modulio of conics
projectively, only 1 (class of) conics
affinely over complex, 2 (classes of) conics:
parabola(e) and hyperbola(e)
(how many points at infinity (1 or 2);
tangent to line at infinity or not)
affinely over reals, 3 conics:
ellipse(s), parabola(e) and hyperbola(e)
(how many (real) points at infinity?
0, 1, or 2?)
Over complexes, $x^2 + y^2 = 1$ and $x^2 - y^2 = 1$
equiv via multiplying $y$ by $i$
(geometrically, rotate $y$ in complex direction)
translation,
rotation,
dilation
(reflection not necessary, as have symmetry;
not obvious!)
shear not necessary
(shear ellipse and get ellipse -- not obvious!)
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parameter space, moduli space
parameter space: vertex, axes
circle has $O(2)$ symmetry;
(physically, 2 dimensional Euclidean space)
unit hyperbola has $O(1,1)$ symmetry
(physically, 1+1 Minkowski space-time)
(their complex equivalence shows that $O_\bC(2) \cong O_\bC(1,1)$)