Singlecurves

Singlecurves is a term used in mathematics, computer graphics, and design to refer to curves that are defined by a single parametric expression. In its broad use, a singlecurve is the image of a continuous map γ from an interval I ⊆ R into Euclidean space R^n, typically written as γ(t) = (x(t), y(t)) in 2D or γ(t) = (x(t), y(t), z(t)) in 3D. The curve is traced as t varies over I, producing one continuous line without splitting into separate branches.

In practice, singlecurves are contrasted with curve families that require several parametric components or multiple curves to describe a shape. Common examples include a line r(t) = (a + vt), a circle x(t) = r cos t, y(t) = r sin t, and a helix x(t) = R cos t, y(t) = R sin t, z(t) = ct, each defined by a single parameter t. The curvature κ(t) and speed |γ′(t)| derive from the same parameterization and influence sampling for rendering.

Applications include computer-aided design, typography, animation, and robotic or CNC path planning where a single continuous stroke is advantageous. Advantages of singlecurves include straightforward representation, predictable sampling, and ease of analytical manipulation; limitations include limited ability to express multi-branch or highly angular shapes without reparameterization or combining multiple singlecurves.

Variant concepts include reparameterization for uniform spacing and polycurves, sequences of singlecurves joined at breakpoints. See also parametric curve, Bezier curve, and spline.