Singlecurve

Singlecurve is a term used in mathematics and related fields to denote a single, continuous curve defined by one primary equation or a single parametric description, as opposed to a set of curves or a family of curves. It may be given explicitly as y = f(x) on an interval, parametrically as r(t) = (x(t), y(t), z(t)) with t in I, or implicitly as F(x) = 0. The concept emphasizes continuity and a single relation rather than comparisons across multiple curves.

Examples include a straight line y = mx + b, a parabola y = ax^2 + bx + c, or a circle described parametrically by r(t) = (cos t, sin t). In three dimensions, a helix r(t) = (cos t, sin t, t) is also a single curve.

Singlecurves are commonly constructed by data fitting, interpolation, or geometric modeling. Techniques include polynomial regression, spline fitting, and Bezier or B-spline representations that produce one continuous curve approximating or interpolating a point set.

Applications span computer graphics, computer-aided design, robotics path planning, and data visualization, where a single curve represents a relationship or trajectory rather than a set of competing curves. Related concepts include parametric curves, implicit curves, and spline curves.