There is a well-known model of 2-dimensional projective geometry points and lines including points at infinity and line at infinity in terms of three dimensional Euclidean geometry, namely, ordinary projective points represented by Euclidean straight lines through the origin and intersecting the Euclidean plane z = 1; ordinary projective lines by Euclidean planes through the origin intersecting the Euclidean plane z= 1; and the projective points at infinity by Euclidean straight lines through the origin but lying on the Euclidean plane z = 0; and the line at infinity by the Euclidean plane z = 0, itself. This model also shows that parallel projective lines on the Euclidean z = 1, do meet in a point at infinity, namely, the Euclidean line of intersection of the two Euclidean planes representing the two parallel projective lines. Is there a similarly explicit representational model that shows that any two circles in the complex projective plane intersecting at two fixed imaginary points?


To get a perfect complex geometric analogy with the mental picture you’ve drawn would involve setting the whole mess in complex three-space, so to our real eyes, six-dimensional. You can do it, especially if you’re very geometrical.

But the usual way is to think of a point in the projective plane as given by a vector $v=(x,y,z)\in\mathbb{R}^3$, or equally well by any nonzero scalar multiple of $v$. (This gives you your line through the origin.) The traditional way is to write $(x:y:z)$. The ordinary point $(x,y)$ in the finite plane has the “homogeneous coordinates” $(x:y:1)$. A line now is described by a nonzero linear homogeneous form in the three variables $X$, $Y$, $Z$, I mean $AX+BY+CZ$, not all coefficients to be zero. Then the line at infinity is represented as the zero-set of the linear form $Z$.

A circle or any other conic section will be represented by a nonzero quadratic linear form, for instance the unit circle is given by $X^2+Y^2-Z^2$. and the circle whose ordinary equation we write $(x-a)^2+(y-b)^2=r^2$ will be represented by the quadratic form $(X-aZ)^2+(Y-bZ)^2-r^2Z^2$. You see that the unit point on the $x$-axis has homogeneous coordinates $(1:0:1)$, for example. You see also that both of these circles contain the points $(i:1:0)$ and $(-i:1:0)$ on the line at infinity once we allow complex coordinates.

I know that this isn’t what you asked for, but it is the best way to look at things unless you have redoubtable geometric intution.


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