Euclidian projection on ellipsoid and conic

Projecting a point on ellipsoid or conic in n-dimensional space

Description

Find the projection of point P in R^n on the ellipsoid

E = { x = x0 + U*(z.*radii) : |z| = 1 }, where U is orthogonal matrix of the orientation of E, radii are the axis lengths, and x0 is the center.

Or on generalized conic E = { x : x'*A*x + b'*x + c = 0 }.

The projection is the minimization problem:
min | x - P | (or max | x - P|) for x in E.

Method: solve the Euler Lagrange equation with respect to the Lagrange multiplier, which can be written as polynomial equation (from an idea by Roger Stafford)

• MATLAB release: MATLAB 7.10 (R2010a)
• Other requirements: Should work from v 2097A (bsxfun required)

[Mark Shore]

It was very interesting to watch this unfold from a simple newsgroup question to this FEX submission. No rating at this point because Bruno has used a number of techniques I'm not familiar with (so my rating would be uninformed), but it seems to work as advertised.

One quick comment: I added alpha(0.5) to his example script to render the ellipsoid semitransparent and make the projection more obvious.