infinite points

      We want to generate points on the line at infinity as we generated those on G—P; i.e., find the basic forms (the "primes") from which the others can be generated.

We get lots of help in this because lines and conics hide infinite points which can easily be displayed.

If P0 = (x0:y0:z0) is  on [ l : m : n], then  [ l x0 : m y0, n z0] is on the line at infinity.

  Using G—Io = : c - a: , we have these points (red is prime forms)

    G         c - a                     c+a-2b

     Io       b(c-a)

    So       cc - aa

     No      sb (c - a)

     pSo    bb(cc - aa)

                   b(bc+ab-ca)(c-a)

                       b sbb (c-a)

                   (cc-aa)SB

                   (bb-ca)(c-a)

                   (aa-bc+cc-ab)(c-a)

     ∞         (c-a)(c+a-2b)

This is actually more than we need to generate most of the infinite forms.

Nice!  Lots of significance with little work.