Then the key idea of GPCA is to directly find the polynomials that fit all the data points simultaneously. In the above example, it first identifies the quadratic polynomial that fits the two lines. Then the derivative of the polynomial evaluated at any point x on the first line is proportional to b1; and the derivative evaluated at any point x on the second line is proportional to b2. In this way, we can retrieve the model of the two lines from such fitting polynomials.
Many of the ideas can be generalized to an arbitrary number of subspaces of arbitrary dimensions. One can find a complete and general theory of GPCA in the literature listed in the Publications section.