Math 4800/6080. Week Six Starter
1. Show that every line:
(ax + by + cz = 0)
in RP2 is the image of a 1-1 “linear” map from RP1 to RP2 of the form:
Φ(u : v) = (s0 u + t0 v : s1 u + t1 v : s2 u + t2 v)
for some constants s0 , t0 , s1 , t1 , s2 , t2 .
2. Show that the standard unit circle, hyperbola and parabola:
(x2 + y 2 − z 2 = 0), (x2 − y 2 − z 2 = 0) and (yz − x2 = 0)
are all images of 1-1 “quadratic” maps from RP2 to RP2 of the form:
Ψ(u : v) = (q0 (u, v) : q1 (u, v) : q2 (u, v))
where q0 , q1 , q2 are homogeneous quadratic polynomials.
3. Play with triples:
(c0 (u, v) : c1 (u, v) : c2 (u, v))
of homogeneous cubic polynomials in u, v and see what the image looks
like. Is the map from RP1 to RP2 ever 1-1 onto its image?
1