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Real curves, or real algebraic curves, are one-dimensional geometric objects defined by polynomial equations with real coefficients. In the plane, a real curve is the set of real points of a plane algebraic curve in the real affine plane or, more generally, in the real projective plane. The study of real curves distinguishes their real locus from the complex curve obtained by extending scalars to the complex numbers.

A real curve carries a real structure: the complex points form a complex algebraic curve C, equipped with the anti-holomorphic involution given by complex conjugation. The real curve is the pair (C, σ), and its real points are the fixed points of σ. This framework allows the use of both algebraic and topological techniques to study the curve.

In the nonsingular case, the genus g of a complex projective plane curve of degree d is g = (d−1)(d−2)/2. For real curves, the real locus has at most g+1 connected components (Harnack’s bound). Curves attaining this maximum are called M-curves. In real projective plane geometry, real components can be classified as ovals (contractible in RP^2) or, if present, a noncontractible component known as a pseudoline.

Singularities add complexity; the normalization of a singular real curve yields a nonsingular model that mirrors its essential features while simplifying local behavior.

Examples range from real conics, which typically have a single oval of real points, to higher-degree curves with multiple real components and various arrangements. Real curves play a central role in real algebraic geometry and have applications in areas such as optimization, computer-aided design, and geometric modeling.