Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Projective Geometry question.

  1. Aug 8, 2009 #1


    User Avatar
    Gold Member

    I have this question from Nigel Hitchin's notes in PG:
    "Let V be 3-dimensional vector space with basis v1,v2,v3 and let A,B,C be points in the projective space P(V) be expressed in homogeneous coordinates relative to this basis:
    A=[2,1,0], B=[0,1,1], C=[-1,1,2].
    Find the coordinates wrt to the dual basis of the three points in the dual space P(V') (where V' is dual space of V) which represent the lines AB, BC,CA."

    The dual basis would be f1,f2,f3, where f:V->F for some F field.
    such that [tex]f_i(v_j)=\delta_{ij}[/tex] (the delta of kroncker).
    So for A: v=2v1+v2, [tex]f=\sum_{i=1}^{3}c_i f_i=\sum_{i=1}^{3}f(v_i) f_i[/tex]
    B: v=v2+v3, the line AB satisfies: 2v1=-v3
    C: v=-v1+v2+2v3 the line CA satisfies 3v1=2v3 and BC satisfies v1=v3.

    So for AB=[1,a,-2] for some a scalar, BC=[1,b,1] and for CA=[2,c,3].

    Is my answer valid or not?

    Thanks in advance.
  2. jcsd
  3. Sep 4, 2009 #2


    User Avatar
    Gold Member

    I see that there were some views, but no one dared to reply, I wonder why.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook