Invariant points of low dimensional curve families

From noisy observations of a finite family of functions an approximation in a lower dimensional space can be constructed using the method of principal components. If certain restrictions are to be satisfied by the approximation, e.g. being densities, this leads to a modified estimation procedure. It is shown that in certain dimensions this will produce a family of curves which all intersect in the same points. This property may be interpreted in some cases as a characteristic feature of regularity of the data or as an artificial creation by the device in others.