semicubical

Semicubical is an adjective used in mathematics to describe curves that are related to a semi-cubic form, most often referring to the semicubical parabola. The semicubical parabola is the plane algebraic curve defined by y^2 = x^3. Real points lie in the region x ≥ 0, and the graph has a cusp at the origin. It can be parametrized by x = t^2, y = t^3 for real t, which provides a simple rational parametrization and shows the curve is rational of genus 0.

The curve is cubic, having degree three in x and y, and its real locus displays a distinctive cusp at (0,0). The cusp has a single tangent line, namely the x-axis, and the curve is symmetric with respect to the x-axis since y changes sign with t. As a singular cubic, the semicubical parabola serves as a standard example of a cusp singularity: the origin is a simple cusp with multiplicity 2 and a delta-invariant equal to 1.

In mathematics, the semicubical parabola is studied in algebraic geometry and singularity theory as a fundamental example of a plane curve with a cusp. It is often discussed alongside smooth cubics and other cuspidal curves to illustrate concepts such as parametrization, tangents at singular points, and the behavior of curves near singularities. While primarily of theoretical interest, semicubical curves also appear in historical treatments of curve construction and in introductory discussions of algebraic curves.