CU-IP-P1 Infinity points of CU

According to Bézout’s theorem, a line (a curve of degree 1) and a cubic curve (degree 3) intersect in exactly three points, counted with multiplicity.

In the context of the complex projective plane, there exists a line at infinity, which contains all points “at infinity” — both real and imaginary. Since this is a line, it must intersect any cubic curve in three points. These are referred to as the points at infinity of the cubic curve CU.

These points at infinity may be real or imaginary. Algebraic analysis shows that imaginary points at infinity always occur in conjugate pairs. As a result, a cubic curve must have either zero or two imaginary points at infinity, and consequently either one or three real points at infinity.

Case of a Circular Cubic

When the cubic is a circular cubic, it contains the two circular points at infinity — which are complex conjugate and imaginary. Therefore, a circular cubic always has exactly one real point at infinity, and thus one real asymptote.

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