid (all axes and centerpoint) and the rotation it has undergone from a position where its principle axes are aligned with the XYZ-axes.

I need a faster way to solve this problem. I can easily find all points on the surface of the ellipsoid and project these on the XY-plane (not knowing which points define the contour). However, then, I need to find the points that define the contour in the XY-plane. See picture on dropbox.

My questions:

1) Is there a fast way to find the points defining the contour as in image 2?

2) Or, is there another numerical methods to find the contour of a projection of a rotated ellipsoid?

3) Or, is there a way to solve the analitical solution in Matlab without using symbolic values?

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